SYSTEM DYNAMICS IN HYDROPOWER PLANTS

by

Dag Birger Stuksrud

This thesis is submitted to the Norwegian University of Science and Technology in partialfulfilment of the requirements for the Norwegian academic degree

Doktor Ingenwr

Department of Thermal Energy and Hydropower Norwegian University of Science and Technology

Trondheim

NTNU

September 1998

iDisce quasi semper victurus;

vive quasi eras moriturus. (Anon.)

This book is dedicated to my wife, Maibritt, and to my son, Markus, who made it all possible with their patience, understanding and cooperation throughout this work

i

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

-T7TT

ACKNOWLEDGMENTS

This work has been carried out at the Department of Thermal Engineering and Hydropower, Faculty of Mechanical Engineering, at the Norwegian University of Science and Technology (NTNU).

This Dr. Ing. work started in January 1995 as a part of the research programme “EFFEN-PRODUKSJON”, that was financially supported by the Norwegian Electricity Federation (EnFO) and the Research Council of Norway (NFR). In January 1997, this research programme ended and the doctoral work was transferred to a new programme, called “EFFEKT”. The financial support was from that time given by NFR and some of the largest participants in the Norwegian hydropower supply sector. I wish to express my sincere thanks to all the financial supporters.

During the three and a half years of work with this thesis, numerous people have given their valuable contributions in form of useful comments, guidance and suggestions. All the contributors cannot be mentioned by name, but my general gratitude is meant for all of them. However, some people have been of great importance to me and should be mentioned and thanked by name:

My academic supervisor, Professor Hermod Brekke, for giving me the opportunity to do this research and for offering me his confidence. His support and guidance have been very valuable for me throughout the time I have worked on the thesis.

Torbjeim Kristian Nielsen and Finn Olav Rasmussen, Kvaemer Energy as., for their interest in my work and for being the most important supervisors for me outside the University. Their cooperation and support have meant a lot to me.

Bjorn Harald Bakken, SDN'l'EF Energy Research, Arivd Elstrom, ABB Power Generation and Roald Sporild, ABB Corporate Research for discussions and guidance concerning the electric part of the hydropower system.

Stewart Clark, Student and Academic Section, NTNU, for reading through the thesis and giving me the necessary linguistic advice.

I would like to thank my new employer, Kvaemer Oil & Gas a.s., for allowing me time to prepare for the defence of this thesis.

Finally, I want to thank all my colleagues at the Department throughout the study for valuable discussions.

Sandefjord, September 1998

Dag Birger Stuksrud

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ABSTRACT

This thesis presents a study of system dynamics in hydropower plants. The main intention has been to establish new models of a hydropower system where the turbine/conduits and the electricity supply and generation are connected together as one unit, in order to study possible interactions between the two power regimes. This subject is of current interest due to the new deregulated energy market in Norway, the new submarine cables to the European continent and the expected introduction of new technology and controller units. Seen from the aspect of power plants, larger and more rapid loading are expected. How this will affect the conduits and the governors is of great importance, and a simulation tool, capable of handling these situations, will be greatly appreciated. Seen from the aspect of electric side, the new situation may cause capacity problems in the electric grid and cause unstable conditions. Almost invisible oscillations, originating from the conduits, may be superposed and threaten the system performance. In order to foresee the influences of introducing new technology, as new controllers and generators for instance, an improved mathematical model of the total system is needed. New linear and non-linear mathematical models are therefore established in the thesis and new simulation programs are developed.

In order to describe the system dynamics as well as possible, a previously developed analytic model of high-head Francis turbines [33] is improved. The acceleration resistance in the turbine runner and the draft tube are included in the model (hydraulic inertia). Expressions for the loss-coefficients in the model are derived in order to obtain a pure analytic model The necessity of taking the hydraulic inertia into account is shown by means of simulations. Unstable behaviour and a higher transient turbine speed than expected may occur for turbines with steep turbine characteristics or large draft tubes. The turbine model has earlier on been verified with regard to a high-head Francis turbine. In this thesis an experimental verification is performed on a low-head Francis turbine and the measurements are compared with simulations from the improved turbine model The result shows that the dynamic turbine model is after adjustment, capable of describing low-head machines as well, with satisfactory results.

In this thesis a method called, the “Limited zero-pole method”, is used to provide new rational approximations of the elastic behaviour in the conduits, with frictional damping included. The new approximations are used to provide an accurate state space formulation of a hydropower plant.

Simulations performed by means of the new computer programs show that hydraulic transients, such as water-hammer and mass oscillations, are reflected in the electric grid. Unstable governing performance in the electric and hydraulic parts will also interact with each other. This emphasizes the need for analysing the whole power system as a unit.

IV

PUBLICATIONS

Three papers, which are part of this thesis, have been published and presented duringthe study:

• Stuksrud, Dag Birger: “Simulation of turbine governing in time domain”,Contribution to the XVm International Association of Hydraulic Research (IAHR) Symposium on Hydraulic Machinery and Cavitation, Valencia, Spain, 1996.

• Stuksrud, Dag Birger: “Dynamic simulation of a governing turbine connected to a strong electric grid”, Contribution to the Japan Society of Mechanical Engineering (JSME) Conference on Fluid Engineering Towards the next Century, Tokyo, Japan, 1997.

• Stuksrud, Dag Birger; Nielsen, Torbjom Kristian; Rasmussen, Finn Olav: ”Analytic model for dynamic behaviour of Francis turbines”, Contribution to the VHI International Association of Hydraulic Research (IAHR) Work Group Meeting on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, Chatou, France, 1997.

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NOMENCLATURE

Roman lettersa wave propagation speed [m/s]A cross sectional area [m2]As surge shaft cross section [m2]a swirl angle [degrees]b inlet turbine height [m]bp permanent speed drop [-]bt transient speed drop [-]Cs turbine constant [-]c absolute velocity [m/s]c„ swirl velocity component [m/s]Cm meridional velocity comp. [m/s]cosij) power factor [-]D pipe diameter [m]D torque in per unit [-]Dtmbine turbine torque in per unit [-]Dciccmc generator torque in per unit [-]Dt tunnel diameter [m]E’ ’ subtransient internal voltage in per unit [-]E’ transient internal voltage in per unit [-]Eg electromotive voltage in per unit [-]Egen rated terminal voltage [V]Ek terminal voltage in per unit [-]Eqf field voltage in per unit [-]Em correction parameter for saturation in the windings [-]Ecqu equivalent voltage in per unit [-]E, stiff voltage in per unit [-]En linearized efficiency in respect of the angular speed [-]Eq linearized efficiency in respect of the flow [-]Etp transformer voltage primary side in per unit [-]By transformer voltage secondary side in per unit [-]Eiocai local load voltage in per unit [-]f frequency [Hz]ft synchronous frequency [Hz]g gravity [m/s2]H head [m]H generator inertia constant [1/s]Hgmss gross head [m]hw Allievis constant [-]i stator current local frame in per unit [-]I hydraulic inertia [kgm2]I stator current global frame in per unit [-]j complex operator [-]J polar moment of inertia [kgm2]k friction parameter [1/s]k$ steady-state friction parameter [1/s]

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ka dynamic friction parameter [s]K pressure feedback gain H

K, gain of voltage controller H

Kq linearized ratio between servo strike and guide vanes[-]Kn gain of derivative damping unit H

L length [m]Lt tunnel length [m]Ln, blade length [m]ms specific torque H

M torque [Pa]M Mannings friction number [m1/3/s]n turbine speed [rpm]n$ synchronous turbine speed [rpm]NP number of generator poles H

P power [W]

Pg produced active power in per unit H

Pk active power on the terminal in per unit HQ flow [m3/s]Qr reactive power [VAr]Qig produced reactive power in per unit [-]Qrk reactive power on the terminal in per unit H

Q» linearized flow in respect of the angular speed [-]Qy linearized flow in respect of the guide vanes opening [-]r radius [m]ra anchor resistance H

R, turbine loss coefficients H

R) ohmic line resistance [Q]

S apparent power EVA]S geometric parameter [m2]Strans rated transformer power [MV A]Sgen rated generator power in per unit [-]s„ short circuit power elect, grid [MVA]s Laplace operator H

t time [s]Ta generator time constant [s]Td integral time [s]T* integral time speed governor [s]Ty actuator time constant [s]To servo time constant [s]Tw pipe time constant [s]T. pressure feedback time constant [s]Tq„ hydraulic inertia time constant [s]Too” subtransient open circuit time constant [s]Tqo” subtransient open circuit time constant [s]Td0’ transient open circuit time constant [s]Tf power amplifier time constant [s]Tr voltage controller time constant [s]Tn derivative damp, time constant [S]

u angular velocity [m/s]

vrn

w relative velocity [m/s]C zero/pole constant [1/s2]zg characteristic impedance HZp potier reactance in per unit [-]Zequ equivalent impedance in per unit [-]Ziine transmission line impedance in per unit [-]Ztrans transformer impedance in per unit [-]Zgnd electric grid impedance in per unit [-3Zjocal local load impedance in per unit HZ complex friction factor Hx" subtransient reactance in per unit [-]x’ transient reactance in per unit [-]X, line reactance [Q]Y Guide vane or valve opening [degrees] [m]z elevation [m]

Greek lettersa swirl angle [degrees]P blade angle [degrees]P rotor angle [elect, degrees]Sr ohmic transformer short circuit [Q]Sx reactive transformer short circuit mn efficiency [-]<p machine constant HK opening degree [-]X Moody’s friction factor [-]Xf dynamic friction factor [Ns/m2]p density [m3/s]CT self-governing parameter [-]T shear stress [Pa](0 angular frequency [1/s]*£2 speed number [-]Qs shock losses [-]V machine constant H

Subscriptn0BDQdq

nominal condition (best efficiency point) initial condition base valueD-axis component (global)Q-axis component (global) d-axis component (local) q-axis component (local)

IX

Superscript’ transient condition” subtransient condition0 rated condition (full load)

Dimensionless propertiesh = H/Ho headq = Q/Qo y = Y/Y0D = M/MoP = P/Po l = L/LoQ = co/too

flowguide vane or valve openingtorquepowerlengthspeed/frequency

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CONTENTS

1. INTRODUCTION..................................................................................................11.1 MOTIVATION FOR THIS WORK............................................................... 11.2 DEFINITIONS................................................................................................. 21.3 PURPOSE AND SCOPE OF WORK............................................................ 41.4 OUTLINE OF THESIS...................................................................................91.5 CONTRIBUTIONS EST THE PAST..............................................................17

2. CLASSIC TURBINE GOVERNING ANALYSIS........................................ 192.1 THE HEAD/FLOW TRANSFER FUNCTION, b/q.................................. 192.2 APPROXIMATIONS, BY MEANS OF THE LIMITED

ZERO-POLE METHOD, OF THE TRANSFER FUNCTION, h/q,WITH FRICTIONAL DAMPING INCLUDED.........................................242.2.1 The elastic transfer function, h/q, with frictional

damping included.................................................................................242.2.2 Approximations of the transcendental term.......................................262.2.3 Approximations of the non-linear square root term......................... 302.2.4 Complete approximations................................................................... 302.2.5 Verification of the approximations.....................................................32

2.3 LINEAR DYNAMIC MODEL OF A HYDROPOWER PLANT............ 382.3.1 Turbine unit connected to an isolated grid....................................... 382.3.2 Turbine unit connected to a stiff electric grid................................... 42

3. MODERN TURBINE GOVERNING ANALYSIS....................................... 473.1 IMPROVEMENT OF A DYNAMIC FRANCIS TURBINE

MODEL..........................................................................................................473.1.1 Flow through the turbine runner........................................................483.1.2 Dynamic draft tube model................................................................. 533.1.3 The turbine torque equation...............................................................593.1.4 The steady-state condition of the dynamic turbine model.............. 613.1.5 Consequences of the hydraulic inertia.............................................. 65

3.2 EXPERIMENTAL STUDY......................................................................... 763.2.1 The experimental set up....................................................................763.2.2 Steady-state measurements................................................................793.2.3 Transient runaway measurements..................................................... 823.2.4 Steady-state measurements compared with simulations..................833.2.5 Transient runaway measurements compared

with simulations................................................................................. 853.3 NON-LINEAR DYNAMIC MODEL OF A HYDROPOWER

PLANT CONNECTED TO A STIFF ELECTRIC GRID..........................913.3.1 The turbine and the conduits............................................................ 923.3.2 The synchronous generator...............................................................943.3.3 The speed governor...........................................................................993.3.4 The voltage controller..................................................................... 1003.3.5 The electric grid............................................................................... 1023.3.6 The electric load...............................................................................1043.3.7 Per unit representation.....................................................................106

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3.3.8 Steady-state conditions........................................................................ 1073.3.9 Simulation program..............................................................................108

4. INTERACTIONS BETWEEN THE HYDRAULIC PART AND THEELECTRIC PART OF A HYDROPOWER SYSTEM................................... 1114.1 LINEAR TIME DOMAIN SIMULATIONS................................................111

4.1.1 Simulation of a turbine unit connected to an isolated grid................ 1114.1.2 Simulation of a turbine unit connected to a stiff electric grid............118

4.2 NON-LINEAR TIME DOMAIN SIMULATIONS.....................................1234.2.1 Simulations compared with SIMPOW................................................1244.2.2 Local load connection.......................................................................... 1274.2.3 Downloading.........................................................................................1304.2.4 Undersized surge shaft cross section.................................................. 1344.2.5 Uploading with line rejection...............................................................1364.2.6 Short circuit with sudden synchronizing.............................................139

5. CONCLUSIONS AND FURTHER WORK.......................................................1435.1 RESULTS OF THE INVESTIGATION.......................................................1435.2 RECOMMENDATIONS FOR FURTHER WORK......................................144

BIBLIOGRAPHY......................................................................................................... 147

APPENDICES................................................................................................................149

Appendix 1 A description of turbine governing systems............................................ IAppendix 2 Deriving the Allievi Equations...............................................................XIIAppendix 3 Deriving the pressure/flow transfer function, h/q.............................XVIIAppendix 4 Frictional damping model by Kangeter/Brekke................................... XXAppendix 5 Surge shaft model in the frequency domain.......................................XXIAppendix 6 Method of Characteristics (MOC)....................................................XXHIAppendix 7 Load flow analysis..............................................................................XXIX

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1. INTRODUCTION

1.1 MOTIVATION FOR THIS WORK

The hydro-electric power system in Norway will undergo a lot of changes in the near future. This is mainly a result of the deregulated open energy market that came into force with the Energy Act in 1992. The philosophy, that the price of the power will control the power flow in the transmission system, has resulted in a less predictable situation and a new way of operating turbine units. More rapid up- and downloading is expected and capacity problems and breakdowns in the electric grid may become a real problem. To make matters worse for planners, six subsea HVDC cables from Norway to the Continental Europe, with a capacity of 600-800 MW each, are either commissioned or will be under construction at the beginning of the next century. The idea is that surplus power is exported from Norway as peak loads during the day. This will be compensated by cheap inports from Europe at night. Hus will also provide an energy reserve in case of dry years in Norway. Taking this into consideration, Norway is meant to function as a pump storage for Continental Europe. An additional challenge is that the continual increase in power consumption in Norway has resulted in situations where the load almost has reached its maximum limitation (theoretically: 27000 MW, practically: 23000 MW).

In the 1970s stability problems caused many disconnections and breakdowns in the transmission system. Problems of the same type were also detected in the USA and New Zealand at the same time. There was considerable discussion among engineers around the origin of the oscillations. It was believed that interaction from the turbines and conduits may be significant, but it was never verified experimentally or theoretically. The problem was controlled at that time by the introduction of a new type of voltage controller. After the new open energy market was commissioned in 1992 the stability problems seemed to increase again. The focus is again on whether transients from the hydraulic part of the power system (turbine and conduits) may propagate into the electric grid and cause the “unknown” stability problems. This seems reasonable when taking the new way of operating turbine units into consideration. Also the introduction of new technology, that recently has been released or is in the design phase, may reduce the system stability. This is mainly new controllers and new generator types, such as ASD units (Adjustable Speed Drive).

In order to meet the new challenges, it is highly relevant to focus on a more complete model of a hydropower plant and its transmission system. Closer cooperation between the traditionally separated engineering disciplines in hydropower engineering (electric power and hydraulics) has been started through a common research programme called “EFFEKT”, which this thesis is part of Attention has been placed on the whole power system, popularly speaking; from “water to wire”. There is a common agreement among hydropower engineers that the available mathematical models and tools for system dynamics analysis of hydropower systems are insufficient. Improved, extended models are necessary in order to predict the consequences of the new power balance in Norway and the interconnected countries in Europe.

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1.2 DEFINITIONS

This section lists and explains technical terms and concepts used in the thesis, sorted in alphabetical order.

Allievi’s EquationsBasic equations of one-dimensional transient pipe flow. Described in Appendix 6.

ASD generatorAdjustable Speed Drive. A generator that is capable of changing the turbine speed in the range of approximately ± 10% of nominal speed in synchronous operation. This is advantageous for the hydraulic efficiency and also for the electric grid stability.

Coordinated electric gridAn electric grid or generating system where several turbine units are interconnected.

Classic turbine governing analysisStability analysis by means of linearized equation of the hydropower system in the frequency domain. Time response or system dynamic analysis using state space formulations and inverse Laplace transformation from the frequency to the time domain.

Electric part of the hydropower systemThe synchronous generator, the voltage controller, the electric grid and the electric load.

Frequency dependent frictional dampingTerminology used for one-dimensional models of transients in the conduits, where the damping effect, caused by the variation in the velocity profile, is approximated.

Hydraulic inertiaAcceleration resistance in transient pipe flow. For one-dimensional flow in a straight pipeline for instance, Newton’s Second Law gives; pL/A.

Hydraulic part of the hydropower system The turbine, the speed governor and the conduits.

Isolated electric gridAn indication used for a turbine unit that is disconnected from the coordinated grid. It may also be referred to as a turbine running under isolated load conditions. Traditional model used by hydraulic engineers, where only the hydraulic part of the system is modelled.

Large load variationsTurbine loading larger than approximately ± 5-10%. Also up- and downloading.

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Mass oscillationsSlow surge oscillations, in the areas of seconds/minutes, between all free water levels in the conduits, caused by movements of the guide vanes. For instance in the supply tunnel between a surge shaft and the upper reservoir.

Method of Characteristics (MOC)A general solving technique for partial differential equations. Used here to solve Allievi’s Equations.

Modern turbine governing analysisTime domain simulations by means of non-linear equations. Focus on the dynamic performance of the total system, including the turbine, the conduits, the governors, the generator and the electric grid connected to a total model

OutputThe same as power; turbine output = turbine power etc.

Pressure feedbackA control technique used for improved turbine governing. The pressure across the turbine is measured and used as a second feedback loop in the system (cascade control). Described in Appendix 1 (Al.2.3).

RunawayThe steady-state condition of a turbine that is being disconnected from the electric grid with the guide vanes in fixed position.

SIMPOWA powerful simulation tool used for analysis of electrical power systems.

Small load variationsTurbine loading less than approximately ± 5-10%.

Standard shaped draft tubeDraft tube shaped with a cone, a bend and a diffuser. The most common way of arranging a draft tube.

Steady-state frictional dampingTerminology used for one-dimensional models of transients in the conduits, where the damping effect, caused by the variation in the velocity profile, is neglected. Only wall- friction that can be measured in steady-state flow is modelled.

Stiff electric gridA notation used for a turbine unit that is connected to a stiff coordinated electric grid. A stiff grid is a generating system that is of large capacity and the generator is “small” in relation to the system grid. The voltage and the frequency in the coordinated grid will not be affected by the turbine dynamics. This is a common model used by electrical engineers. The turbine and the conduits are as often strongly simplified.

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Straight draft tubeDraft tube with a cone and a straight circular part. This is often used in connection with pump-turbines.

Strong/weak interconnection to the power gridTransmission line from a unit to the coordinated grid with high/low transfer capacity. The transmission line has small/large reactances compared with the unit.

Transcendental function A function that is not algebraic.

Transient braking zoneThe area outside the runaway line (r| = 0) in a turbine performance diagram.

Transient runawayThe transient time domain history from an arbitrary steady-state operational point torunaway.

Turbine characteristicsA diagram of a turbine showing the steady-state connection, for all operation conditions, between the pressure head, the flow and the turbine speed. Another common notation is “turbine performance diagram”, where the efficiency as often is included in the diagram.

Turbine unitIndication of the turbine runner and the synchronous generator.

Water-hammerRapid surge oscillations in the conduits caused by waves, travelling with the speed of sound, between the turbine and the nearest free water surface.

13 PURPOSE AND SCOPE OF WORK

The purpose of this work is to:

A) Improve models for system dynamics analysis of hydropower plants.

B) Investigate, by means of the improved models, whether interactions between the hydraulic part and the electric part of the hydropower system are present or not.

Both A and B, will be described below.

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A) Improve models for system dynamics analysis of hydropower plants.

System dynamics analysis is a necessary part when a hydropower plant is designed or upgraded. In order to obtain complete simulation models for this task, several improvements are presented in the thesis.

First a classification of the commonly used analytic techniques and models of hydropower systems is needed. System dynamics analysis of hydropower systems can roughly be divided into two groups. In the thesis they are named “Classic” and “Modem” turbine governing analysis. Both types of analysis are very important and from my point of view they should supplement rather than compensate each other. Effort is therefore spent on improving models of both types of analysing methods.

Classic turbine governing analysisThe Classic analysis is the traditional way of analysing the system stability and is the most frequently used method among engineers and scientists. It is based on linearized transfer functions of the hydropower system and is solved for small load variations around an operation point (linear theory). The system stability is analysed in the frequency domain and the system performance is studied in the time domain by means of a “state space transition” from the frequency domain (inverse Laplace transformation).

A main problem with the hydraulic models in this research field has been to establish satisfactory models of frictional damping in the conduits. For analysis in the frequency domain a number of non-linear and iterative models are established with good results, but they are not compatible with state space formulations and therefore cannot be used in time domain analysis. An increased use of computers and digital techniques has led to a number of commercial computer programs for mathematical modelling, simulations and control They are mostly based on linear theory with state space modelling for time domain solutions. The frictional damping has been an obstacle for providing accurate software algorithms of hydropower systems for this task. The problem is attacked in the thesis in order to:

• Establish new approaches to frictional damoine in the conduits that make it possible to produce state space solutions of hydropower systems with more accuracy than today.

The elastic behaviour of the conduits is described by partial differential equations in the time domain and transcendental transfer functions in the frequency domain. Because of their nature, transcendental functions cannot be implemented in programs based on state space formulation. Li [30], presented a promising method of transforming the transcendental equations of the conduits from the frequency domain to the time domain by means of rational expressions and state space formulations. The method was called the “Limited zero-pole method”. This was a breakthrough in hydraulic research since a state space solution of transcendental equations up to then was done with great uncertainty. Li's point of origin was the transfer function between the pressure and the flow, h/q, without frictional damping in the conduits included. Since the frictional damping is important to account for in stability calculations, strength evaluations and efficiency

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estimations it would be desirable to include this in the model. The Limited zero- pole method will be used in the thesis in order to derive new rational approximations of the transfer function, h/q, in the conduits with the frictional damping included.

A linear modal of the whole hydro-electric system is of current interest. Since new technology is planned, more exact models for the connections between the generator/grid and the turbine/conduits are desirable. For instance one main purpose of the ASD generator is to control the speed of the turbine in such way that the hydraulic efficiency is optimized. This cannot be done without a model where both the hydraulic and electric parts of the power system are one unit. Satisfactory models of this kind do not exist today. This is mainly because hydraulic engineers and electric power engineers have been working separated in their own disciplines. This will be treated in the thesis and the scope is to:

• Establish a linear simulation model of a hydropower system where the hydraulic part and the electric part of the system are connected together as one unit

The new approaches of the conduits with frictional damping included will be connected to existing linear models of the turbine, the turbine governing system, the synchronous generator, the voltage controller and the electric grid.

Modern turbine governing analysisWhile the Classic analysis considers both analysis in the frequency domain and in the time domain, the Modem analysis is a pure time domain solving method. The physical equations of the system are solved directly in time domain without linearization and transition to the frequency domain. Earlier on, Modem turbine governing analysis was a complicated task to perform and linearized simplifications were the only choice in many cases. Therefore, the Classic analysis was almost absolute. Today, improved computer technology has made it possible in a much larger scale to perform a directly solution in the time domain.

Although the Classic analysis is based on linearized equation, the parameter settings in the speed governor have turned out very well up to now. A reason for this may be that the common design criteria, especially for Francis turbines, has been subject of only small load variations around the best efficiency point. So far, this has been a common way of operating turbine units and linear analysis has therefore been sufficient in most cases. Because of the new open market conditions many turbine units will be exporting peak loads and use a wider range in their operating areas (off-design condition). They will also be exposed to more rapid up- and downloading. The non-linear effects in the system will therefore be more significant and system dynamics analysis based on linearized equations (Classic method) may be insufficient. Non-linear models are desirable because of this.

A challenge in Modem analysis has been to establish an accurate model of the turbine characteristics. A common way has been to digitalize a measured model turbine characteristics, but this is done with lot of uncertainty. Besides this, a model based on measurements cannot be used in the design phase of a hydropower plant. Measured

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turbine characteristics are also often hard to provide. A promising turbine model was presented by Torbjom K. Nielsen [33] in a doctoral work in 1990. A dynamic model of a high-head Francis turbine was derived. It showed very good results compared with measurements. Some magnitudes in the model have to be guessed or determined by trial and error in order to fit measured turbine characteristics. The turbine model is therefore not purely analytic. This is treated in this thesis and the purpose is to:

• Improve an existing dynamic turbine model 1331 in order to provide a simulation model that is more accurate and less dependent on measurements than the existing model

In order to make the dynamic turbine model less dependent on measurements, expressions will be derived for the loss coefficients in the turbine model. Earlier on these coefficients had to be determined by trial and error.

One main conclusion in the doctoral work by Nielsen [33] was that the acceleration resistance of water in the turbine, so-called hydraulic inertia, strongly affected the dynamics of the turbine, especially in high-head turbines and pump turbines. He verified this experimentally, but analytic expressions of the hydraulic inertia were not derived. In this thesis approaches of the hydraulic inertia in the turbine rurmer and the draft tube will be derived. In order to obtain this the turbine model has to be derived from an “earlier physical level” than Nielsen did. Until today, the hydraulic inertia effect is not implemented in any simulation programs. This will be done in the thesis and the consequences of neglecting the hydraulic inertia will be shown by means of simulations.

An experimental investigation will be performed in order to test the derived hydraulic inertia approaches and also test the accuracy of turbine model against more “difficult” turbine characteristics than the well-defined characteristics of a high-head machine. An experimental study will therefore be performed on the runaway performance of a low-head Francis turbine, with two different kinds of draft tubes.

The need for an improved dynamic turbine model in connection with the electric grid is, as mentioned under the Classic analysis, very important in order to be able to analyse the system performance with the new open market conditions and to predict consequences of the introduction of new technology. A non-linear model of the hydropower system will be suited for studying whether transients from the conduits will be reflected in the electric grid or not. Today’s models have a lot of simplifications and are strongly characterized by either the hydraulic or the electrical engineering disciplines. A model where the most commonly used ways of respectively analysing the hydraulic and electric parts of the hydropower system connected as one unit does not exist. In the thesis this issue is addressed in order to provide a new simulation tool where both parts of the system are connected and each discipline’s way of representing its part of the system is taken into account. The goal is to:

7

• Establish a new non-linear simulation model of a hydropower system where the hydraulic part and the electric part of the system are connected together as one unit

The hydropower system can mainly be divided into seven elements; the turbine, the conduits, the speed governor, the synchronous generator, the voltage controller, the electric grid and the electric load. The turbine will be represented by means of the improved turbine model. The dynamics in the conduits will treated by means of the Method of Characteristics (MOC) [61]. Models of the controllers, the generator, the electric grid and the load is taken from well-known literature on electrical issues [16], [24].

A challenge is to investigate how all the elements should be connected together as one unit. In order to solve this a common per unit representation of the total system is needed. The traditional load flow analysis, that calculates the steady- state condition of the electric part of the system has to be extended. This is necessary in order to connect the two power regimes in a correct physical way.

B) Investigate, by means of the improved models, whether interactions between the hydraulic part and the electric part of the hydropower system are present or not.

In Norway, the interest in studying the interaction between dynamics in the conduits and in the electric grid started in the middle of the 1970s. Unexplainable load oscillations caused a lot of disturbances in the power supply in western Norway.

Up to the 1960s there were many separate coordinated transmission systems in Norway and stability problems rarely occurred. In the middle of the 1960s the separate systems were connected and connections to Sweden and Jutland (Denmark) were commissioned in the so-called Nordel system. This large transmission system was made up of strong subsystems and weak transmission lines. Stability problems started to be a problem. Power oscillations with time constants around 1-2 seconds and increasing amplitudes were often observed, especially in western Norway and in the connections to Sweden. This caused many disconnections and breakdowns in the transmission system. It was believed, among engineers, that interaction from the turbines and conduits may be a significant factor in the origin of the oscillations. A committee from the Norwegian energy supply industry, the Norwegian Electric Power Research Institute (EFT) and the Norwegian Water Resources and Energy Administration (NVE), called The “Derivat utvalget”, investigated this exhaustively. They did not manage to verify that hydraulic transients caused the stability problems, but they found that the problem could be controlled by the introduction of a new type of voltage controller with an additional unit. After some of the largest power stations (larger than 50 MW) had this installed, the stability problem almost vanished [16]. The question whether the transients in the conduits affected the electric grid was still unsolved.

After the new open energy market was commissioned in 1992 the stability problems seemed to increase again. Today, the combination of large imports and very low

8

production in Norway is critical Unfamiliar production and transmission relations are believed to be the main reasons for this. A closer study of the interaction between the conduits and the electric grid is desirable in order to predict large stability problems in the electricity supply.

Stability problems in the electric grid, caused by possible interactions from the hydraulic part, have also been observed in the USA and especially on North Island in New Zealand, on a large scale.

Interactions between the hydraulic and the electric parts of the power system have not been proven by means of simulations before. This is mainly because of the lack of simulation models for this purpose. In the thesis the developed and improved simulation models will be used to investigate whether interactions between the two power regimes are present or not.

Traditionally, hydraulic engineers perform stability calculations without a thought of the electric grid. They are "learned” to study models of a turbine running under isolated load conditions. Taking into account that all hydropower plants are working together in a coordinated grid and are helping each other with the governing, the stability requirements may be reduced for some of the units. Electrical engineers concentrate their stability calculations on the coordinated electric grid. Usually, they put the boundary on the turbine shaft and thereby neglect the turbine and conduits. It is obvious that instability in the hydropower system will affect both sides and this will be shown by means of simulations. The Classic analysis is suited for this task and the new linear model will therefore be used for this purpose.

The linear model will also be used to perform time domain simulations of the "pressure feedback’’ technique [8]. This technique is a new turbine governing method where the pressure across the turbine is measured and used in the control algorithm. Time domain simulations of this method are required in order to clarify the advantages of this method.

The Modem analysis is suited for simulations that may indicate whether transients in the conduits, such as "water-hammer ” and “mass-oscillations ”. will be reflected or not in the electric grid. It is convenient to study this by means of the non-linear model, because larger load variations can be simulated. Subjection to larger load variations will “visualize” the transients in the system in a better way than a linear model is capable of and will therefore make it easier to study possible influences.

9

1.4 OUTLINE OF THESIS

This section presents a brief overview of all the chapters and sections in the thesis. The purpose and the goal of each section will be described. The thesis is divided into three main chapters. The improvements of the mathematical models, Classic and Modem, are described in the two first chapters (Chapters 2 and 3). The simulation part concerning interactions between the hydraulic and electric parts of the power system is covered by the third chapter (Chapter 4).

Chapter 2: Classic turbine governing analysis.This chapter presents:

• New linear approaches to frictional damping in the conduits that make it possible to perform state space solutions of hydropower systems with more accuracy than today.

• A linear simulation model of a hydropower system where the hydraulic part and the electric part of the system are connected together as one unit.

First the approaches of the frictional damping in the conduits will be established. The approaches will then verified and tested against the Method of Characteristics (MOC). An improved linear model of the hydraulic part can then be established. Finally, a connection between the hydraulic model and a linear model of the electric part is established.

Section 2.1: The head/flow transfer function, h/q.This section gives an overview of typical transfer functions between the head and the flow, h/q, in the conduits. This function is needed in a state space formulation of the hydropower system and it is the basis of the Limited zero- pole method, where the frictional damping approaches will be included. Different types of the function are discussed and models of frictional damping is treated. The Rayleigh damping method is presented. This method is used as a frequency-dependent frictional damping model in the thesis.

Section 2.2: Approximations, by means of the Limited zero-pole method, of the transfer function, h/q, with frictional damping included.This section presents first the Limited zero-pole method and second the derivation of the new frictional damping approaches.

Section 2.2.1: The elastic transfer function, h/q, with frictional damping included. This section clarifies the basis of the frictional damping approaches. The head/flow transfer function is expressed as two part functions, the transcendental term and the non-linear square root term, that will be approached in the next sections.

Section 2.2.2: Approximations of the transcendental term.The transcendental term of the h/q transfer function is approached.

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Section 2.23: Approximations of the non-linear square root term. An approach of the non-linear square root term is established.

Section 2.2.4: Complete approximations.This section presents the complete approaches of the head/flow transfer function with factional damping included by means of the approximations given in Sections 2.2.2 and 2.2.3. The approximations are suitable for time domain analysis, and take both the elasticity and the frictional damping in the conduits into account.

Section 2.2.5: Verification of the approximations.The intention with this section is to perform a time domain verification of the approaches by comparing them with the Method of Characteristics. Various simulations are performed in order to demonstrate their accuracy and limitations. Before the verification can be done the approaches have to be transformed to the time domain by means of a state space formulation.

Section 2.3: Linear dynamic model of a hydropower plant.This section presents linear dynamic models of a hydropower system where the hydraulic part and the electric part are connected together as one unit. Models of respectively a turbine running under isolated load conditions and a turbine connected to a coordinated stiff electric grid are established. The models are built up by means of existing block diagrams, with the new frictional damping approaches included.

Section 2.3.1: Turbine unit connected to an isolated grid.A linear model of a turbine running under isolated load conditions is presented. The derived approaches of the transfer function, h/q, with frictional damping included will be connected with a model of the turbine and speed governor presented in [4]. The model will be extended further by including a separate model of a surge shaft and a algorithm of pressure-feedback speed governing [30].

Section 23.2: Turbine unit connected to a stiff electric grid.Hus section presents a linear model of a turbine connected to a coordinated stiff electric grid. The point of origin is the model presented in the section above (isolated grid). This model will be connected to existing linear models of the synchronous generator, the electric grid and the voltage controller presented in [16], [24]. The connection between the models will be on the turbine shaft. The synchronous generator model needs an expression of the turbine torque as input, but the turbine model [4] gives the turbine power as output This will be solved by rewriting the turbine model

11

Chapter 3: Modern turbine governing analysis. This chapter presents:

• Improvement of an existing dynamic turbine model [33] in order to provide a simulation model that is more accurate and less dependent on measurements than the existing model

• A new non-linear simulation model of a hydropower system where the hydraulic part and the electric part of the system are connected together as one unit.

Improvements in the dynamic turbine model will be performed first. Afterwards, consequences of hydraulic inertia will be shown by means of simulations. Then the experimental study will be described and at last the non-linear simulation model will be presented.

Section 3.1: Improvement of a dynamic Francis turbine model This section covers improvements in the dynamic turbine model. The turbine model consists of two differential equations, one for the flow through the runner and one for the turbine torque. First the equation of the flow through the runner will be derived from an “earlier physical stage” than in [33]. In this way an expression of the hydraulic inertia in the runner can be found. A dynamic model of the draft tube will then be established in order to find an expression of the hydraulic inertia which can be used together with the turbine runner. This is necessary because it seems reasonable that most of the hydraulic inertia is placed in the draft tube. The turbine torque equation will then be improved by deriving expressions of the loss coefficients in the model The improved model will finally be used in simulations in order to show consequences of the hydraulic inertia in the hydropower system

Section 3.1.1: Flow through the turbine runner.This section presents the first stage of deriving an improved version of the dynamic turbine model [33]. The equation of the flow through the runner will be derived from an earlier physical point of origin. An expression of the hydraulic inertia in the runner can then be found.

Section 3.1.2: Dynamic draft tube model.This section covers the derivation of a dynamic draft tube model An expression of the hydraulic inertia will be established and combined with the equation derived in Section 3.1.1. An improved version of the equation of the flow through the runner is then obtained.

Section 3.1.3: The turbine torque equation.This section presents the improvements of the turbine torque model The goal is to reduce the number of “unknown” magnitudes by establishing analytic expressions of the loss coefficients in the model. After this is obtained, the improved version of the dynamic turbine model is completed.

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Section 3.1.4: The steady-state condition of the dynamic turbine model This section discusses the features of the dynamic turbine model with respect to the turbine characteristics. The purpose of this section is to describe the limitations in the turbine model due to the shape of the turbine characteristics. Low-head Francis turbines have more “difficult” characteristics than high head machines. The turbine model is not able to produce such characteristics without a correction parameter. The correction parameter is important in Section 3.2.2, wheremeasurements of a low-head Francis turbine are compared withsimulations of the improved dynamic turbine model

Section 3.1.5: Consequences of the hydraulic inertia.The purpose of this section is to show by means of simulations the consequences of the hydraulic inertia in the hydropower system. This phenomena is unknown for many engineers and might be the answer to unpredictable stability problems with plant. A computer program with a turbine connected to an isolated grid, based on the improved dynamic turbine model will therefore be derived.

Section 3.2: Experimental Study.An experimental study on a low-head Francis turbine will be treated in this section. The purpose is to test the hydraulic inertia approaches and compare the turbine model with unpredictable turbine characteristics.

Section 3.2.1: The experimental set-up.The outline of the experimental set-up is described in this section. The basis is a low-head Francis turbine tested with two different draft tubes. The measurements will firstly be presented and afterwards they will be compared with simulations performed by means of the improved turbine model

Section 3.2.2: Steady-state measurements.This section presents measurements of the steady-state turbine characteristics. They are needed in order to evaluate whether a correction parameter has to be used in the dynamic turbine model

Section 3.2.3: Transient runaway measurements.This section covers the transient runaway measurements. The turbine speed is measured during a disconnection of the generator.

Section 3.2.4: Steady-state measurements compared with simulations. In this section simulations are compared with the steady- state measurements. The accuracy of the turbine characteristics computed by the improved turbine model will be discussed.

Section 3.2.5: Transient runaway measurements compared with simulations. This section compares the transient runaway measurements with simulations. A discussion about whether the hydraulic inertia approaches are satisfactory or not is included.

& 35

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Section 3.3: Non-linear dynamic model of a hydropower plant connected to a stiff electric grid. This section presents a non-linear dynamic model of a hydropower system where the hydraulic part and the electric part are connected as one unit. Mathematical models of all the elements in the power system are first treated separately and second connected to a total model by means of a per unit representation (dimensionless form). The steady-state conditions of the total model will then be treated. An extended version of the traditional load flow analysis will be derived. This is necessary in order to connect the power regimes in a physically correct way. A flow chart of a derived computer program of the system is shown and described at the end of the section.

Section 3.3.1: The turbine and the conduits.The model of the conduits in the thesis consists of an upper and lower reservoir, a supply tunnel, a surge shaft and a penstock. The Method of Characteristics [61] will be used to solve the transients in the conduits. This section presents equations of boundary conditions that are needed in MOC (equation of the flow through the runner, a surge shaft and a reservoir).

Section 3.3.2: The synchronous generator.A 5th order model of the synchronous generator [24] will be presented in this section. The model is well-known and regarded as basic and one of the most accurate models of the generator. Some definitions concerning the rotating fields in the generator will also be given. This isneeded in order to clarify the difference between the mechanical and electrical angular speed. This section also covers a “spin-off’ example of a “hydraulic analogy” of reactive power.

Section 3.3.3: The speed governor.This section presents common models of speed governors [4], [33]. Governor models with and without permanent speed drop will be shown.

Section 3.3.4: The voltage controller.This section presents a basic model of a voltage controller [24].

Section 3.3.5: The electric grid.Mathematical models of a stiff coordinated electric grid are presented [16]. This model is regarded to be sufficient for studying the dynamics of turbine units.

Section 33.6: The electric load.This section covers the mathematical representation of the electric load. The electric load is the most difficult part to model with accuracy. This section discusses common ways of modelling the load.

Section 3.3.7: Per unit representation.A per unit representation is treated in this section in order to connect the elements in the system together as one unit. A common power basis

14

is needed for the whole system in order to provide a dimensionless representation of the system.

Section 3.3.8: Steady-state conditions.This section describes the steady-state conditions of the model A rewriting of the traditional load flow analysis is necessary and will be presented in this section.

Section 3.3.9 Simulation program.This section presents a new developed computer program of a hydro­electric system. A flow chart of the program is presented and the routines and the calculation procedures are explained.

Chapter 4: Interactions between the hydraulic part and the electric part of a hydropower system.

This chapter presents:

• Investigation, by means of the improved models, whether interactions between the hydraulic part and the electric part of a hydropower system are present or not.

First the linear dynamic model will be used to study interactions between an isolated grid and a stiff grid due to stabilitv/instabilitv performance. The purpose is to show how instabilities in the hydropower will affect both parts of the system. Simulations are performed in the time domain by means of the linear model derived in Chapter 2. Time domain simulations of the “pressure feedback” governing technique is shown since the linear model is a suitable analytic tool for this task. Finally, the non-linear model presented in Chapter 3 will be used in various realistic cases in order to study whether transients in the conduits, such as “water-hammer” and “mass-oscillations”, will be reflected in the electric grid or not.

Section 4.1: Linear time domain simulations.. This section covers simulations by means of the new linear model presented in

Chapter 2. The derived models will be used to study how a stable/unstable turbine unit that is running under isolated conditions will behave when it is connected to the coordinated electric grid. Simulation examples are shown in order to study governing stability and pressure feedback.

Section 4.1.1: Simulation of a turbine unit connected to an isolated grid. Various simulation cases of a turbine unit connected to an isolated grid will be performed in order to compare the results with simulation of a unit connected to the coordinated grid. Simulations of pressure feedback governing will also be shown.

Section 4.1.2: Simulation of a turbine unit connected to a stiff electric grid. The turbine unit, simulated in the previous section, will be simulated after the unit has been connected to a coordinated stiff grid. The results will be discussed in relation with Section 4.1.1.

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4.2 Non-linear time domain simulations.This section presents simulations by means of the new non-linear model presented in Chapter 3. The new computer program will be used to study whether hydraulic transients are affecting the electric grid or not.

4.2.1 Simulations compared with SIMPOW.The intention with this section is to verify the new computer program with respect to the electric part. SIMPOW is one of the most powerful computer programs for power systems in the world. A verification will be done by comparing results from a simulation case where the effect on the hydraulic side is small.

4.2.2 Local load connection.This section deals with a simulation of a local load connection on the transmission line between the turbine unit and the electric grid.

4.2.3 Downloading.Downloading of a turbine unit connected to the electric grid is shown in this section.

4.2.4 Undersized surge shaft cross section.In this section the consequence of having undersized surge shafts in the conduit system is shown for a turbine unit connected to the electric grid.

4.2.5 Uploading with line rejection.Uploading of a unit connected to the electric grid that is being exposed to a sudden line rejection is shown in this section.

4.2.6 Short circuit with sudden synchronizing.This section shows how the new computer program can be used to study synchronizing of turbine units.

The Conclusion of the thesis and recommendations for further work are presented in Chapter 5. This is followed by the Bibliography with complete references sorted in alphabetic order.

There are also seven appendices included in the thesis. These contain supplementary information and explanations.

Appendix 1 gives a description of turbine goveminp systems. The main intention with this appendix is to give the reader a perspective on different turbine governing techniques and suggested improvements. Turbines connected to a coordinated grid are described and speed governing actions are treated.

Appendix 2 presents a complete derivation of the Allievi’s Equations. These equations form a fundamental base for all water-hammer calculations.

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Appendix 3 describes the derivation of the transcendental transfer function between the pressure head and the flow in the conduits. This function is the origin of the linear approaches made by the Limited zero-pole method.

Appendix 4 describes a quasi non-linear frequency dependent frictional damping model by Kemgeter/Brekke [4].

Appendix 5 considers a surge shaft model in the frequency domain. The model is implemented in the state space model in Chapter 3.

Appendix 6 describes the theory of the Method of Characteristics (MOC). This shows how a frequency dependent damping term can be included by means of an additional diffusion term, so-called Rayleigh damping. Also necessary boundary conditions are treated.

Appendix 7 describes how a load flow analysis is carried out for an electric grid and how linearized electric magnitudes can be found.

1.5 CONTRIBUTIONS IN THE PAST

The most basic contributions in the field of system dynamics analysis of hydropower plants were presented several years ago. Today, there are few contributions in this field of research compared with other fields in hydropower research. Chaudhry [11], Raabe [42] and Siervo and Leva [48] have published basic literature concerning mathematical modelling of hydropower plants in the 1960s and 70s. The modelling was concentrated on the hydraulic part of the power system. Due to the low computer capacity that time, effort was spend on providing simplifications of the models in order to solve them.

The first doctoral thesis in Norway, in this field, was written in 1984 by Professor Brekke [4]. He developed the theory of the Structural Matrix Method (SMM) for turbines and conduits. Stability calculations for complex hydropower plants with several tunnels, surge shafts and creek intakes were now possible. The turbine characteristics were also included in the model and he suggested a new frequency dependent damping model which was verified on a large scale with good results. Brekke’s work can be placed in the category of Classic turbine governing analysis.

Li [29] presented his Dr. ing. contribution in 1989. Stability analysis for hydropower systems with throttled versus unthrottled surge tanks and air cushion surge tanks were treated. One main purpose was to investigate large amplitude surge oscillations in relations to the critical Thoma area. Much of the theory was based on [4] and the thesis was a valuable contribution to a complex field. Li also presented several publications in the beginning of the 1990s [8],[30].

Contributions to the topic of transient analysis of hydropower plants are mostly concerning dynamics in the conduits. Transient pipe flow in general is a large research field today with many contributions each year. Among many interesting publications, the works by Jaeger [23] and Wylie and Streeter [61] are well-known and regarded

17

as fundamental. Hydraulic transients modelled by means of the Bond graph method is, among many publications, presented by Filho [17]. During the last few years the use of this method has increased a lot in hydraulic research. A reason for this may be the well arranged way of modelling large systems.

Various models of frequency dependent frictional damping are established. Zielke [62] established an analytic model of frequency frictional damping in laminar oscillating pipe flow. Kengeter [27] presented a well-known approach of damping in turbulent flow for pipes with small diameters. In 1996, Svingen [55] presented a one­dimensional model of transient flow in a pipe, including frequency dependent Rayleigh damping. Because of its “easiness” and sufficient accuracy, this model is suitable for water-hammer calculations in connection with the Method of Characteristics.

In order to establish models of the turbine characteristics, contributions have been given by Ramos and Almeida [43]. Their work is concentrated on overspeed effects of small Francis turbines, where the turbine characteristics from nominal operating condition to runaway speed is approached. Pejovic [41] has carried out investigations of pump turbine characteristics with focus on their typical S-shape. In 1990, Nielsen [33] presented his doctoral thesis in the field of modem turbine governing analysis. A dynamic model of a Francis turbine, for use in non-linear time domain simulations, was presented. Nielsen showed by theory and experiments that the hydraulic inertia in the turbine had a great influence on the transient behaviour for high-head Francis turbines.

Very few publications are presented in order to establish models of both the hydraulic and electric parts of the hydropower system connected together as one unit. In the 1950s Paynter [40] established models of general hydraulics by use of electric analogy. He also spend effort on connecting several energy regimes together by means of the Bond graph method. In Johannessen [25] the dynamic behaviour of the conduits is also described by an electric analogy, though with a simplified hydraulic representation. Nielsen [35] has presented simulations of two turbines interconnected, though with a strongly simplified representation of the generator and the electric grid. Weber [59] has presented several papers where the hydraulic part and electric part of the power system are connected together. The hydraulic part of the system is treated as a “black-box” which has to be parameter estimated against measurements on plants before the total model can be solved. In this way the models are not suited in the design phase but only for existing plants where measurements can be performed. The results of Weber’s investigation of a load rejection on a high-head turbine in the Swiss Alps showed good agreement with measurements. One main reason for this may be that the turbine in this case was equipped with a relive valve (not commonly used in Norway). Large transients in the hydraulic system will therefore be damped in a large scale and the sufficiently accuracy of modelling the hydraulic part will be decreased.

One of the most reliable computer programs for large power systems is SIMPOW. Considerable technical and mathematical expertise lies behind its development and dates back to the late 1970s. Its functions are being continuously developed and modernized. The program is used world-wide and especially in universities and corporate research. The electrical part of’he power system is modelled in detail but the hydraulic part is strongly simplified, fhe program is therefore not capable of calculating water-hammer and mass oscillations.

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2. CLASSIC TURBINE GOVERNING ANALYSIS

Classic turbine governing analysis is here considered to mean;

• Stability analysis by means of a linearized equation of the system in the frequency domain.

• Time response or system dynamic analysis using state space formulations by means of inverse Laplace transformation from frequency to time domain.

This chanter presents:

1) New linear approaches to frictional damping in the conduits that make it possible to produce state space solutions of hydropower systems with more accuracy than today.

2) A linear simulation model of a hydropower system where the hydraulic part and the electric part of the system are connected together as one unit.

First the approaches of the frictional damping in the conduits will be established. These will then verified and tested against the Method of Characteristics (MOC). Finally, an improved linear model of a hydropower plant is established.

2.1 THE HEAD/FLOW TRANSFER FUNCTION, h/q

In Appendix 2 the fundamental equations for one-dimensional transient and elastic flow in a pipe are derived. They are called the Allievi Equations and in dimensionless form they can be expressed as:

The Continuity Equation:

dq AgH0 oh(ZI)

The Equation of Momentum:

(Z2)

The equations are hyperbolic partial differential equations. In Appendix 3 the transfer function between the pressure head, h, and flow, q, in front of the turbine is derived:

(2J)

where the Allievi constant, hw is given by:

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K (2.4) Qoa

" 2Xg^o

z is a complex friction parameter given by:

z = -Js2 + ks

where k is the friction parameter:

. ®Dr k = ——

Qo<IP

(2.5)

(2.6)

The friction parameter may be modelled both as a linearized steady-state term or as frequency dependent term, depending on the way of modelling the shear forces, t.

The transfer function in Equation (2.3) is transcendental because of the hyperbolic tangent term. Both the elasticity and the frictional damping are included. A number of simplifications and also improvements exists. Table 2.1 presents a ranking of some of them due to their complexity.

Table 2.1 Different types of the head/flow transfer function.

1h z AL— = -2hw -tanh —z q s \a J

uIIN Steady-state +

Frequencydependent

Yes

— = -2 hw - tanhf — z] q s \a J2 z = 'Js2 + kss Steady-state Yes

3— = -2 hw tanhf—z| q \a J z = s No friction Yes

4 z=s + ks Steady-state No

5r"26"tiz)

z = s No friction No

In Figure 2.1 the transfer functions given in Table 2.1 are drawn in a Bode diagram (amplitude-frequency diagram). As we can see the inelastic version (number 4) is only

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good for low frequent disturbance, m < 0.2 rad/s. It means in practice that only slow disturbances can be studied. Figure 2.2 shows an example of the influence of a linear closing valve in frequency and time domain. A fast closing characteristic will contain more high frequent components than a slow closing one. In fact an instantaneous closing valve contains all frequency components.

- Pressure/flow transfer function

3) Elastic with

10

D*g Saikmi4TniNTW.1T

Figure 2.1 Amplitude/frequency plot of various versions of the h/q transfer function.

By including a steady-state damping term to the inelastic function (number 5 in Table 2.1) the function will be affected at low frequencies.

The amplitudes of the elastic version without damping (number 3 in Figure 2.1) will reach infinity at the resonance frequencies. The elastic transfer function including a steady-state damping term (number 2 in Figure 2.1) will have equal damping at each resonance top and the amplitudes will reach a finite value at the resonance frequencies.

The friction parameter, ks, is derived in [4] by substitute the shear stresses in Equation (2.6) with an expression for the linearized shear stresses in turbulent flow:

A/Xg2 ZpQo{Q«q)(2j)= 4 )

Then:

K = 42oAD

(2.8)

This linearized damping model has been referred to as “the traditional damping term”.

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5

a [rzdJs]

Figure 2.2 The connection between a linear closing valve in frequency- and time domain.

Frequency dependent friction in turbulent pipe flow is a complicated physical phenomenon that is not well understood today. The terminology “frequency dependent damping” is used for one-dimensional models, where the damping effect, caused by the variation in the velocity profile, is approximated. Many models are established but none of them seem to be universal, they have their limitations, as for instance large/small pipe diameter and rough/smooth pipe walls. Kongeter [27] developed a frequency dependent model by adding a frequency dependent term to the steady-state one (number 2 in Figure 2.1). The model was later modified by Brekke [4], The model demands iteration and cannot be used in a state space model. The model is described briefly in Appendix 4.

In this thesis a model called stiffness proportional damping or Rayleigh damping is used as an example of frequency dependent friction. The model is well described in[55], [56]. The model is obtained by adding a diffusion term to the Equation of Momentum in Equation (2.2). It is appropriate to define the diffusion term as:

Qo 4^9

AgH0 p a1 (2.9)

The extended Equation of Momentum then becomes:

_ Qo f tcDt [ dq | tfqa AgH0{Q0p a p a2, (2.10)

The Rayleigh model has many advantages-.• It shows good results compared with experiments

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• Fast calculations (one-dimensional)• It may easily be implemented in MOC for time domain analysis• Explicit expression in the frequency domain is obtained (no iteration)• Compatible with state space formulations

There are also some disadvantages:

• Largest damping is obtained at the lowest flow rate (physically incorrect)• Underestimates the damping in the first couple of oscillations• The friction parameter, Xf, has to be determined by trial and error

The reason for choosing Rayleigh damping as a frequency dependent model in the thesis, is first and foremost because of its simple and explicit form that makes it possible to express the model as a rational approach by means of the Limited zero and pole method. Although, the dynamic Rayleigh friction parameter, Xf has to be determined by trial and error, it may easily be calibrated to match the model of Kemgeter or Brekke for instance in the frequency domain. If this is done in the first instance, the Rayleigh damping model can easily be used both in the frequency- and the time domain.

In Figure 2.1 the frequency dependent transfer function including Rayleigh damping (number 1 in Figure 2.1) is shown. As we can see, the damping will increase with increasing frequency, which is physically correct. The friction term z, is given by [56]:

is2 +kssZ~i 1+kds (2.11)

where the friction factors are given by:

(2.12)

(2.13)

ks is the linearized steady-state damping term and fa is the dynamic damping term. X is Darcy-Weisbach’s friction factor and Xf is a dynamic friction factor.

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2.2 APPROXIMATIONS BY MEANS OF THE LIMITED ZERO-POLE METHOD OF THE TRANSFER FUNCTION, h/q, WITH FRICTIONAL DAMPING INCLUDED

Classic state space formulations demand rational transfer functions in the frequency domain and a set of ordinary differential equations in time domain. Since the elastic behaviour of the water column in conduits is governed by partial differential equations in the time domain and by transcendental equations in frequency domain, this has been an obstacle for applications of new software tools for automatic control Rigid water column theory satisfies the state space criteria, but the theory is only valid for slow surge oscillations and for plants where the penstock is relatively short. Most of the hydropower plants in Norway have long pressure shafts and the elasticity in the water column will strongly affect the system.

Some techniques for establishing rational expressions for the elastic transfer function, with no frictional damping, are obtained. In [30] the method of Taylor Expansion is studied. A first order Taylor approximation is exactly the same as using rigid water column theory or the inelastic transfer function (number 4 in Figure 2.1). As we can see from Figure 3.1, this approximation will only be good for the frequency range up to co = 0.2 s'1. A second order approximation gives satisfactory results up to to = 1 s'1. No further improvements were found for higher than the second order Taylor approximation.

A new way of establish rational approximations to the elastic h/q transfer function was presented by Li [30]. The method was called the Limited zero-pole method. The transfer function is approximated by using a selected number of its zeros and poles. The method showed very promising results for the transfer function with no frictional damping (number 3 in Figure 3.1). Originally, the transfer function has infinite poles and zeros along the imaginary axis. The investigation showed that a selection of four poles and three zeros gave accurate approximation up to os = 4 s'1. That is considered to be good enough for turbine governing stability analysis [30]. The method will give accurate approximations up to any frequency we want. The only price is an increased order of the system

Since the Limited zero-pole method seems to give very good approximations for the transfer function with no frictional damping, the method could also be applicable when frictional damping is included. This section presents new approximations for this case.

2.2.1 The elastic transfer function, h/q, with frictional damping included Figures 2.3 and 2.4 present the amplitude and the phase of the transcendental transfer function, h/q, with frequency dependent Rayleigh damping included. Data for the conduits presented in [30] are also used in this chapter. They are described in Table 2.2.

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Table 2.2 Data used in simulations.L 1000 mHo 250 mQo 5 m3/sA 5 m2a 1000 m/s

The transfer function can be expressed as two part functions as follows:

(2.14)

(2.15)

(2.16)

The steady-state friction parameter is chosen to be X = 0.013, which is a realistic value in this case. The linearized friction term can then be calculated to be: K, = 0.005 s'1. By choosing the Rayleigh friction parameter to A/= 5*106 kg/ms a realistic amount of frequency dependent frictional damping will be present. The dynamic friction term is then calculated to Kd=0.005 s. The Allievi constant becomes; Hw=0.20 and the time constants of the pipe is; Tw= (Q0L/gAHo) = 0.41 s.

Equation (2.13) is the transcendental term of the transfer function. It has an infinite number of zeros and poles and causes the resonance peaks shown in Figures 2.3 and 2.4. Equation (2.14) is a non-linear square root term with a finite number of zeros and poles. In order to find approaches for the h/q transfer function, the two terms can be treated separately and afterwards be combined to a total approximation.

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1= 1003 = 1000 m/s

Figure 2.3 Amplitude of the transfer function, h/q, with Rayleigh damping included.

m [rjd/*]

Figure 2.4 Phase of the transfer function, h/q, with Rayleigh damping included.

2.2.2 Approximations of the transcendental termBy substituting s = jco in Equation (2.14) the following expressions can be established:

-J = tanha \ \ + kdjco

= j tan(w) (2.17)

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(2.18)Lu-—co\

a | 1 + kdjco

As we can see from Equation (2.17) the hyperbolic tangent term is expressed as a complex tangent function.The poles can be found by letting: (—

—> 00

By studying a tangent function, this will occur at the resonance peaks when:

h = ±—(l+2«), n = 0,1,2,3,.... (2.19)

By combining Equations (2.18) and (2.19), and solving with respect to ©, the following expression can be established:

m + Ckdup(n)2)j±y4Cup(n)2 - k]-Cup{n)2kd[Cup{n)2kd +2*,) (2.20)

Where:

up (n) = (l+2n), n = 0,1,2,3,.... (2.21)

The zero-pole constant is defined by:

Kir(2.22)

The poles can now be found by letting s =jco:

s = ~^[ks + Ckdup{n)2}± ^j^4Cup(n)2-kj-Cup(n)2 kd{cup{n)2 kd +2fc5j (2.23)

As we can see, the poles are appearing in complex conjugated pairs. Equation (2.23) can be expressed in a more convenient way suited for state space solutions:

Cu („)4*2 +{ks + Cup(nYkd)s+CuP(n)2) = 0 (224)

Finally, by substituting for Up(n) an expression for the poles can be found:

■-1 -2- f/+(^ + C(1 + 2nf kd)s+C{\ + 2n)2) = 0, n = 0,1,2,3,.... (2.25)C(l + 2«) v x ' '

27

Likewise, the zeros can be found by letting: l ->0

According to the tangent function this will happen when:

u = 0, n = 0 (2.26)

u = ±—{2n), n = 1,2,3.... (2.27)

In order to simplify the mathematical derivation, Equations (2.26) and (2.27) are combined:

k , , [mz(«) = 0, « = 0(2.28)

By combining Equations (2.18) and (2.28) and solving with respect to co, the following expression can be found:

® = Ckduz{nf )j±^4CU;(n)2 -k] -Cuz{n?kd{Cuz(rfkd +2ks) (2.29)

The zeros can be found by letting s = jco:

s = + Ckduz(n)2j±-j^4Cuz(n)2 -k2 -Cuz(n)2kd{cuz(n)2kd + 2fcs) (2.30)

Rewriting Equation (2.30):

_ ~7~o P + (K + Cuz(nf kd)s + Cuz(nf) = 0 (2.31)Cuz(n) ' ’

By inserting uz(n) an expressions for the zeros can be found:

s(s + ks) = 0, n = 0 (2.32)

-r-rr(s2 +{K + 0»2kd)s+c(27i)2) = o, « = 42,3,.... (2.33)C(2n)

As we can see there are two real single zeros for n = 0 in s = 0 and s = -ks. The other zeros will appear in complex conjugated pairs.

An approximation of the transcendental term of the h/q transfer function can then be established:

28

zeros<?/, poles

q),

+(*, +kdC(2n)2)s+C(2nf)

r|c(i+2«)2 +(^+^c(i+2w)2)'y+c(i+2”)2)(2.34)

Where N: is the limited number of zeros and Np the limited number of poles in the approximation.

Frequency dependent damping Steady-state damping

No damping

* Complex conjugated POLES

O Complex conjugated ZEROS O Real ZEROS

Figure 2.5 The placement of the zeros and poles of the transcendental transferfunction, (h/q)i, with no frictional damping, steady-state damping and frequency dependent damping.

In Figure 2.5 the zeros and the poles are sketched in the left complex half plane. With no frictional damping included ( h = fe = 0) the zeros and the poles will be placed along the imaginary axis and a double zero is placed in origo.

With a steady-state damping term included (fa= 0) the zeros and poles will be placed along a straight line {-kJ2) in the left complex half plane, except of two single zeros placed in s = 0 and s = -kj.

If frequency dependent damping is included an additional damping effect, formed as a horizontal parabola starting at the straight line will be present. The two singlezeros will be unaffected.

29

2.2.3 Approximations of the non-linear square root term The non-linear square root term can be expressed as:

4 + &J _ | s+k,

l + kds wys(l + kjs) (2J5)

The function has one real zero, s% = -ks and two real poles; Spi = 0 and Sp2 = -1/kd- The non-linear effects dominates when the frequency is low, and therefore small error is obtained by neglecting the dynamic friction term. This can be identified by studying the pole, Sp2, which is placed far from origo in the left complex half plane (this indicates a very fast response).

(2.36)

The function has then one zero, Sz = -ks and one pole, Sp = 0.

The following rational approach can then be established by means of the zero and pole:

(2.37)

2.2.4 Complete approximationsA complete approximation of the transfer function, h/q, can now be obtained by combining Equations (2.34) and (2.37). The approximation must also satisfy the limiting value (see Figure 2.1):

vJhlim—q

= ~T„{s + ks) (2.38)

Only one of the double zero in s = -ks is therefore used and the term 2hw is substituted by Tv,. The following new approximation of the transfer function, h/q, with frictional damping included can then be established:

- = —----------- ^ K)-------------------------------------------- (2.39)

This new approximation can be reduced to exactly the same function as presented by Li [30], if frictional damping is negelcted; ks=fa = 0.

30

Li [30] showed that a 4 pole-3 zero approximation (4th order), of the h/q function without frictional damping, would give satisfactory accuracy up to to = 4 s'1. In this case a 4th order approximation can be established by selecting N:=NP = 1:

4-f-| r.c),, WMwMc)----- (2.40,? ^ 4 J (s2 + (*$ + kdc)s+Cj(s2 + (*, + 9kdC)s+9C)

In Figures 2.6 and 2.7 the amplitude and phase of the 4th order approximation (Equation 2.40) compared with the original transfer function (Equation 2.16) are plotted. As we can see from the amplitude/frequency curve in Figure 2.6, the approximation is almost accurate in the frequency range up to to = 5 s'1. The phase/frequency curve is shown in Figure 2.7. The approximation shows very good agreement with the exact curve up to m = 5 s'1. There will be a certain phase shift between the two curves at very low frequencies, but this effect will be very small in transient time domain analysis.

------[tVq| 4 pde-3 zero approximation

Figure 2.6 Amplitude/frequency characteristics for a the head/flow transfer function and its 4th order zero-pole approximation.

31

. —<Wq

■<h/q 4 pda-3 zero approx mat on

.100 a

e> [red/s]

Figure 2.7 Phase/frequency characteristics for a the head/flow transfer function and its 4th order zero-pole approximation.

Concluding remarksThe new approximations of the transfer function, h/q, with frictional damping included seems to be successfully derived in the frequency domain. They will be verified in the time domain by means of a state space formulation in the next section.

2.2.5 Verification of the approximationsThe intention with this section is to perform a time domain verification of the derived approximations by comparing them with the Method of Characteristics (MOC). The rational transfer functions are written in canonical form and transferred to a set of 1st order differential equations (State space formulation). A 4 pole-3 zero approximation will lead to 4 differential equations in the time domain. The rational transfer functions are based on linearized functions and we are limited to study small variations around the steady-state operation point. It is also necessary to have in mind that approximations with a limited band width will have restrictions concerning how fast the variations can be, see Figure 2.2.

In order to perform a time domain solution of the derived approximations the head/guide vane transfer function, h/y, is used as origin. The guide vane is here regarded as a valve linearized around (Ho,Qo,Yo). The influence from the turbine characteristics is neglected. According to [4] the head/guide vane transfer function can be expressed by means of the h/q transfer function:

hk- = ^T

y i_I*2 q

(2.41)

32

By inserting the 4 pole-3 zero approximation in Equation (2.41) the following transfer fhnction, written in canonical form, can be established:

h b3s3 +b2s2 +bls+bQ y sA + a3s3 + a2s2 + axs+a0 (2.42)

Where bt and a/are canonical constants.

The transmission from frequency to time domain is well described in [3], The time domain model is often called a Design Plant Model (DPM). The time domain egression of Equation (2.42) can be given by the following equations (Canonical controllable form):

AxiAx2

Ax3

Ax 4

0 0 0 ~ao Ax, b01 0 0 -a, Ax2 b,0 1 0 ~a2 Ax3

+h

0 0 1 -°3. _Ax4_

•Ayto (2.43)

Ah{t) = [0 0 0 i]-[Ax, Ax2 Ax3 Ax4]T

Ht) = — to = Ah(t) +1 ■“0

AyW+i-S)

(2.44)

(2.45)

(2.46)

In order to verify the state space model, computations performed by MOC are used as comparison. The theory of the MOC is described in Appendix 6. It is a pure time domain method and ft includes non-linear effects in the system. The state space model is programmed and solved by a Classic Runge Kutta method of 4th order.

In Figure 2.8 an example of calculated pressure head in time domain is shown. The valve closes linearly from y - 1 to y = 0.9 in 2 seconds. Data from Table 2.2 are used in the simulations.

33

f- woe0.92 t

- - Ope^ng degree

Seconds

Figure 2.8 Comparison between MOC and the 4 pole-3 zero approximation. Linear valve closure from.}' = 1 to y = 0.9 in 2 seconds.

As we can see there is an excellent agreement between MOC and the 4 pole-3 zero approximation. This closing history corresponds to the restrictions in the state space model; relatively slow closure time and small variations in the opening degree.

In Figure 2.9 the closing from y = 1 to y = 0.9 is performed in one second. The agreement is still satisfactory, but small oscillations can be seen on the first square pulse of the approximation.

Figure 2.9 Comparison between MOC and the 4 pole-3 zero approximation. Linear valve closure fromy = 1 to y — 0.9 in 1 second.

34

—"' MOC•4 poto-3 zero approximation

- - Opertng degree

Figure 2.10 Comparison between MOC and the 4 pole-3 zero approximation. Instantaneous valve closure from j = 1 toy = 0.9.

Instantaneous valve closure results in a square first pulse. In Figure 2.10 an instantaneous closing history from y = 1 to y = 0.9 is shown. The pulsation in the approximation is conspicuous and the agreement is rather bad in the first three pulsations. The explanation is the limited band width the 4 pole-3 zero approximation has (to = 5 s'1). An instantaneous valve closure demands theoretically an approximation with an infinite band width, see Figure 2.2. Although, better agreement with MOC can be obtained by using a higher order zero-pole approximation.

In Figure 2.11 the principle of increasing the order of the approximation is shown. An increased number of zeros and poles will make the pressure curve more like a square. The oscillations will decrease in amplitude an increase in frequency. The analogy is Fourier cosine- and sine- series.

Increasing number of zeros/poles

Figure 2.11 The principle of increasing the order of the zero-pole approximation towards a square pulse.

35

So far, the examples have dealt with small changes in the opening degree of the valve. The only limitation has then been how fast the opening degree can be changed. A large change in the opening degree will have a limitation according to the linearized equations used in the state space model. In Equations (2.47) and (2.48) the non-linear valve equation and the linearized version are respectively given [5]. As we can see, the linearized function is uncoupled from the opening degree. The steady-state valve characteristics are shown in Figure 2.12.

Q

On

y \ H rr

To V Ho(2.47)

1 H lf. -x (2.48)

From Figure 2.12 we can see that towards full closing the linearization error will be tremendous. Even if the valve is fully closed the linearized model will relieve the system with half the steady-state flow.

In Figure 2.13 a relatively large closing from y = 1 to y = 0.5 is performed during 2 seconds. The agreement with MOC is not satisfactory. The early damping of the approximation is caused by the above argument; more water is going through the valve and the system will be relieved in a false way.

y = 0.9Linearization point

Linearized curve

04 j

Figure 2.12 Linearized and non-linear valve characteristics.

If the closing period is increased the agreement with MOC is more acceptable because of reduced water-hammer effects. This is shown in Figure 2.14 where a closing from y = 1 to y = 0.5 is performed during 4 seconds.

36

Figure 2.13 Comparison between MOC and the 4 pole-3 zero approximation.Linear valve closure fromj = 1 to y = 0.5 in 2 seconds.

MOC-' 1 '4 poifr3 zero approximation- - Opening degree

Figure 2.14 Comparison between MOC and the 4 pole-3 zero approximation. Linear valve closure fromj = 1 to y = 0.5 in 4 seconds.

Concluding remarksVerification of the derived approximations of the transfer function, h/q, is now successfully performed. They show very good results compared with MOC. It is though necessary to take the prescribed restrictions, concerning limitations in band width and linearization error, into account. The approximations will be used in the next section in order to establish linear dynamic model of a hydropower plant.

37

2.3 LINEAR DYNAMIC MODEL OF A HYDROPOWER PLANT

In this section a dynamic model of a hydropower plant subjected to small load and frequency variations are established and a computer program is developed. The system is build up by means of the mathematical program MATLAB (ver.5) and the additional package S1MULINK (ver.2), a commercial software package for modelling, simulating and analysing dynamic systems. A system can be established by means of pre-defined boxes for several built-in transfer functions and solved by means of state space formulations in time domain. In this way a physical block-diagram can be programmed directly and output curves can automatically be generated for each state space variable. The user interface is very good and a model can easily be changed and extended. In Sections 2.3.1 and 2.3.2 models of a turbine unit receptively connected to an isolated grid and a stiff electric grid are presented. The new approaches derived in Section 2.2 will be used to describe the dynamics in the conduits with frictional damping included.

2.3.1 Turbine unit connected to an isolated gridFigure 2.15 shows the main structure of a model where a turbine is running under isolated load conditions. The system can be built up of three main blocks or sub­systems;

• The speed governor system• The turbine-conduits system• The generator-grid system

They are connected to each other by means of the turbine speed, n, the turbine torque,Dt, and the opening degree of the guide vanes, y.

The common way of exciting the dynamics in this system are either by changing the electric load (torque) or by changing the reference speed in the speed governor. This system can be regarded as a typical model used by hydraulic engineers. The electric part is neglected and the turbine unit has to be stable on an isolated grid. This is in fact regarded as “worst case” seen from a hydraulic engineer’s point of view.

SpeedGovernor GeneratorTurbine/

Conduits

Figure 2.15 The main model structure of a turbine connected to a isolated grid.

In Figure 2.16 the sub system of the speed governor is shown. The unit consists of a PI governor, a block for the electro-hydraulic system (actuator) and a block for the servo motor. The system may also have a permanent speed drop, bp. The speed reference, nref, may be kept constant or be excited to several in-built signals. Included in the

38

model is also an opportunity of adding a pressure feedback loop to the speed governor. In Appendix 1 (Al.2.3) pressure feedback is mentioned in connection with speed governing improvement.

n_refSpeed

reference

Electro/hydraulicsystem

ServomotorPIGovemor

Speed

Permanent speed drop

Pressure Feedback Pressure head

Figure 2.16 The structure of the speed governor sub-system.

In [8] the following transfer function of the pressure feedback system is suggested:

_______ ___________ (2 49)(z;s+i)(rrfs+iX05s+i) 1;

Where the gain, K= 0 - 1 and the time constant, Ti = 10D - 502* In Figure 2.17 the Pl-govemor is shown with a transient speed drop, bt, and a compensated integral loop (Td is the integral time). Figures 2.18 and 2.19 show the electro-hydraulic system and the servo motor system. Ty is the electro-hydraulic actuator time constant and To is the servo motor time constant.

integral loop

Transient speed drop

Figure 2.17 The Pl-govemor unit.

39

Electro-hydraulic time constant

Figure 2.18 The electro-hydraulic system.

Servomotor time constant

Figure 2.19 The servo motor system.

In Figure 2.20 the turbine-conduits system is shown. The linearized coefficients from the turbine characteristics are explained in the Nomenclature. Usually, the turbine power is used as output signal of the turbine-conduits unit. The disturbance from the generator is treated as the active generator power. In order to be physically correct the output signal should be the turbine torque. Since the electric grid is neglected the turbine power may be used as well It is though more convenient to use the torque as output in order to match the model of a generator connected to a stiff electric grid, shown in the next section.

The linear connection between turbine power, p and turbine torque, D, can be established in the following way. The turbine output can be expressed as:

p= D'Q

Linearization around an operation point (po = D,0Qo) gives the following connection in dimensionless form:

t£l =>p=Dl+Q.=>D'=p — £2 (2.50)

40

Etficency*

Guide vjntsutfo

7*0—*©

vqConduits

Figure 2.20 The modified turbine-conduits system.

The transfer function between the pressure head and the flow in front of the turbine, h/q, is shown in Figure 2.21. The pressure rise caused by the elastic effects in the penstock (water-hammer) and the inelastic long-term pressure rise caused by a water level rise in the surge shaft (mass oscillations) are added together (superposition) to a resulting pressure head in front of the turbine.

(h/q)1Conduits

Pressure head

(h/(fl2 Surge shaft

Figure 2.21 The pressure head/flow sub-system.

The elastic (h/q)i transfer function can be expressed by means of 4 poles and 3 zeros according to the developed approximation given in Section 2.2.4:

(h\ bzs3 +b2s2 +bts \q) x s4 +a3s3 +a22 +ats (2.51)

In Appendix 5 a second order inelastic transfer function describing the pressure rise in front of the turbine, caused by mass oscillations, is derived.

41

2

(2.52)'h

<q.PxS + Po

s2 +als + a0

The turbine torque, D, is the input signal to the generator-grid system. In a steady-state condition the electric torque, De is equal to the turbine torque. If the electric torque is changed the torque difference will accelerate the unit and change the speed. The generator is usually represented by a 1st order model in an isolated grid presentation. T„ is the time constant of the synchronous generator.

Electric Torque

Turbine Torque

Figure 2.22 The generator-grid system (isolated grid).

Concluding remarksA linear model of a hydropower plant connected to an isolated grid is nowestablished. In the next section it will be shown how this model can be extendedfurther by connecting it to a model of the electric grid.

2.3.2 Turbine unit connected to a stiff electric gridA single turbine unit connected to a stiff electric grid (infinite bus) is the most common way of performing stability calculation of synchronous generator and the grid. The extended version of the main model structure, shown in Figure 2.15, is presented in Figure 2.23.

VoltageController

SpeedGovernor Turbine/

Conduits

Figure 2.23 The main model structure of a turbine connected to a stiff electric grid.

42

Compared with the isolated grid model, we can see that a voltage controller here is added. The generator-grid system gets input signals from both the turbine torque, Dt, and the field voltage, Eqf. The output signals are the speed, n, and the terminal voltage, Ek. Input signals to the voltage controller system are the terminal voltage, Ex, and the speed, n. The speed signal is used for an additional derivative damping term. Newer and more complicated controllers also have an input signal from the active power of the generator. The output signal of the controller is the field voltage, Eqf.

Field current

Terminalvoltage

Voltage

Active Output

Turbine torque Rotor angle

Reactiv Output/Current

Figure 2.24 The generator-grid system.

The generator-grid system is shown in Figure 2.24. The presented model is a third order generator model connected to a stiff electric grid [16]. The model will not be presented further in this section because the synchronous generator and the electric

43

grid will be treated in details in the Modem turbine governing analysis in Section 3.3. The system consists of four sub-systems; the field current, the voltage, the active power and the reactive power. The frequency (or electric angular speed) in the stiff grid, <%y can be excited to in-built signals (pulse, ramp etc.). The linearized constants in the model have to be determined for each operation condition. In Appendix 7 the procedure of calculating them are shown [14].

Eq1Transient q-axis volatge

Rotor angle

vu - w_grid Speed difference

Speed

PgActive output

Electric torque

Figure 2.25 The active power system.

In Figure 2.25 the active power sub-system is shown. The electric torque, De, and the active power, Pg, of the generator are both output signals of the system. The linearized constants, the generator damping constant, D, and the steady-state electric torque,Meo, are calculated in Appendix 7.

ElfTransient q-axis

voltage

Speed

QgReactiv output

Figure 2.26 The reactive power system.

44

Figure 2.26 shows the reactive power system where the reactive power, Qg, and thecurrent, zg, of the generator can be determined.

Terminal voltage

+ + +

Transient q-axis voltage

Rotor angle(Speed)

Speed

Figure 2.27 The voltage system.

In Figure 2.27 the voltage system is shown. The output signal is the terminal voltage, Hit, governed by the voltage controller. Figure 2.28 shows the field current block.

(Beta)

Figure 2.28 The field current system.

Various models of the voltage controller exist. The presented model, showed in Figure 2.29, is a basic model [24] with a controller unit, a power amplifier, an internal stabilizer and a additional derivative stabilizer. The input signals are the terminal voltage, Ei, and the speed, n. As mentioned above more sophisticated models also have an input signal from the active power. The reference voltage, Ei/ef, can be excited in the same way as the reference speed and the grid frequency. The output signal is the field voltage, E#.

45

Terminalvoltage

EqfField voltageEk_ref

Reference terminal voltage

Controller Power amplifier

KoTo

Speed Derivativestabilizer

Figure 2.29 The voltage controller system.

Concluding remarksA linear model of a turbine unit connected to a stiff electric grid is established. In Chapter 4 the model will be used together with the isolated grid model in order to study stability performance in time domain.

46

3. MODERN TURBINE GOVERNING ANALYSIS

The terminology Modem turbine governing analysis, means:

• Time domain simulations of equations including non-linear terms, without linearization. Non-linear terms may for instance be found in the equations for the conduits and the turbine- and the electric torque.

• Focus on the dynamic performance of the total system, including the turbine, the conduits, the governors, the generator and the electric grid connected to a total model

This chapter presents:

1) Improvement of an existing dynamic Francis turbine model [33] in order to provide a simulation model that is more accurate and less dependent on measurements than the existing model

2) A new non-linear simulation model of a hydropower system where the hydraulic part and the electric part of the system are connected together as one unit.

Improvements of the dynamic turbine model will be performed firstly. Afterwards,consequences of the hydraulic inertia will be shown by means of simulations. Then anexperimental study will be described and at last the new non-linear simulation modelwill be established.

3.1 IMPROVEMENT OF THE DYNAMIC FRANCIS TURBINE MODEL

There are many advantages of having an analytic turbine model for dynamic simulations. In an early stage of design or upgrading a hydropower plant system stability and performance are of great importance. In this phase, measured model turbine characteristics are usually not available. Therefore a model describing the mam effects and behaviour of the turbine, would be very helpful Usually, model turbine characteristics are digitalized and stored in discrete points. To be able to perform simulations, it is necessary to interpolate and extrapolate between the stored values. This process is often done with lot of uncertainty. Problems with discontinuity outside the measured area followed by numeric error may easily occur. An analytic model has no problem with this.

In Nielsen [34] a model of the dynamic behaviour of Francis turbines is presented. The turbine model exists of two differential equations; one for the flow through the turbine runner and one for the turbine torque. The model is an analytic approach based on the one-dimensional Euler equation. The turbine is defined by its main geometry, general loss coefficients and its design values. The goal was to describe the dynamic performance of a Francis turbine, as simple as possible, in a system where the turbine is one of many elements. Nielsen also proved that the hydraulic inertia in the turbine must be included in a dynamic model in order to calculate a correct dynamic

47

performance. This inertia causes a 1st order lag for the flow through the runner and the draft tube and the transient turbine characteristics will differ from its steady-state during a transient runaway history. Nielsen verified his model against measurements on a high-head Francis pump turbine at the Water Power Laboratory, NTNU and the agreements were very good for this kind of turbine.

The goal in this section is to cany out a further improvement of the dynamic part in the turbine model. Since the steady-state condition of the model already seems to give a good picture of the flow through a Francis turbine, effort is not spent on adjust this part. The one-dimensional modelling describes the most dominating effects in the turbine behaviour and three-dimensional effects seem to have more adjusting consequences [33].

In the following the turbine model will be improved in three ways:

• Section 3.1.1The model will be derived from a more physical point of origin (Equilibrium of forces along a mean stream line) instead of the steady-state Euler turbine Equation, used in [33]. In this way, an analytic expression for the hydraulic inertia in the runner can be found. In [33] the hydraulic inertia had to be found by trial and error.

• Section 3.1.2Establish a dynamic model of a draft tube in order to find an expression of the hydraulic inertia, which can then be included in the turbine model This is necessary because it is likely to believe that most of the hydraulic inertia is caused by the flow and its swirl-component in the draft tube.

Section 3.13• Make the turbine model more analytic compared with the existing model by

deriving expressions for the loss coefficients in the turbine torque equation.

Section 3.1.4-3.1.5• Show the consequences of hydraulic inertia in hydropower plants by means of

simulations.

Section 3.2• Perform an experimental study on a low-head Francis turbine with two different

kinds of draft tubes. This is done in order to test the turbine model against more complex turbine characteristics than a pure high-head machine have. Also, to identify the difference in hydraulic inertia by changing to another draft tube with a larger length-area ratio and with a more well-defined geometry.

3.1.1 Flow through the turbine runnerAn approach of the flow through the turbine runner is derived in this section. In [33] the Equation of the flow through the runner was derived from the I dimensional steady-state Euler equation. Afterwards, Nielsen used the Bond graph method to show that a hydraulic inertia term must be present. The intention with this section is to

48

derive the same equation by calculating the equilibrium of forces along a mean stream line. In this way the same equation as derived in [33] can be obtained, but with a new analytic expression of the hydraulic inertia included.

Figure 3.1 shows a sketch of a Francis turbine with inlet and outlet velocity triangles. A differential element, dm, can be set up inside the runner and the pressure drop can be calculated along a mean stream line, Lm, from inlet to outlet of the runner, n and r2 are respectively the inlet and outlet radius of the runner blade, c: is the meridional component of the water velocity which determines the flow and cu is the swirl component, u is the angular velocity, w is the relative velocity and ai is the inlet angle governed by the guide vanes.

Figure 3.1 Meridional sectional drawing of a Francis turbine.

The following assumptions are made:

• One-dimensional, inelastic flowThe turbine element is short compared with the conduits and it may therefore be regarded as inelastic.

• No changes in potential energyThe difference in elevation between the inlet and the outlet of the runner is small

• No frictional lossesThe frictional losses are neglected in the derivation since they will be accounted for in the turbine torque equation, Section 3.1.3.

Equilibrium of forces in meridional direction gives:

Y.F = pasdAdm = PA — dAdm (3-D

49

The acceleration term, as is given by:

dw dw 2 dr dw 2 dr+ w------ co r— = -co r—dt dm dm dt dm

(3.2)

The first term is the relative acceleration and the second one is the centripetal acceleration. There are no contributions from the Coriolis acceleration along the streamlines (the relative velocity vector).

Combining Equations (3.1) and (3.2) and introducing the head, H = —:PS

( dw 2 dr\ dp dw p 2 dr2 dHfbr* -2* (3.3)

By integrating Equation (3.3) from inlet to outlet of the turbine along the mean length of the streamline (runner blades), the following approximation can be derived:

pLm Yt ~ 2 ^^~M^l - H2) (3.4)

Introducing the geometric parameter, S and the pressure drop trough the turbine, H as follows:

'tt

(3.5)

h = hx-h2

pLm^ = pSa>2 -pgH

Introducing the opening degree of the turbine (guide vanes), k [33]:

O

K =-p-gH sin a,a sma lopt

when:

co = co H

(3.6)

(3.7)

(3.8)

(3.9)

Solved with respect to the head gives:

50

(3.10)

Inserting Equations (3.9) and (3.10) into (3.7) gives:

r

(3.11)

Nielsen [33] assumed that the transference between velocity head and pressure head outside the best efficiency point will be encumbered by huge diffusion losses. The assumption seems to give a reasonable picture of the shape of the Francis turbine characteristics. By using this assumption, the following approach can be established by means of Equations (3.7) and (3.11):

(3.12)

The relative acceleration term has to be evaluated in order to find the dynamic term and the hydraulic inertia. From the velocity triangle, shown in Figure 4.1, the relative velocity can be expressed in terms of the flow, Q, the angular speed, o», and the swirl angle, a:

2 2orQ Ataxia

(3.13)

Letting:

(3.14)

where u is a dummy variable.

Then:

(3.15)

Differentiation of Equation (3.14) gives:

du 2 Q - or A sin(2g) dO dt (a sin a)2 dt(A sin a)

(3.16)

51

Then, the result shows that the acceleration term can be expressed as:

dw \ du dOA 2d

Where the hydraulic inertia term is given by:

(3.17)

4 -pLm (20- armAmsmaJ

2Am sinam^Q2 + wrxAm{ccrmAm sin2 a, -2sin2am)(3.18)

We can see that the inertia term depends on the flow, the angular speed and the swirl angle. rm, Am and <%, are respectively the mean value of the radius, the cross section and the swirl angle of the runner:

4, = ^(4 +4) am=^(al+a2) (3.19)

The dynamic equation of the flow through the turbine becomes:

(3.20)

Introducing dimensionless magnitudes (per unit):

Q_

&Q = -

£0

C0„(321)

Equation (3.20) can then be written in dimensionless form where index R denotes runner:

-oO’ (322)

Where cris called the self-governing parameter:

<»„ r, <a„<j = S—~ =gH„ 2gHn l r{1-4 (323)

This parameter has a great influence on the slope of the turbine characteristics. This will be discussed in Section 3.1.4. In the dimensionless form the hydraulic inertia is represented by means of the time constant Tqr, which introduces a 1st order lag to the system.

52

(3.24)

3.1.2 Dynamic draft tube modelThe draft tube is a large part of the turbine, and it seems reasonable that most of the hydraulic inertia appears here. Including the draft tube in a dynamic turbine model is not an easy task, especially when the goal is to make it one-dimensional We have to keep in mind that the draft tube is only one of several elements in the total system and only the main effects concerning the transient behaviour are of interest. Flow analysis in draft tubes are not completely understood today, even in steady-state conditions. A difficult geometry and often unpredictable inlet conditions fed from the runner are the main reasons for this.

In this section a one-dimensional dynamic model of a draft tube is established in order to find an expression for the hydraulic inertia which can then be connected to the previously derived equation for the runner. An important goal is to include the swirl component of the flow, since it is likely to believe the hydraulic inertia varies with the degree of the swirl flow. In order to establish a relatively simple model of a complicated draft tube behaviour several assumptions have to be made. Figure 3.2 shows a sketch of a straight and conical draft tube with a surrounding control volume. The inlet conditions (in position 2) are regarded as the same as the outlet conditions of the turbine runner, though the angular speed is zero.

Figure 3.2. Straight conical draft tube with a surrounding control volume.

53

The Energy Equation can expressed [60]:

^\({u+l-c2+^dV = -fy{u+^c2+g{H+ Z)}cndA + Q-W (3.25)

W = W s+ ^pghSrldA (3.26)A

u is here internal energy, c is absolute velocity and Z is the elevation above datum. Q is

the heat transfer and IF is the work, divided into pressure and shaft works (not present here), n is the area unit vector.

The following assumptions are made:

• No frictional losses; ^pghCfldA = 0A

The frictional losses will be implemented in the turbine torque equation, Section 3.1.3.

• No heat transfer with the surroundings; Q=0 and constant internal energy through

the draft tube;«/ = u2, ~r = 0 dt

Assuming that the water temperature is constant through the draft tube and thatheat generation due to frictional losses and turbulence is neglecteble.

• Inelastic flow; p = constant and — ~ —dt aThe draft tube is short and can therefore be regarded as inelastic.

The Energy equation the becomes:

j|cW = -|Qc2 + gfZ+H^vndA =>

+ g(Z, + *,)) (327)

From the velocity triangle the following connection can be established:

-f— (328)Asma

Combining Equations (3.27) and (3.28):

54

M-QLAg(sma) dt 2g\A^(siaa2) A3(sma3)

(% + %) fc + z3)sin a2 sine,

(329)

Rearranging and introducing the swirl rate, y:

sin a,r=-—-sma3

(330)

£gsina2 dQ Ag(sm.af dt

Q22gA%(sma2)

f z y n2\1-73

V \A3J y+ H2 + Z2~y(H3 + Z3) (3.31)

By introducing dimensionless properties the equation for the flow through the draft tube can be expressed as:

tqd^=ko92 +z2-r{h +zi) (3.32)

Index D denotes draft tube. The dimensionless constant, fe> is given by:

kD —Ql

2gA2(sma2) H„ i-rU)<A3j

(3.33)

Tqd is the time constant expressing the effect of the hydraulic inertia in the draft tube (1st order lag):

rr _ LDsma2QnJQD ~ . T. , . \2 (334)

Equation (3.34) shows that the length/area ratio (Lr/AmD) has a great influence on the size of the hydraulic inertia. In order to get a picture of how the ratio varies with different kinds of Francis turbine types, data from a database of 175 units in Norway[9] are studied. Geometrical data for the draft tube are archived by using the speed number and interpolation functions based on statistical data [47]. The interpolation functions are based on data from American turbines and cannot directly be compared with Norwegian turbines. Although it gives a guidance of the variations. The result is shown in Figure 3.3, where the length/area ratio, for a standard shaped draft tube (cone, bend and diffuser), as a function of the speed number *£2 (or specific speed nq) is shown. As we can see the data is spread, but a trend curve shows that the ratio increases towards lower speed numbers (high-head turbines). The speed number is defined as follows:

55

CO, (3.35)*Q = co„ a 7m „

J2gHn }l 42gHn 30(2g)%

Speed Number a

Figure 3.3 Estimated length/area ratio (Lc/AmD) for a standard shaped draft.

OmD is the mean value of the swirl angle in the draft tube. It can be estimated roughly by means of the mean value of the inlet and outlet swirl angle:

«m£> =j(a2 +“3) (3.36)

The inlet swirl angle of the draft tube is the same as the outlet angle of the turbine runner. From the outlet velocity triangle we have:

tan/?, = ^--------- (3.37)U2~cui ar2A2-Qcota2

Assuming cU2~ 0 in best efficiency point. From Equation (3.36) we get:

tan/?2 = ——— (3.38)

The angle, /%, is given by the runner geometry and is regarded as constant for dynamic analysis. Combining Equations (3.37) and (3.38) and introducing dimensionless magnitudes gives:

a2 =atanVWjG),

(3.39)

56

The approach gives correct pro-rotation swirl in part load and contra-rotation swirl in full load.

In [12] experimental study of swirl flow in draft tubes is presented. Pressure, axial- and tangential velocity measurements were carried out by the use of a Pitot tube and a LDV (Laser Doppler Velocimeter). For a straight conical diffuser (%/% = 1.5) the experiments showed that the inlet swirl angle will be unaltered through the draft tube. In this case the mean value of the swirl angle in the draft tube becomes:

amD = «2 (3.40)

For a straight tube connected to a 90° bend and a difiiiser, close to a typical standard shaped draft tube, the experiments show that the inlet swirl angle will decrease through the draft tube. In Figure 3.4 measured data is shown in connection with a linear approach, which seems to fit very well

■ Measured data (O.C. Dehfaug)

— — Modified approach

Vortex Breakdown Zone

Iafatswk1*ig)»,ct2 »I

Figure 3.4 An approximated connection between the inlet swirl angle, <X2, and the outlet swirl angle, a;, for a standard shaped draft tube, based on measurements.

In [12] measurements are not carried out for inlet swirl angles below 55°, but according to [13] the linear approach may be assumed to yield down to approximately 45°. Below this swirl angle vortex breakdown (backflow in the tube centre) may occur and the flow conditions will be unpredictable and almost chaotic. Therefore, this author has chosen a% = 45° as the lowest legal value for the inlet swirl angle (also for a straight draft tube). The following approach can then be used for calculation of the mean value of the swirl angle in a standard shaped the draft tube:

=\(a2 + 6122 + 0.65(<z2 -55.01)), a2> 45° (3.41)

The modified approach adopted here is shown in Figure 3.4 with a dotted line.

57

In Figure 3.5 the time constants of the hydraulic inertia for the draft tube as a function of the inlet stream angel is shown. The draft tube inertia is shown relative to the inertia in the best efficiency point (irrotational flow) for a straight- and a standard shaped draft tube with the same L/A-ratio. We can see that the inertia increases towards decreasing inlet angles and the inertia in the straight draft tube is generally larger than in the standard shaped draft tube. When we also have in mind that the L/A-ratio in most cases is larger for a straight draft tube the difference will be even larger.

Figure 3.5 Relatively hydraulic inertia in draft tubes as a function of the inlet swirlangle, a2.

The time constant of the draft tube, Tqd, can now be combined with the time constant in the runner, Tqr. The dynamic turbine model then becomes:

Tqr and Tqd are given by Equations (3.24) and (3.34). The frictional losses in the draft tube are not included in Equation (3.42). This will be accounted for in one of the loss coefficients in the turbine torque equation, presented in the next section.

Concluding remarksAn extended version of the Equation of the flow through a Francis turbine is derived. Approaches of the hydraulic inertia in the runner and the draft tube are established. The following conclusions of Sections 3.1.1 and 3.1.2 can be made:

• The hydraulic inertia will increase with an increased swirl flow in the draft tube.

• The hydraulic inertia is generally larger in a straight draft tube than in a standard shaped one.

58

In Section 3.1.6 the necessity of including the hydraulic inertia effect in the dynamic turbine model will be shown, due to calculation of a correct transient runaway speed and for stability considerations.

* The hydraulic inertia depends strongly on the L/A-ratio, which in fact increaseswith a decreasing speed number.

3.1.3 The turbine torque equationIn [33] an one-dimensionally expression based on the Euler turbine equation is derived. Losses in the turbine were originally described by constants and they had to be determined by trial and error in order to match measured turbine characteristics. According to [34] and [44], the turbine efficiency and turbine torque can be expressed as follows in dimensionless form:

l-Rn -R,Qqh

(3.43)

(3.44)

The Ru, Rn and Rn constants are loss coefficients, respectively accounting for the wall friction (runner and draft tube), the shock losses (off-design effects) and the mechanical friction in bearings and disk friction. In [44] it is shown that an additional coefficient, Ru, is necessary in order to satisfy the requirements in the best efficiency point.

ms is called the specific torque at zero speed of rotation [33] given by:

ms = t/r—(costtj +tana,„ shuz,) (3.45)

The machine constants <p and yr can be expressed as [34]:

1—aP=I+<r (3-46)

H 'Z? + a, 1 R S' (3.47)

<7is the geometric dependent self-governing parameter given in Equation (3.23). T2S is called shock-less speed and it is the speed when the inlet incident angle is zero. It can be expressed as [44]:

(3.48)

59

c = f tana, Vtana,^ + tan/).Vtana,„Jl tan a, +tan /?,

The inlet flow angle, j3j, can be foimd by means of the velocity triangle:

c-. 2,9

(3.49)

tan/9, = ■ Snl2,9

A, tana,

(3.50)

In the best efficiency point the following must be valid:

-%- = o => 2Rn -RtA = o da,

(3.51)

— = 0 => 2%,(r-24%+ =0 (3.52)

7J„=t-Rl,- RI3 - J?;4 (3.53)

This leads to expressions for all the loss coefficients as functions of the best efficiency, except for the shock-less coefficients, Rtf.

R«=-Rn *«=2*„ (3.54)

Since no requirements are obtained for Rn, this parameter can be regarded as a free variable which can be used to adjust the turbine characteristics if simulations differs from measurements. In order to improve the turbine model further an expression for R12 will be derived.

If we have a known or assumed operation point outside the best efficiency point we are then able to find an expression for Rq. A convenient operation point may be the runaway speed, Qnm at 77 = 0, with nominal guide vane opening, ay,. This point may as often be assumed with sufficient accuracy.The steady-state version of Equation (3.42) gives:

(l + <7%, - = 0 (3.55)

For the conduits the Bernoulli Equation yields in a steady-state operation:

u -™ H.

^0

^S^ppe^-pipeQnm R~nm\ -R-mr.2 9 2 (3.56)

60

Combining Equations (3.55) and (3.56):

„ = l(l+<rK*-oal 9"1 1+0+O-K-. (3.57)

Equation (3.43) and (3.48) finally gives the expression for the loss-coefficients, ifo:

RI2 = ^rml ~ Rrm4 (3.58)

Where:

^m.3 —

T"—Rn^L~ ritPL,"run

•^ran4 “RI4^nm

^m,(9™,(l + (Z>)-^ro,)(nn

(3.59)

(3.60)

Concluding remarks77ze loss-coefficients in the turbine torque equation can now be approached by means of the nominal efficiency and the nominal runaway speed. An improved version of the dynamic turbine model [33] is then completed In the next section the steady-state conditions of the model will be discussed.

3.1.4 The Steady-state condition of the dynamic turbine model This section discusses the steady-state condition of the dynamic turbine model Figure3.6 shows an example of a typical performance diagram for a high-head Francis turbine [54]. The turbine characteristics, the runaway line (q = 0) and the efficiency are all simulated by means of Equations (3.42), (3.43) and (3.44) and an equation for the flow in the conduits (the Bernoulli Equation in steady-state operation).

61

Oj6 0.7 0Z 03 ID 11 12 12 14 IS IS

Figure 3.6 Simulated performance diagram in dimensionless form.

The most important parameter concerning the inclination or shape of the turbine characteristics is the geometrical dependent self-governing parameter, given in Equation (3.23). For high-head Francis turbines the inlet radius of the turbine, rj, is greater than the outlet radius, n. The self-governing parameter a then becomes positive and is causing falling turbine characteristics. A simulated falling characteristic is shown in Figure 3.7. A negative feedback between the angular speed and the flow is obtained, which is favourable in a turbine governing view.

For very low-head Francis turbines the situation is nearly opposite, with n < negative self-governing parameter and rising turbine characteristics. Then we have a positive feedback between the angular speed and the flow and a more complicated turbine to govern. A simulated rising characteristic is shown in Figure 3.7. Kaplan turbines, with adjustable runner blades, have even more rising characteristics than the low-head Francis turbines. They are often regarded as the most difficult turbines to govern. The presented turbine model is not valid for this kind of turbines. Another expression for the turbine torque, using ‘lift and drag’-philosophy, most then be derived.

The third simulated characteristic in Figure 3.7 has no feedback between the angular speed and the flow. This is a typical characteristic for a Felton turbine. The turbine model can be used for this kind of turbines, but the turbine torque and efficiency can not be calculated by means of Equations (3.43) and (3.44). They are only valid for Francis turbines. In [33] separate equations for the Felton turbine are derived.

62

Figure 3.7 Simulated turbine characteristics with different self-governingparameters.

Francis turbines can roughly be divided into high-head, medium-head and low head turbines by means of the speed number, given in Equation (3.35). From [6] the following classification can be established:

High-head Francis turbines 0.15 < *Q < 0.30Medium-head Francis turbines 0.30 < *£2 < 0.70 Low-head Francis turbines 0.70 < *52 < 1.20

As we can see in Figure 3.7 the turbine model is only able to generate either falling characteristics, rising characteristics or straight characteristics. Many medium-head and low-head Francis turbines and especially high-head Francis pimp-turbines have turbine characteristics with both a rising part and a tailing part. In the first instance the presented turbine model is not able to describe this kind of characteristics. Nielsen [33] was aware of this and, after performing measurements on a pump turbine, he suggested an additional term for this kind of turbines, Rq. This parameter was accounting for all kind of losses in connection with transference from velocity head to pressure head (diffusion losses) [33]. By including the diffusion losses, Equation (3.42) becomes:

(%: + = (1 + <?)&--oQ2-Rq{q-qc)

Where qc is the flow at zero incident angle given by:

Yl + cot a,„ tan/lb?) _ coqc ~ a[ 1+cota, tan A, J * °

(3.61)

(3.62)

63

The i^-term had to be determined by trial and error in order to fit measurements. In [34] a further development of the approach is suggested after a intensive study on measured pump-turbine characteristics; the i^-term seemed to be dependent of the opening degree:

k - 9c)2 =^“^(9- 9c)2 (3-63)

In this case, the equation of the flow through the runner becomes:

[tqr + TQd)^ = 0 + °)h - (”) - 0Q2 - Rq (q -qc)2 (3.64)

In Figure 3.8 a modified falling turbine characteristic is shown in order to describe characteristics for medium/low-head and pump-turbines.

Figure 3.8 Simulated turbine characteristics with and without modification.

Concluding remarksThis section has discussed the steady-state conditions of the dynamic turbine model. The turbine model is only able to generate either falling characteristics, rising characteristics or straight characteristics. It has been shown how a parameter, correcting for diffusion losses, can be used to generate turbine characteristics with both a falling and a rising part. In Section 3.2 an experimental study of a low-head Francis turbine is performed. Due to the expected turbine characteristics of this machine, it is likely to believe that this parameter must be included in the dynamic model. This will be discussed further in Section 3.2.

64

3.1.5 Consequences of the hydraulic inertiaIn this section simulations of a high-head Francis turbine are performed in order to show the influence of the hydraulic inertia. A computer program in Visual Basic 4.0 is developed and it is based on the theory presented so far in Chapter 3. The computer program is made for transient runaway performance and speed governing on an isolated grid. Figure 3.9 shows a flow chart of the computer code.

Load variations D,toceift) ,

Turbine torque Hydraulic inertii Calculations

FrequencygovernorPI/PID

Fixed guide vanesk= constant

Manual governed guide vanes

Steady-stateCalculations

Figure 3.9 Flowchart of the developed computer code.

Firstly, the initial steady-state conditions of the system are calculated. The transient behaviour of the system appears by changing the value of the generator torque (or electric torque). In a steady-state condition the turbine- and generator torque are equal The program has to decide whether the guide vanes should be kept in a fixed position (runaway) or whether a PI/PID speed governor, or a manual governing controller, should be connected. An eqdicit RK4 routine then solves the torque balance equation, Equation (3.74), and eventually additionally differential equations for the speed governing. The angular speed and guide vane opening are sent as input values to the MOC-routine. The presented turbine model is used as boundary condition at the right-hand pipe end. This will be shown in more details in Section 3.3.1. A reservoir is used as boundary condition in the left-hand pipe end. The routine calculates the static pressure difference over the turbine and the flow through the turbine. An additional routine then calculates the turbine torque, the hydraulic inertia

65

and other secondary magnitudes as for instance the output and the efficiency. Then the values are written to an output file before a new time step can be taken.

j fc*=034m P„= 153.1 MW I |LC/Ab=is|Lm= 1.3 m 06=12°

I n=0.27 %r=oa29

Figure 3.10 Input data for dynamic simulations.

Data for the power plant used in the simulation examples are shown in Figure 3.10. The turbine is a high-head Francis turbine with speed number *Q = 0.27 and nominal output of P„ = 153.1 MW. Output in full load is P° = 180.2 MW. Data for the penstock and the runner are shown in the figure. The draft tube has a length/area-ratio of Lr/AmD =15 and has a standard shaped draft tube with cone, bend and diffuser. The draft tube model in Equation (3.41) is therefore used. The time constant of the servo motor is To — 0.1 s. and the permanent speed drop is put to zero.

Figure 3.11 Generated turbine characteristics.

66

A number of generated turbine characteristics are shown in Figure 3.11. The abscises and the ordinate axes are respectively reduced angular speed and reduced flow:

^reduced[K

o>JH ’(3.65)

The speed and the flow are reduced with the square root of the total pressure difference over the turbine (the net head). The total pressure is the sum of the static pressure difference over the turbine, hi - fa, the level difference, zi- Z2 and the kinetic pressure difference:

H = fa -fa +z, -z2 +—6= (3.66)

Another common way of representing the turbine characteristics is by use of the Hill diagram. The abscises and the ordinate axes are then respectively unit speed, Nu, and unit flow, Qu. The connection between the reduced values and unit properties are:

Nn = to.’reduced 3 fin =0,

Q.•reduced (3.67)

Where n„ is the nominal speed and Dj is the outlet cross sectional area of the turbine.

The falling characteristics, with steep gradients for higher opening degrees, are typical for a high-head Francis turbine. The shock-losses constant Rn is calibrated against a desired nominal runaway speed, n„Jnn = 1.45, according to Section 3.1.3. The runaway line (q = 0), shown in the figure, is the steady-state boundary where the turbine will end up if the generator and the speed governor are disconnected.

If a speed governor is connected and the turbine aggregate runs on an isolated grid with zero permanent speed drop, the steady-state conditions will be at the synchronous line, shown in Figure 3.12. The angular speed is equal to the synchronous speed in all points at the line, but the head and the flow changes due to the variations in the opening degree of the guide vanes.

67

•>c=1.2

'*=1-° REP.

/ Synchronous Line

ic=0.8 ;

*=0.6

*=0.4

I oz a

098 0985 099 0995 1.005 1.01 1.015 192

Figure 3.12 The synchronous line in the performance diagram.

A transient runaway simulation is performed for the high-head Francis turbine, with data given in Figure 3.10. The turbine is running in the best efficiency point (b.e.p) when the generator is suddenly being disconnected and the guide vanes are kept in fixed position. The angular speed will immediate accelerate towards the runaway speed. In Figure 3.13 the transient history is shown for the cases with and without hydraulic inertia included in the model. The results show that the angular speed will reach a higher value in the transient period if the hydraulic inertia is included. A phase shift, caused by a 1st order lag, is an another important consequence of the hydraulic inertia. It can clearly be seen on the figure and this may reduce the stability margins in the system. This will be shown later in this section.

5 t.«

Figure 3.13 The angular speed during a transient runaway, with and without hydraulic inertia included in the simulation model

68

In Figure 3.14 the runaway history of the two cases, with and without hydraulic inertia included, is shown in a traditionally performance diagram. If the hydraulic inertia is neglected the runaway history will follow the same path as the defined steady-state characteristic, though it will oscillate around the runaway point due to water-hammer effects in the penstock. The area outside the steady-state runaway line may be called the transient braking zone. In this area the turbine may only run in short transient moments. The efficiency is in fact negative and energy is fed to the turbine at the expense of the rotational energy. Therefore, the angular speed will decrease in this area. Especially if a speed governing turbine is subjected to a large load rejection, the transient braking zone has to be used on a large scale in order to decrease the angular speed properly.

Steady-Stata Operation Zone

Steady-State Runaway Line

Inducting Hy&svOc Inertia

Transient Braking Zone'

Figure 3.14 Transient runaway, with and without hydraulic inertia included in the turbine model, drawn in a traditionally performance diagram.

By including the hydraulic inertia in model the transient runaway path will diverge from the steady-state characteristic and will be curling around the steady-state point. This is shown experimentally on a high-head Francis turbine in [33] and will be shown in the next section where experimental investigation is performed on a low-head Francis turbine. The divergence between the curves results in a higher transient speed and introduces a phase shift in the system. In the figure points on the two runaway paths are marked at the same time of moment, t — T„. The curve including the hydraulic inertia “leads” on the other one until maximum speed is achieved for both of them. After this the curve including the hydraulic inertia will be time delayed during the oscillations towards the steady-state condition.

This way of presenting the transient runaway path in a turbine performance diagram gives a good picture of the influence of the hydraulic inertia. Since the water-hammer effects already are included in the net head, a more correct presentation is achieved by

69

including the additional dynamic pressure contribution, caused by the hydraulic inertia, in the net head. By redefining the net head the runaway path will follow the steady- state characteristics. This is shown in Figure 3.15. By means of [33] the redefined reduced flow and angular speed may be expressed:

co.reduced

CO

CO,

H.Qreduced ^

Hm(3.68)

An important spin-off conclusion in [33] was that the hydraulic inertia in this way could be determined experimentally if the steady-state characteristics are known. This way of presenting the transient runaway path projects the influence of the hydraulic inertia along the steady-state characteristic. As we can see from Figure 3.15 the inertia forces the characteristic further out in the transient braking zone than the one without inertia included does.

Stsidy-Sferte OptrttionZone

„>0 Study-Sun Runaway Lina

' Inducing HydrauBc Inertia

Transient Braking Zone

1.445 1.455

Figure 3.15 Transient runaway, with and without hydraulic inertia included in the turbine model, drawn in a performance diagram where the net head is redefined.

Figure 3.16 shows the time constants caused by including the hydraulic inertia for different guide vane openings during a transient runaway from zero speed of rotation up the runaway speed. We can see that the inertia increases towards lower guide vane openings. The difference in time is very small though. The dip in the curves occurs because the turbine runs through the best operational point for the actual guide vane opening. The inertia will then decrease because of the low swirl degree. In this way the hydraulic inertia reflects the turbines off-design conditions. Though, the phase shift is

70

less for large guide vane openings than for small openings, the observable effect is larger. This is because the slope or gradients of the characteristics usually increase with increased guide vane openings. The hydraulic inertia has therefore to be seen in connection with the gradients of the characteristics. Especially high-head Francis pump turbines with s-shaped characteristics may feel the consequences of the hydraulic inertia very heavily and it may explain a lot of unpredictable operation and governor problems for these kind of turbines.

-7 » "

Figure 3.16 Time constants caused by hydraulic inertia during a transient runaway with different guide vane openings.

In Figure 3.17 the ratio between the time constants in the draft tube and the in runner is shown. The contribution from the runner grows with smaller guide vane openings. Around the best efficiency point the hydraulic inertia in the draft tube is over 25 times the size of the runner inertia and in off-design conditions between 5-10 times.

71

Time [sec.]

Figure 3.17 The ratio between the time constants of hydraulic inertia in the draft tube and runner, during a transient runaway with different guide vane openings.

The next examples concern speed governing on an isolated grid. Figure 3.18 and Figure 3.19 show respectively the opening degree and the angular speed after a sudden load connection of 10% from the nominal operation point. Cases with and without hydraulic inertia are shown in the figures.

Figure 3.18 The opening degree, with and without hydraulic inertia included, after a sudden load connection of 10%. Parameter settings: bt = 0.25, 7^= 10.

72

1.005

Hf^asjUsiMitakKkjC»C

Figure 3.19 The angular speed, with and without hydraulic inertia included, after a sudden load connection of 10%. Parameter settings: bt — 0.25, Td — 10.

The angular speed will decrease because of the immediate shortage of power. The governor is trying compensate this by a larger guide vane opening. The parameter setting in the governor (b, = 0.25, Td = 10) is tuned to give a relatively good time response and sufficient stability margins. The parameter setting is based on the case without hydraulic inertia included in the model But, as we can see the hydraulic inertia will give another response than we in the first hand had expected. The oscillations become more significant with higher amplitudes. This is typical for a system where the phase margin is reduced, which in feet is the result of the hydraulic inertia. The parameter settings should therefore be reset in order to match this.

Figure 3.20 The path in the turbine performance diagram, with hydraulic inertia included, after a sudden load connection of 10%.

Parameter settings are; b, = 0.25, Tj= 10 and stability is achieved.

&

73

Though the hydraulic inertia decreases the stability margins the system is still stable. In Figure 3.20 the transient path is drawn in the turbine performance diagram. As we can see the picture is zoomed up a lot, because the governing takes place on a very small area of the total performance diagram The new steady-state point is placed on the synchronous line (since bp = 0) with a higher opening degree. The turbine will then balance the required output and speed.

If the parameter in the speed governor in the first hand were very badly tuned, the hydraulic inertia may actually make the whole system unstable. In Figures 3.21 and 3.22 the opening degree and the angular speed is shown for the same case as above but the proportional gain is increased (b, = 0.19, Td — 10). Without hydraulic inertia included the system is stable but the oscillations are though unacceptable. Including the hydraulic inertia will result in an unstable governing system, with increasing amplitudes. In Figure 3.23 the unstable governing is drawn in the turbine performance diagram We can see that the path will go through continual growing circles and gradually move away from the new synchronous point.

HyGnutc hwoa induced

Figure 3.21 The opening degree, with and without hydraulic inertia included, after a sudden load connection of 10%. Parameter settings: b, = 0.19, Td = 10.

74

HyCndtelnetbatoctoM

Figure 3.22 The angular speed, with and without hydraulic inertia included, after a sudden load connection of 10%. Parameter settings: bt — 0.19, Td — 10.

Syndrome Uw

Figure 3.23 The path in the turbine performance diagram, with hydraulic inertia included, after a sudden load connection of 10%. Parameter settings are; bt = 0.19, Td - 10 and the system is unstable.

Concluding remarksIn this section the major consequences of the hydraulic inertia in the system are shown by means of simulations of a high-head Francis turbine. During transient runaway performances and speed governing the angular speed will be higher than expected in the transient period and a phase shift is introduced in the system. This decreases the stability margins in the system and should be taken into consideration when the parameter in the speed governor is determined. The hydraulic inertia varies with the length/area ratio and the off-design conditions. The observable effect is larger if the gradients of turbine characteristics are large.

75

3.2 EXPERIMENTAL STUDY

This section compares results from an experimental study with simulations. Transient runaway and steady-state measurements were carried out for a low-head Francis turbine (*/2=0.9) with two different draft tubes.

• Since the turbine model already has been verified against a high-head turbine, with promising results [33], it was also interesting to determine how well it could represent a low-head turbine. Since a low-head turbine may have both rising and falling characteristics, a relatively large deviation between simulations and measurements should be expected. The turbine model handles at first only pure felling or rising characteristics. But it is possible to adjust the characteristics with the diffusion loss coefficient. This is shown in Section 3.1.4. Although, relatively large deviations between the measurements and the turbine model much be expected, especially since the runner has been subjected to a lot of modifications without any documentation of it to obtain.

• The test was carried out with two differently shaped draft tubes, one standard shaped (cone, bend, diffuser) and one straight and conical shaped. In this way the hydraulic inertia approaches for the draft tube could be tested. The standard shaped draft tube was the original designed draft tube and it has a relatively small length/area ratio (Lc/Ad = 11). It was likely that the hydraulic inertia will have minor influence on the dynamics. On the other hand it was believed that the straight draft tube would affect the system dynamic in a larger scale, because of the large length/area ratio (Lc/Ad = 22). This draft tube was originally designed for a high- head Francis pump-turbine.

3.2.1 The experimental set upThe experimental investigation was carried out at the turbine test rig in the Water Power Laboratory, at the Norwegian University of Science and Technology. The Laboratory is equipped with an upper and a lower reservoir with a pumping system in between. The turbine is placed between the reservoirs, see Figure 3.24.

The standard shaped draft tube is connected to a pressure tank containing an overflow pipe. In this way the head in the outlet of the draft tube could hold an approximate constant value. The outlet of the straight draft tube goes right down in the lower reservoir. In order to avoid low pressure at the outlet of the runner and the possibility of water column breakdown, the outlet of the draft tube was throttled. A DC- generator (Ta = 0.6 sec.) is connected to the turbine and made it possible to run the turbine in a wide operation area. The generator is connected to the outside stiff grid through a thyristor circuit.

76

Figure 3.24 Principle drawing of the experimental set up.

Data for the pipes, the draft tubes and the level differences in the experimental set up are given in Figure 3.24. The turbine runner has been subjected to several modifications during the last years and therefore reliable data for the design conditions were difficult to obtain. Geometrical data for the runner are given in Figure 3.25 [63].

rtt=0.116 m

0.175 m

%,= 0.176 m

Figure 3.25 Main dimensions of the turbine runner

77

By means of investigation and measurements presented in [45], [50], [57] and [63] the nominal values of the turbine test rig can be approximated. They are presented in Table 3.1.

Table 3.1 Nominal values of the low-head Francis runner.

Ipppplllfii

Head H„ 10.3 m 15.4 mFlow On 0.39 m3/s 0.44 m3/sSpeed Dn 750 rpm 825 rpmGuide vane opening Ctln 24.2 deg. 24.2 deg.

Steady-state turbine characteristics:The flow was measured by means of a magnetic inductive flowmeter (Krohne), which was firstly calibrated against an overflow channel (weir). The differential pressure over the turbine was measured by means of 8 pressure transducers (UCC transmitters, 11 kHz, 0-5 bar). Four of them were placed symmetrically along the circumference of the inlet section of the spiral case and the other four were mounted symmetrically at the outlet section of the draft tube. Comparison with measurements done with a differential pressure gauge showed excellent agreement. Though, for transient measurements the transducers were not suited and produced a lot of noise. The angular speed was measured with an optical sensor (“optical fork”) acting on wheel with 60 slots. Through a current/voltage-converter and A/D-converter all the signals were sampled and stored in a computer governed by the program LabView. Before each measurement point was read, the system had been running in the steady-state condition for around 30 minutes, in order to obtain stable conditions.

Transient runaway experiments:Measurements were carried out by disconnecting the generator. The intention was to study the interaction between the hydraulic inertia and the speed. The experiment was performed in practice by use of the emergency button. After the steady-state runaway condition was achieved, the guide vanes were closed down to a very low opening degree and the generator was connected again. During the transient period the speed was measured and sampled by means of the computer program LabView. Each measurement series was performed three times and the mean values are presented with calculated uncertainties.

From Figure 3.24 the gross head can be calculated to be; Hgmss = 13.5 m with the standard shaped draft tube connected and Hpoa = 16 m with the straight draft tube connected. Earlier experiments [57], [45] showed that the friction parameter and the wave propagation speed respectively could be estimated to be X — 0.02 and a = 1200

m/s. The system is in fact also inelastic since the Allievi constant, h„ » 1. Large water-hammer effects therefore do not occur in the system.

The level difference between the runner and the draft tube outlet changes from = 0.55 m to Z1-Z2 = 5.21 m due to the two draft tube types. Further the inlet cross section

78

area of the turbine is Ai = 0.13 m2 and the outlet cross section area of the standard shaped draft tube is A2 = 0.495 m2. The outlet of the straight draft tube was throttled and the cross section during the measurements was Ai = 0.192 m2. Expressions for the total pressure drop over the turbine can the be established:.

Standard shaped draft tube:H = (hl-h2) + 055+2.80 • Q2 (3.69)

Straight draft tube:H=(hl-h2)+521 +1.62 • Q2 (3.70)

Restrictions:The return pump was one of two bottlenecks in the system. It could only deliver a maximum flow rate of 0.45 m3/s. If the flow through the turbine exceeded this limitation the water level in the upper reservoir started to decrease tremendously. In order to avoid this water level variations the turbine was restricted to run on relatively low guide vane openings (below the best efficiency point), especially in the case with the straight draft tube connected (because of a high flow rate). The other bottleneck was the generator. It has an upper and a lower speed limitation. If the speed drops below approximately 180 rpm, the generator will be disconnected because of limitations in the thyristor circuit. The generator had never been run at higher speed than 1300 rpm, and the supplier’s recommendation was not to run the machine at higher speed than this for longer periods of time.

3.2.2 Steady-state measurementsSteady-state characteristics for the turbine with the standard shaped draft tube connected are measured for the guide vane openings <Xi = 18°, ai = 22°, ai = 24.2° and ai = 30°. The characteristics are shown in Figure 3.26. In Figure 3.27 the measured characteristics for the turbine with the straight draft tube connected are shown. They are carried out for the guide vane openings a, = 15°, ai = 18°, ai = 22° and ai = 24.2°. In Figure 3.28 a comparison between the two cases are shown for the guide vane openings ai = 18° and a% = 22°. The flow-speed (Q,n) characteristics are shown in Figure 3.29 for the same guide vane openings. As we can see from the figures, the characteristics in the measured area have a rising part firstly and are then followed by a falling part. The shape of the characteristics for the two cases seems to be quite equal. The straight draft tube forces the characteristics upwards and to the right in the diagram. The estimated runaway lines in Figure 3.29 are based on the transient measurements, presented in the next section, and results from [50]. The steady-state runaway condition could not be measured properly because the limitation in the speed of generator, as mentioned above.

79

Standard shaped draft tut

| 0.9

0.7 -dfa1 = 30; measurements -affa1 = 24.2; measurements -alfal = 22: meastremertts -alfal a 18; measurements

04 05 0.6 07 06 0.9 1.1 1.2 13 1.4

Figure 3.26 Measured turbine characteristics with the standard shaped draft tube connected.

Straight draft tube

—8tfa1=24 2; measurements

alfal = 22; measurements alfal = 16; measurements

-o-alfal = 15; measurements

Figure 3.27 Measured turbine characteristics with the straight draft tube connected.

80

Q [m

/S]

Qfe

duw

d

0.95' •

0.85 ■-

0.8 •

Standard fhapcd draft tubo |

0.75 • •<•alfal o 18; measurements

•alfat = 22; measurements

»18; measurements

•atfat = 22; measurements

re 3.28 Measured turbine characteristics. Comparison between the standard shaped and the straight draft tube connected, for a.\ = 18° and at = 22°.

a!fa1=18; measurements

* "alfal°22; measurements

atfa1=18; measurements

a!fa1°22; measurements

fl • Estimated runaway

* • Estimated runaway

Stoddard shaped draft tube

n[ipm]

Figure 3.29 Flow-speed (Q-n) characteristics (at = 18°, at = 22°) for the standard shaped- (blue) and the straight (red) draft tube connected. Estimated runaway lines.

81

3.2.3 Transient runaway measurementsThe transient runaway measurements were performed for the guide vane openings a, = 18°, ai = 22°, a, = 24.2°. The measured runaway speed is shown in Table 3.2. In Figures 3.30 and 3.31 the transient runaway history for the turbine with respectively the standard shaped and straight draft tube connected is shown for cti = 18° and a, =22°.

Standard shaped draft tfibe

= 18; measurements

= 22: measurements

Figure 3.30 Transient runaway speed for ai = 18° and ai = 22°. The standard shaped draft tube is connected.

: Straight draft tube :

•affal = 16; measurements

------aJfat = 22; measurements

£■ 990

Time [see.]

Figure 3.31 Transient runaway speed for (%, = 18° and a, = 22°. The straight draft tube is connected.

82

Table 3.2 Measured runaway speed with calculated uncertainties based on three measurement series.

Standard shaped draft tube Straight draft tubeCCl runaway speed uncertainties runaway speed uncertainties18" 1458.1 rpm ±13.1 rpm 1613.6 rpm ± 5.4 rpm22" 1486.0 rpm ± 8.3 rpm 1614.2 rpm ± 5.3 rpm

24.2" 1472.5 rpm ± 0.3 rpm 1602.4 rpm ±0.1 rpm

As we can see from the measuring results the straight draft tube shows a great influence on the dynamics compared with the standard shaped one. A large transient deviation from the steady-state runaway speed occurs, see Figure 3.31. As believed at first, this is reasonable since the hydraulic inertia is larger in the straight one. The measured curves starts all at the lowest possible speed because of restrictions in the thyristor circuit.

3.2.4 Steady-state measurements compared with simulations In order to simulate the turbine characteristics the inlet and outlet radius of the runner are needed. For low-head machines the inlet radius varies a lot, see Figure 3.25 and the mean value of the inlet and outlet radius are used in the simulation (r, = 0.146 m, r% = 0.114 m ). By testing several combinations this seems to give the best approach. The turbine model will then generate a set of weakly falling characteristics, see Figure 3.32 (straight draft tube). As we can see, there is some deviation between the simulated curves and the measuring points. Although, the deviation is small when it is taken into account that the curves are generated by means of a one-dimensional approach.

Since the turbine has both a rising and a falling part the Rq (diffusion losses, see Section 3.1.3) correction parameter can be used to fit the measurements even better. Very good agreement between simulation and measurements can be obtained in the measuring area by letting Rq = 0.15 for the straight draft tube and Rq = 0.075 for the standard shaped draft tube. The shock-losses constant, Rn, are calibrated to match the measured runaway speed, see Section 3.1.3. In Figure 3.33 (with straight draft tube) and Figure 3.34 (with standard shaped draft tube) simulated turbine characteristics are shown in connection with the measured points. As we can see, the deviation is very small and for turbine governing analysis this agreement is almost excellent. For higher speed, near the runaway point, and in the low speed area larger deviations should be expected. It was not possible to perform measurements in these areas, as explained above. The turbine model gives slightly falling characteristics in the high speed area, but according to earlier experiments [50] they seem to have a straighter section in between two falling parts. In the low speed area the characteristics seem to be straighter shaped [50] than the simulated curves. The turbine model is not capable of modelling this exactly, so both these limitations can cause some deviations between the runaway measurements and the simulations. This will be discussed in the next sections.

83

Straight dr ft tube

n alfal = 22; measurements•alfal - 22; simulation

a alfal = 16; measurements= 18. simulation

a alfal = 15. measurements—alfal = 15; simulation

Figure 3.32 Measured and simulated turbine characteristics with the straight draft tube connected (Rq = 0).

—alfal = 22; simulation » alfal = 22; measurements

—alfal = 18; simulation ° alfal = 18. measurements

—alfal = 15, simulation - alfal»15, measurements

0.4 0,8 “redirad 1.2 1.4

Figure 3.33 Measured and simulated turbine characteristics with the straight draft tube connected (Rq = 0.15).

84

Standard Shaped draft tube

—-aifat = 3% simulation = measurements

—aifat a 242; simiiation n alfal = 242; measurements

n alfal e 22; measurements•aifat = 22; simulation

—aifat = 16; simiiation a aifat * 18; measurements

Figure 3.34 Measured and simulated turbine characteristics with the standard shaped draft tube connected (Rq = 0.075).

3.2.5 Transient runaway measurements compared with simulations Simulations and measurements are compared for the guide vane openings ai = 18° and at = 22°. The length/area ratio is respectively Ld/Ad = 11 and Ld/Ad = 22 for the standard shaped draft tube and the straight draft tube. The draft tube model, taking the swirl component into account (Section 3.1.2), is implemented in the turbine model. It is possible to choose between straight draft tube and draft tube with a bend in the simulation program. Figure 3.35 and Figure 3.36 show simulations compared with measurements for the turbine with the standard shaped draft tube connected. In Figure 3.37 a zoomed picture of the speed amplitude for ai = 22° is shown. As we can see, the agreement is excellent with only small deviations in the speed amplitude. The transient speed rise is approximately 1.98 times the nominal speed.

n [rp

m]

n lrP

ml

Standard shaped draft tube

= 18; measurements

1 3 5 7 9 11 13 15

Time [sec.]

Figure 3.35 Measured and simulated transient runaway speed with a, = 18° and the standard shaped draft tube connected.

Standard shaped draff tube

•alfal = 22; measurements

0 2 4 6 8 10 12 14 16 18

Time [sec.]

Figure 3.36 Measured and simulated transient runaway speed with a, = 22° and the standard shaped draft tube connected.

86

8* 1470

Standard shaped draft tube

——atfal = 22; smdaKon

—atfal = 2% measurements

Figure 3.37 Measured and simulated transient runaway speed with ccj = 22° and the standard shaped draft tube connected. Zoomed picture of the speed amplitude.

In Figures 3.38 and 3.39 the transient runaway speed for the turbine with the straight draft tube is shown for respectively ai = 18° and cti = 22°. A zoomed picture of the speed amplitude for ai = 22° is shown in Figure 3.40.

Straight draft tube

—atfal = 18; simdation

------atfal«18; measurements

1 3 5 7 9 11 13 15

Time [sec.]

Figure 3.38 Measured and simulated transient runaway speed with ai = 18° and the straight draft tube connected.

87

1790 T

1590 j

Straight-draft tube1390-

1190 atfai = 22 simulation

------affal = 22 measurements

6 990

390 --j

Figure 3.39 Measured and simulated transient runaway speed with ai = 22° and the straight draft tube connected.

Straight draft tube

affal = 22 sanitation1640 -

------atfal» 22 measurements

1630 j

1620 -f

1610 4

Time [sec.]

Figure 3.40 Measured and simulated transient runaway speed with a, = 22° and the straight draft tube connected. Zoomed picture of the speed amplitude.

As we can see, larger deviations between measurements and simulations are obtained for this case. The maximum speed rise is simulated well with a value of approximately2.02 times the nominal speed. The measured curves seem to have more hydraulic inertia than the simulated ones. An explanation may be a deviation between the steady- state characteristics in the high speed area, as mentioned above. A more straight shaped characteristic will give a more bodied speed amplitude, like the measurements.

88

The maximum speed peaks also occurs at different times. Since the aggregate in this case has a low generator time constant and a large mass of water in a straight draft tube under, it was decided to see if this time constant was affected by the mass of the water in the draft tube. By regarding the water in the draft tube as a massive cylinder, the following rough approach of the polar inertia variation due to the flow, AJW can be established:

dhdt At 2

A7W = ^pQr2At (3.71)

where r is the radius (here mean value) of the water column. The generator time constant is defined as follows [5]:

(3.72)

Where °P is the maximum power. Inserting the contribution from the water:

1 a corrected

Letting:

(3.73)

Then the corrected generator time constant becomes:

Ta,^emJ=Ta+KwQ (3.74)

In Figure 3.41 the simulated curve for ai = 22° with corrected Ta is shown together with the measurements and simulation in Figures 3.39 and 3.40. As we can see, the deviation and the phase shift between the maximum speed peak becomes smaller. The simulated speed peak also decreases in value. It seems therefore reasonable that the generator time constant is affected by the amount of water in the draft tube in this case. Another explanation for the deviations between the measurements and the simulations for the straight draft tube may be the draft tube geometry. Since the dimensions of the original cone, mounted at the outlet section of the turbine, was larger than the cone of the straight draft tube, a short diverging joint flange had to be used, see Figure 3.42 Though the flange is rather small compared with the whole draft tube, it could have affected the measurements and especially the shape of the steady- state characteristics in the high- and low speed areas. The nominal values are also uncertain and the swirl component could be larger than calculated from that point of view. Irrotational flow in the nominal condition is also unobtainable in this case, which in feet is assumed by the turbine model. But here it is assumed that the overall reason

89

for the deviation is the divergence between original and simulated steady-state characteristics in the low- and high speed area.

1650-

—-alfal = 22; stmtiation

•alfal = 22; simulation modified Ta

—alfal = 22; measurements

1620 j.

Time [sec.]

Figure 3.41 Measured and simulated (with and without Ta modification) transient runaway speed with oci = 22° and the straight draft tube connected. Zoomed picture of the speed amplitude.

Figure 3.42 Sketch of the straight draft tube

Concluding remarksThis section has presented results from an experimental study of a low-head Francis turbine. In spite of all the uncertainties concerning the turbine, I consider that the results are satisfactory. The improved dynamic turbine model, with the new derived hydraulic inertia approaches, gives better results than expected at first for a low-head Francis turbine. Though, the turbine characteristics have to be tuned more towards measurements than a high-head machine has to.

90

3.3 NON-LINEAR DYNAMIC MODEL OF A HYDROPOWER PLANT CONNECTED TO A STIFF ELECTRIC GRID

This section considers a new non-linear model of a hydropower plant connected to a stiff electric grid. The model is suited for large and rapid load variations, in contrast with the linear model presented in Section 2.3. A new computer program is developed based on the theory presented in Chapter 3. Figure 3.43 shows a sketch of a simplified hydro-electric system.

Mrerieemtir aigaStit

Timd

SAtiromsgracter

Figure 3.43 Hydro-electric power system.

The conduit system presented here consists of an upper and a lower reservoir, a supply tunnel, a surge shaft and a penstock. By means of a synchronous generator the turbine unit is connected to a transformer, a transmission line and a stiff electric grid. Local load variations can be connected on the transmission line. Z is the electric impedance in the system, existing of an ohmic and a reactive part.

There are basically seven separate parts in the system which have to be connected together into a total mathematical model:

1. The turbine2. The conduits3. The synchronous generator4. The speed governor5. The voltage controller6. The electric grid7. The electric load

In the following sections these seven points will be presented, with their governing equations, in order to provide a non-linear simulation model in the time domain. The turbine and the conduits are treated in the same section since the Equation of the flow through the runner must be represented as a boundary condition in MOC. The system

91

will at last be connected together by means of a per unit representation, with an equal power base of the system.

33.1 The turbine and the conduitsThe dynamic turbine model has been treated in detail in the Section 3.1 and Section 3.2. The equation of the flow through a Francis turbine is given in Equation (3.42):

The turbine torque is given in Equation (3.44):

qh^turbine ^ltrnbtn

fi turbine

Where the turbine efficiency is given in Equation (3.43)

\-Rn — -R,2h ^turbine

The equations are made dimensionless with respect to the discharge, the pressure head and the synchronous angular speed in the best efficiency point The turbinetorque equation is implemented in one of the equations of the synchronous generator, shown in the next section.

Variations in the pressure and the discharge in the conduits are calculated by means of the Method of Characteristics (MOC) [61]. The boundary conditions of the tunnel are respectively the upper reservoir at the left pipe end and the surge shaft at the right end. The boundary conditions of the penstock are respectively the surge shaft at the left pipe end and the turbine at the right end. In Appendix 6 the Method of Characteristics is treated, where the derivation of the necessary equations with the reservoir and the surge shaft as boundary conditions are shown:

Tunnel (1)Reservoir at the left end:

rjn __ z-t

Tin _ TT r\n___ il----- MLni\ “ ̂gross > S=Li\ — n

Qn ^5 iX)^M2 (Cpi Hn ) At{Cpx CM2j

Surse shaft at the risht end:

{Qn-Qah

A.(x) -+#r

92

Penstock (2)Surge shaft at the left end:

Qa =CPl -QnBPl ~CM2

BM2

Implementation of the turbine as boundary conditions in MOC will be described below.

Turbine at the right end:The Method of Characteristics is a well-known method of solving transient pipe flow. Alternative techniques are for instance based on lumped parameters. As this author sees it, the MOC is the method to prefer if large time gradients are expected (instantaneous valve closure etc.). In these cases numerical instability often occur if lumped parameters are used. In Figure 3.44 the numerical schema of the MOC with a turbine as a boundary condition in the right-hand pipe end. If a long tailrace tunnel is connected to the draft tube, the MOC can be used for this part as well, with the turbine as a boundary condition in the left end. According to the theory of the MOC, the pipe is divided into (roc) parts resulting in (rac+I) calculation nodes. The turbine element in Figure 3.44 will therefore be the last calculation node; i = (nx+1).

Figure 3.44 Numerical schema of the Method of Characteristics with aturbine as a boundary condition in the right pipe end.

!i1I

I

Re-writing the Equation (3.42):

da&

(1 + <r)Ji- _0_2„sr.

— aCl2 (3.75)

The notation (Hi, Ql) means the element number, where (z) is the turbine element and (i-1) is the pipe element in front of the turbine. Remembering also that (H„, Q„) denotes nominal values and have nothing to do with the discretization.

W5

93

Introducing a 1st order finite approximation for the time derivative:

do, o:-ordt At

(3.76)

The notation {ff, H1) refers to the actual time step and iff'1, If'1) refers to the previously time step, t-At. At the downstream end of the pipe the compatibility equation yields [61]:

h:=cp-bpq: (3.77)

Where the magnitudes Cp and Bp are described in Appendix 6. By combining the Equations (3.75), (3.76) and (3.77) the following algebraic relation of second degree, solved with respect to the flow, comes forth:

GT + aHn(0.n)2-{\ + cr)CP- = 0

The equation above can be solved and expressed in a more compact way:

%(&")'+&,#'+r, = 0 (3.78)

(3.79)

(3.80)

(3.81)

-^+V4-4%

2K}(3.82)

h:2=cp-bpq;2 (3.83)

The subscripts Qa, Ha are used to indicate the penstock.

3.3.2 The synchronous generatorThe synchronous generator is usually represented on the network level by its Thevenin equivalent. The Theorem of Thevenin says that a circuit with constant sources and resistance may, seen from two terminals, be described by means of constant voltage source in series with a constant impedance [19]. Figure 3.45 shows a single phase synchronous generator. Eg is the electromotive voltage (internal emf) and Zg is a

94

characteristic impedance. They are determined on the machine level by solving a set of 1st order differential equations. The terminal voltage, E%, is the connection between the generator (machine level) and the electric grid (network level) [24].

Fig. 3.45 Thevenin equivalent of the synchronous generator.

A generator model of 5th order is the most complex model for a synchronous generator. It incorporates both subtransient effects, field- and damping windings. The electromotive voltage is equal to the subtransient internal voltage, E", and the characteristic impedance is given by the anchor resistance, ra, and a subtransient reactance Xd". j is the complex unit vector:

Eg = E"=ED"+j-Ee" (3.84)

Ed"= Ed"-smfl+Eq"-cosfi (3.85)

Eq"= -Ed''-cos0+ Eq"-smfi (3.86)

Zg =ra +j-xd” (3.87)

Eq" and Ed" represent the subtransient internal voltage in the q- and d-axes. The notation (d,q) refers to the local machine level and the notation (D,Q) refers to the global network reference. The transition from local to global reference is given by the generator rotor angle, /?. This is the angle between the q-axis and the synchronous rotating D-axis, shown in Figure 3.46. The angle is very important in stability considerations. If it grows too large, the unit may M out of synchronism with the electric grid. The angle is in fact the connecting link between the electric and the hydraulic parts of the system and will sense possible unstable conditions from both parts.

95

Fig. 3.46 Definition of (3.

The following five differential equations describe the dynamics of the generator:

f“°’N’r

i\

^turbine ^grid ^

dt

dEq' 1 t

dt

II K

dEq" 1 /dt

ii

dE" = 1 1

= —{Eq'-Eq"-(xd'-xd'%)

z;.

(3.88)

(3.89)

(3.90)

(3.91)

(3.92)

©b is the base angular speed defined in Section 3.3.7. Qnnbmc is the dimensionless mechanical angular speed of the turbine (with respect to the base, cob), and not the electric angular speed of the unit; the so-called rotor speed. Therefore the number of poles, Np, has to be used in the equations. This is the most convenient way of expressing the equations, especially since the governors are based on measurements from the mechanical angular speed and the hydraulic equations are using it directly. In the literature on electrical issues the rotor speed is commonly used in the generator representation. This is mainly because the hydraulic part of the system is simplified. The synchronous frequency in the stiff electric grid, Qgnd, is also called the stator speed. Figure 3.47 shows the principle sketch of the rotating fields of a synchronous generator.

96

Figure 3.47 The rotating fields of a synchronous generator.

The steady-state connection between the rotating fields are given by:

^grid turbine,elect. ir ̂curb mejncch.^ P

(3.93)

There is a constant steady-state relation between the mechanical angular speed and the electric rotor speed in a synchronous generator. In an ASD generator for instance, which in feet is an advanced asynchronous generator, the steady-state relation between the mechanical angular speed and the rotor speed can be varied (in a range of ± 10% of nominal speed). The advantage of this, seen from the hydraulic side, is that the nominal efficiency almost can be maintained in off-design operation by adjusting the mechanical speed. The stator speed will still be equal to the synchronous stator speed.

Equations (3.88) to (3.92) are written in dimensionless form. Tdo" and Tqo" are called subtransient open circuit time constants, Tdo' is a transient open circuit time constant, xd and xq are synchronous reactances, Xd" and Xq" are subtransient reactances and Xd'

is a transient d-axis reactance, id and iq are components of the stator current. H is the inertia constant of the generator, which is usually available for each unit:

TaCosq>n(3.94)

Where Ta is the generator time constant and cos<j>„ is the nominal power factor. Eqr is the field voltage or the excitation voltage governed by the voltage controller, see Section 3.3.4. Dtmbine is the turbine torque, described in Section 3.3.1, and Ddcctnc is the electric torque given by the following expression:

D.(ed"Id+Eq"Iq)

electric (3.95)

97

Where, ID and Iq are components of the stator current, described in Section 3.3.5. Produced Active and reactive power of the unit (in the electromotive source) can be found from the following equations:

(3.96)

(3.97)

The electric power in the grid exists of two components, an active part (real), P [MW] and a reactive part (imaginary), Qr [MVAr]. The total power is called the apparent power, S [MVA]:

S = P + j-Qr (3.98)

Usually, the notation Q is used for the reactive power. Since this thesis is written by a hydraulic engineer, Q is already reserved for the discharge. Qr is therefore used instead for the reactive power.

While the active power represents the real power assumptions in the transmission, the reactive power is an energy oscillation caused by the reactances in the system. Magnetic fields are built up in step with the grid frequency and cause voltage oscillations, losses and poor utilization of the electric system, since the ability of producing active power will be reduced. Seen from another point of view, many components (including coils and condensers) connected to the electric grid are consumers of the reactive power [31].

Describing hydraulic magnitudes by means of an electric analogy is a well known technique in order to get an improved physical understanding of the electric magnitudes. For instance the voltage, E can be compared with the pressure head, H and the current I, can be compared with the discharge, Q. The following example is an attempt to give an “hydraulic comparison” or analogy of the reactive power. The analogy is not based directly on the fundamental hydraulic and electric equations. This is in fact impossible because of the imaginary sizes on the electric side. The example must therefore be looked at more as a “philosophic concept” than a physical explanation.

Example: Hydraulic analosv of the reactive newer.Consider a two-dimensional swirl flow in a pipe (draft tube, for instance), with the oblique velocity, C. The mean axial component of the velocity is called, Cm the rotational component is called, C„ and a is the swirl angle, see Figure 3.48.

98

iu

IS

p

Figure 3.48 Hydraulic analogy to active, reactive and apparent power generation.

The connection between the components is: Cm =

By increasing the swirl angle, the axial velocity will decrease and the rotational component will increase, the oblique velocity is still unchanged Theoretically, if a power generation unit is placed in the m-direction, and the mean axial component, Cm, performs the shaft work, the power can be compared with the active electric power.

In the same manner, if a power generation unit can be theoretical placed in the u-direction, and the swirl component, Ca generates the power, the power can be regarded as the reactive electric power:

Because of the phase shift between the reactive and active power, the power generation in the in the two axis has to take place at different places.

The apparent electric power will be a function ofthe oblique velocity:

S = /(c)

Since an increased swirl angle causes a reduction in the axial velocity, the so-called active power will be decreased and the reactive power will increase. The swirl angle, a, will therefore have its analogy in the electric power factor, <j>. The m-axis can be compared with the real electric axis. The u-axis is in fact also real, but in this example it can be compared with the imaginary electric axis.

3.3.3 The speed governorA traditional model of a PI governor without permanent speed drop can be expressed by means of three first order differential equations:

(3.99)

99

dtc2 1 / \(3.100)

die 1 / \(3.101)

Here is bt the transient speed drop and 7* the integral time, k is the opening degree of the guide vanes, k2 is the position of the servo motor piston and h is the signal given from the governor unit. To is the time constant of the servo motor and Ty is the time constant of the electric-hydraulic transfer system (the actuator). As we can see from Equation (3.99) the dimensionless turbine speed, Tiurbine, will be equal to the dimensionless electric grid frequency, Q%rid, in every steady-state condition.

During a simulation performance, the opening degree and the speed of the servo have to be controlled in such way that they are kept inside their permitted range of operation. The opening degree cannot be changed so fast that the pressure rise will exceed the maximum design value, so the gradients also have to be controlled.

If the governor is equipped with a permanent speed drop, bp, it is most convenient to derive the differential equations by means of a state space formulation directly from a block diagram of the system. From [36] the following set of equations can be found (the electric-hydraulic transfer system is here neglected):

die— = *:2 (3.102)

die2 _ F dD.^ < F,

dt

Where:

Fl = bX’

dtjj ^grid ^turbme

p

-F2k2-F3(k^-k) (3.103)

(bpTo+bJt)F =-----— (3.104)

Kref is the load setting of the unit which can be changed according to the description given in Appendix 1 (A1.3). Hysteresis, lost motion and slackness in the speed governing system is not included in this model. The consequence of neglecting them will be that the speed governor will respond to every speed variation that occurs. Normally, a dead zone of approximately 0.02% is present.

3.3.4 The voltage controllerThere are many different models of the voltage governor in use. This section presents a basic model [24] with a simplified frequency dependent damping unit. More sophisticated and up-to-date models are of higher order and also usually have a feedback loop from the generator active power. Figure 3.49 shows a block diagram of the controller. The terminal voltage, E%, is measured and compared with its set point.

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Besides the governor unit, a derivate and an internal stabilization unit are added to the system. The field voltage, AEqr, is the output signal which is secondly added to the steady state field voltage, Eqm. AEm is a corrective accounting for saturation effects inthe windings (described by its Potier reactance Zp). The signal is amplified before the voltage is fed to the rotor through brushes. During a simulation performance, the field voltage has to be kept inside the permitted limits.

Derivate stabilization

Fig. 3.49 Block diagram of a basic voltage governor.

It is possible to establish a set of four 1st order differential equations by reducing this diagram:

dAE,ifdt

= G1[ei-AEgf)

z Q \grief

EkO~Ek+Kn\^lmbme- NX x lyp /

dEp= G'AE. - G.E-,

dt ™5^ U6"2

dE3dt = g7 ^turbine jj ^grid

1 P '-GSE3

(3.105)

- G3M^ -G4E,+ G2E2 (3.106)

(3.107)

(3.108)

Where Ei, E% and E3 are all dummy variables without directly physical meaning. The constants are given by:

(3.109)

Kr and Tr are the gain and the time constant of the governor unit, K<j and Td are the gain and the time constant of the internal stabilizator. Tf is the time constant of the power amplifier and Kq and Tn are respectively the gain and the time constant of the additional damping unit.

The field voltage is then given by:

Eqf = EqfO + t&qf ~ (3.110)

The Potter voltage has to be calculated in order to find the corrective accounting for saturation effects in the windings, AEm, [19]:

EP=\Et +I-ZP\ (3.111)

j,(ep -0.8)

(25-EP) ’Ep > 0.8 (3-112)

AEm = 0, Ep < 0.8 (3.113)

3.3.5 The electric gridIf a three-phase AC system is symmetrical it can be represented by a single phase (line) equivalent system. Symmetrical conditions mean that the voltage and the current are equal in all the three phases. A stiff electric grid, or more precisely, a system connected to an infinite bus, the large global coordinated grid is represented by means of a stiff frequency, a stiff voltage and a short circuit power (active or reactive).The stiff voltage and the electric grid frequency will not be affected by the unit we are modelling in detail. This assumptions gives a reasonable description of the dynamics of the unit [16]. In order to study oscillations between the coordinated grid and the unit at least a two-machine model has to be used. A turbine unit operating under isolated load conditions will not have any connections to the coordinated grid, and the variations in frequency and voltage will totally be governed by the unit it self.

On the network level the electric system is represented by its Thevenin equivalent, see Figure 3.50. All the impedance in the system is represented in an equivalent impedance, Zequ. The voltage in the stiff grid is represented by an equivalent voltage, EcqU. Both the equivalent magnitudes change due to the load disturbance in the systems, but the physical voltage in the stiff grid, En is still constant [24].

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E, I

/

Figure 3.50 Equivalent single phase network of the electric grid.

The equivalent impedance and voltage in the system can be expressed as follows:

Zegu =

^locallyline Zl'trans Zgrid j

Zlocal + {^line + Ztram + Zgrii )

= En - inline + Ztrass + Zgnd )

(3.114)

(3.115)

Where Z^i is the local load impedance, Zii„c is the impedance in the transmission line, Ztrans is the impedance in the transformer and Zgnd is the stiff electric grid impedance.En is the physical stiff voltage in the coordinated grid.

The impedance in per unit in the transformer, the transmission line, the stiff grid and the local load can be expressed as follows [24]:

(3.116)

(3.117)

Z*"='(J:) (l~) (f) (3.118)

(3.119)

Here sr and ex are ohmic and reactive short circuit voltages of the transformer (in Q). Stnms is the rated power of the transformer and E* and Ets are the voltages on the primary and secondary sides of the transformer. Eiocai is the voltage where the local load occurs, and APiocai and AQiocai are the variations in local load, described here as ressitive, see Section 3.3.6. Ri and Xi are the resistance in the transmission line (in £2).

->

103

Eb and Sb are respectively the base voltage and the base power of the system. They are chosen to be the same as the rated values of the unit. They are defined in Section 3.3.7. S„ is the short circuit power in the stiff grid (in MVAr).

The stator current can be expressed in the following way:

/ =Zg +Zequ = Id+J'Iq (3.120)

The components on the local machine level can be found by means of the rotor angle,P:

Id = ID sin/?-Iq cos/3

Iq = ID cos/?+7g sin/?

(3.121)

(3.122)

Input value to the voltage governor is the terminal voltage, E%. It is often expressed inreal value:

Ek=\E"-I-Zg\ (3.123)

Active and reactive power of the generator, delivered to the terminals, become:

%=Re{f*/*} (3.124)

Qrt=lm{EkI*} (3.125)

Where I* is the complex conjugated of the current, I.

3.3.6 The electric loadLoad changes in the electric system appear when the impedance in the system is changed. Z is the electric impedance in the system, with an ohmic and a reactive parts. The reactive part is actually frequency dependent, but due to the small variations in the grid frequency, it may, in most cases except for long-term short circuits, be kept as a constant [24]. The impedance can be expressed as follows [26]:

Z = r + jx = r + j = r + j<i(xc~xL) (3.126)

Where r is the ressitive or ohmic resistance and x is the reactance, consisting of a capacitive, Xc, and an inductive, xL, part.

It is possible to simulate the following cases in the model:

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• Local load variations respectively on the primary and the secondary side of the transformer. With no initial local load present the equivalent impedance and voltage become (in per unit):

%equ — Zmms + ^line + %grid (3.127)

Eequ ~En (3.128)

Connection of a sudden local load will change the equivalent magnitudes according to Equations (3.113) and (3.114). If an additional local load is connected the resulting local load will be a shunt load as follows:

y ^local,local?.

" local <7 , 7*localt\ + t'locda

(3.129)

• Capacity variations in the transmission line (weak or strong interconnection to the coordinated grid) by changing the line impedance, Za*.

• Short circuit or disconnection of the transmission line. This is on the hydraulic side called total load rejection.

• Changing the short circuit power in stiff electric grid.

• Variations in the stiff electric frequency.

The electric load is in tact dependent on both the voltage and the frequency. A common way of modelling the frequency and voltage dependent electric load is given in the following equations [2]:

(3.130)

(3.131)

Here are P and 0 the active and the reactive power outlets at the actual voltage, E, and frequency,/ The notation (Po,Qo,Eo,£>) describe the original load condition. The power exponents describe the sensibility of frequency and voltage dependence in the active and the reactive load.

A typical Norwegian winter load will have the following power exponents:

MP = 1.5, MQ = 2.5, NP = NQ = 0

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The frequency dependency is in other words neglectable. In Norway, it is common to assume the that the load is independent of the frequency. The following three points will emphasize this [24]:

• The frequency dependency of the load will as a general rule not affect the stability conditions in the system. The influence will rather stabilize the system.

• Frequency dependent load is not fully understood today. Basically, because measurements of the interaction between the voltage variation and the simultaneous frequency variation, are difficult to perform.

• Simulations become more difficult to perform and iteration is necessary.

In the European thermal power system, the frequency dependency seems to be more significant (MP = 1.0, MO = 2.0, NP = 0.5, NQ = -1.0) [2],

If MP = MQ = 2, the impedance in the system is constant and the load is resistivity (ohmic), see Equation (3.118). This is the most convenient way of representing the electric load with an eye to the simulations; no iteration is not needed. In the developed computer program the frequency independent ressitive load is used.

3.3.7 Per unit representationOn the network level (the electric part of the system), the rated power (maximal power on the terminals) and rated terminal voltage of the generator are used as base for a per unit representation of the system (notation S), as shown above.

SB=Sl=Sgm [VA]

EB=El=Egm [V]

(3.132)

(3.133)

The hydraulic part of the system has to use the same base to get the right scaling of the magnitudes, when they are connected together. A way of handling this is to choose a base head equal to the gross head and a base angular speed equal to the nominal mechanical angular velocity of the turbine:

HB=Hgms [m] (3.134)

[s'] (3.135)

Were, fs is the synchronous frequency in the grid (50 Hz) and Np is the number of poles in the generator. The base flow and base torque may now be calculated:

106

(3.136)Q‘-1k [m’,sl

Mb =— [Nm] eoB

(3.137)

The turbine torque, for instance, has to be made dimensionless with respect to the base torque before it can be used in the torque balance equation; Equation (3.89). In Section 3.1.3 the presented turbine torque is made dimensionless with respect to the nominal values and must therefore be corrected as follows:

M,

''tvrbine/iewturbmejimb

D,turbine (3.138)

3.3.8 Steady-state conditionsThe steady-state condition of the hydraulic part of the system can be found by means of the Bernoulli equation in connection with the turbine model Since the angular speed is restricted to be equal the synchronous speed in the steady-state condition, the opening degree, Ko, will be the only input parameter that can be changed.

With respect to the nominal values the dimensionless pressure head in front of the turbine, h and discharge, q in the conduits can be found as follows:

Hgross + Cl*0 G^turbine

JT„ + C,Ko2(o-+1) (3.139)

q=- C,(3.140)

Where:

Cx -AL

\2DgA-

I f XL \' —i V2DgA ) wmdj

penstock

Qn (3.141)

The steady-state pressure head has to be found for each element in the Method of Characteristics.

In order to find the initial conditions in the electric system a load flow analysis has to be carried out by means of an iteration procedure, described in Appendix 7.

The produced active power, Pg in the source of the generator (the electromotive voltage) is given by the turbine power. The produced active power is in other words specified. In order to solve the system either the electromotive voltage, Eg, or the produced reactive power, Qrg, has to be specified, see Figure 3.45. In the developed

107

computer program Eg is specified and the generator is, because of that reason, called a PU-machine (U is another common notation of the voltage). By means of a Newton- Raphson iteration procedure the active and the reactive power, the voltage and the electric angle (the phase shift in relation with the electric grid) can be found in the source and on the terminals of the generator.

Usually, common computer programs for load flow analysis of large power systems, as SIMPOW for instance, specify the PU or PQ directly on the terminals of the machines and not in the source. But as long as the hydraulic system is connected and the active power is given directly in the source by means of the turbine power, the specification should be done here. This way of specify the system is therefore implemented in the new computer program.

The initial conditions of the state space variables, %, in the 1st order differential equations can be found by letting:

= 0 (3.142)dt

3.3.9 Simulation programIn Figure 3.51 a flow chart of the new developed computer program is shown. The program is written in Visual Basic 4.0 and it is running in Windows 95/NT modus. The computation time needed for obtaining satisfactory results of a very large system (long tunnel and penstock) is rather small; in the range 5-10 minutes, depending on the processor capacity. The code is basically programmed for the system shown in Figure 3.43 and it contains the mathematical models descried in Section 3.

The code is written with an eye to a possible extension of all modules. All the above mentioned parts of the hydro-electric system are programmed in separate procedures, which easily can be changed in the future.

A new procedure, like an ASD generator for instance, may easily substitute the procedure of the synchronous generator, without destroying the program structure. In the future the program should be extended with a new unit (two-machine model) and the input of the program should be done more general, in order to fit more complicated variants of the conduit system, than shown in Figure 3.43.

As we can see from Figure 3.51, after the input data is read the program will calculate several constants needed for the simulation. The base values and the constants in the differential equations are determined.

The initial conditions of the turbine and the conduits are then calculated.

Then the impedances in the electric system are found and the load flow analysis is carried out.

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The initial state of the synchronous machine can now be determined. The main calculation loop can now be started.

Firstly, the excitation of the system is done. This may for instance be, as mentioned inSection 3.3.6, local load variations or changes in the electric grid frequency.

Electric magnitudes like the electric torque, active and reactive power of the generator, the terminal voltage and the generator current are then determined. The transition from global level to local machine level is performed.

The next procedure calculates the turbine variables, as the turbine torque, the turbine power, the hydraulic inertia and the turbine efficiency.

The differential equations of the generator, and the controllers are then solved by means of an explicit Runge Kutta method of 4th order (ERK4). The angular speed of the turbine, the opening degree, the field voltage, the rotor angle and the internal voltages in the q- and the d-axes are now available.

The opening degree and the field voltage are then controlled in such way that they are kept inside their permitted ranges of operation with respect to maxtmnm/minimnm values, gradients and saturation.

The Method of Characteristics (MOC) then takes over and calculates the pressure and discharge in the conduits. This procedure is the bottleneck in the program in consideration of the calculation time. Small time steps lead to many elements in the conduits.

After the discharge and the pressure head in front of the turbine are determined the results are written to an output file. A new time step can then be taken.

Concluding remarksA new non-linear dynamic model of a turbine unit connected to a stiff electric grid has been presented and a new computer program is developed. The program will be used in various simulations cases in the next chapter in order to study whether transients in the conduits will affect the performance of the electric grid.

109

■Main Calculations-

( )

j Read Input

Base values Diff.Equ. Const.

Initial conditions Conduits/Turbine

Impedances

Load Flow analysis j

Initial conditions Generator/Grid

t = t -At

Internal j Iteration i

Figure 3.51 Flow chart of the developed computer program.

110

4. INTERACTIONS BETWEEN THE HYDRAULIC PART AND THE ELECTRIC PART OF A HYDROPOWER SYSTEM

This chapter presents:

• An investigation of interactions between the hydraulic part and the electric part of a hydropower system by means of the dynamic models derived in Chapter 2 and Chapter 3.

The linear dynamic model will be used to study interactions between an isolated grid and a stiff electric grid due to stability performance. The purpose is to show how instabilities in the electric part or the hydraulic part will aJffect the total system stability. Simulations will be performed in the time domain by means of the linear model derived in Chapter 2.

The non-linear model presented in Chapter 3 will be used in various cases in order to study whether transients in the conduits, as “water-hammer” and “mass-oscillations”, will be reflected or not in the electric grid.

4.1 LINEAR TIME DOMAIN SIMULATIONS

In this section simulation examples by means of the present developed linear models in Sections 2.3.1 and 2.3.2 are performed. The intention has been to compare the two models, isolated grid and stiff electric grid with each other with the focus on governing stability.

Simulations of a turbine unit connected to an isolated grid will be performed firstly. Afterwards, the results will be compared with simulations of a turbine unit connected to a stiff electric grid.

4.1.1 Simulation of a turbine unit connected to an isolated gridFigure 4.1 illustrates the input data for a high-head Francis turbine (speed number *£2 = 0.27, Allievis constant hw = 0.53 and pipe constant Tw = 0.66 s) running under isolated load condition. The conduit system presented here consists of a tunnel, a surge shaft and a penstock. The tunnel is regarded as inelastic and is described by means of the length, Lh the diameter, Dt (circular) and the Manning’s number, M (frictional parameter). The surge shaft is described by means of its cross section area, As. The penstock is regarded as elastic and the input data are respectively the length, L, the diameter, D, Moody’s frictional parameter, A, the nominal discharge, Q„ and the wave propagation speed, a. The turbine is modelled by means of linearized coefficients, described in the nomenclature. In the speed governor the time constants of the servo motor, T0, and the electro-hydraulic system, Ty are included. T& is the integral time and bt is the transient speed drop or inverse proportional gain. The permanent speed drop, bp, is set to zero in the examples. The generator is described by its time constant, Ta and the synchronous speed, ns.

Ill

Figure 4.1 Hydropower plant running under isolated load conditions. Input data for simulations.

First the 4 pole-3 zero coefficients (see Equation 2.51) of the pressure-flow transfer function, h/q, have to be found. In order to find these, the dynamic friction factor, Kd

(see Section 2.1) has to be determined. The Rayleigh damping function is herecalibrated against Kongeter/Brekke’s damping model [4] in the frequency domain. In Appendix 4 a brief resume of the this model is described and the iteration procedure is showed for a simple system with a pipe and a valve.

In Figure 4.2 the Rayleigh damping model is compared with Kemgeter [10]. Excellent agreement is obtained by choosing the dynamic friction factor, Kd = 0.00034 s, by means of trial and error. In Figure 4.3 the derived 4 pole-3 zero approximation is compared with calibrated Rayleigh curve. We can see that the approximation is excellent.

By means of the theory in Appendix 5 and the input data in Figure 4.1 the coefficients in the inelastic transfer function of the mass oscillations can so be found, see Equation 2.52. After the steady-state condition of the hydraulic part is determined, simulations can be performed.

112

Rayleigh Ksnjjeter

e [rad/8]

Figure 4.2 The h/q amplitude calculated by means of the Rayleigh model andKangeter’s model.

Rayleigh

4 pole-3 zero approximation

Figure 4.3 The h/q amplitude for the 4 pole-3 zero approximation compared withthe Rayleigh damping model.

Case 1In order to identify the influence from the surge shaft a sudden increase in the electric torque of 10% is simulated. In Figures 4.4 and 4.5 respectively the speed and the turbine power are shown, in percentage change from initial condition, without and with the surge shaft included in the model. As we can see the system is stable. The surge shaft introduces a long-term oscillation which makes the governing performance

113

poorer. The influence is very small because of a relatively large cross section area in the surge shaft.

-0.005

-0.015

-0 025

-0.005

-0.015

4)025

Time (Seconds) Time (Seconds)

Figure 4.4 The angular speed after a sudden 10% increase in the electric torque. Without (left) and with (right) a surge shaft included in the model. Isolated grid.

Time (Seconds) Time (Seconds)

Figure 4.5 The turbine power after a sudden 10% increase in the electric torque. Without (left) and with (right) a surge shaft included in the model. Isolated grid.

In Figure 4.6 the pressure head and the guide vane opening are shown with the surge shaft included in the model. We can clearly see that the water-hammer effects are dominating the picture in the beginning of the transient period, and the mass oscillation takes over in the long term.

114

Time (Seconds) Time (Seconds)

Figure 4.6 The pressure head (left) and the opening degree (right) after a sudden 10% increase in the electric torque. A surge shaft is included in the model. Isolated grid.

The periodic time of the water-hammer and the mass oscillation are respectively:

7J = — = 123 s. a

2tc

SA

A A

= 50.6 s.

(4.1)

(4.2)

Case 2In Figure 2.16 a pressure feedback loop is shown in connection with the speed governor. According to [8] the pressure feedback compensator will stabilize the total system and hydraulic noise from the penstock will be filtered out. The increased stability margins may for instance make the surge shaft superfluous. In order to study how the compensator works out for our system simulations are performed. The best performance seems to occur with K= 0.2 and Ti = 100, see Equation (2.41). In Figure 4.7 the angular speed is shown with and without the surge shaft (stable Thoma cross section) included. Compared with Figure 4.4 we can see that the compensator filters out the water-hammer effects, but are not capable of handling the mass oscillations properly.

Therefore, a hydropower plant with mass oscillation problems cannot expect to get increased stability margins with the pressure feedback compensator. The intention with a surge shaft is to avoid large water-hammer effects so that the speed governing can be performed as fast as possible. The intention for including a pressure feedback loop is also to substitute a surge shaft. From that point of view the compensator seems to work very well

115

•0 005

•001

•0 015

•0.02

0

0 20 40 60 80 0 20 40 60 80

Time (Seconds) Time (Seconds)

Figure 4.7 The angular speed after a sudden 10% increase in the electric torque without (left) and with (right) a surge shaft included in the model. The speed governor is equipped with a pressure feedback loop. Isolated grid.

Case 3If the parameters in the speed governor are badly tuned the system will become unstable. In Figure 4.8 the angular speed and the turbine power are shown after the transient speed drop, b, is decreased from 0.2 to 0.11. The system is completely unstable. In the next section it will been shown that a stiff electric grid can stabilize the system if the transmission line has sufficient capacity.

Time (Seconds)

Figure 4.8 The angular speed (left) and the turbine power (right) after a sudden 10% increase in the electric torque. The speed governor is badly tuned and the system is unstable. Isolated grid.

Case 4Up to now the system has been subjected to variations in the electric load or the electric torque. As mentioned earlier the dynamics can also be excited by changing the reference speed. In Figure 4.9 the angular speed and the turbine power are shown after a 0.2% drop in the reference speed. Input data for the simulations are like the original

116

ones, taken from Figure 4.1. As we can see the system is stable. The angular speed will after the transient period match the new reference speed. The steady-state turbine power will be decreased in value since the system takes this as the need of power is decreased. In Figure 4.10 the angular speed and the turbine power are shown with a badly tuned speed governor connected, as in Figure 4.8. The system is clearly unstable.

X10

Tene (Seconds)

x10

Figure 4.9 The angular speed (left) and the turbine power (right) after a sudden drop in the reference speed of 0.2%. Isolated grid.

Tene (Seconds)

Figure 4.10 The angular speed (left) and the turbine power (right) after a sudden drop in the reference speed of 0.2%. The speed governor is badly tuned and the system is unstable. Isolated grid.

Concluding remarksVarious simulation cases are performed of a turbine unit connected to an isolated grid. In the next section the results will be compared with simulations of the same system connected to a coordinated electric grid.

117

4.1.2 Simulation of a turbine unit connected to a stiff electric grid The following simulation deals with the same turbine unit as in the previous section, but the isolated grid is connected to a stiff electric grid, through a transformer and a transmission line. In Figure 4.11 input data of the turbine/conduits and the electric part are given. The generator is now described by means of additional properties as transient- and sub-transient reactances and time constants.

Figure 4.11 A single turbine unit connected to a stiff electric grid. Input data for simulations.

Before the simulations can be performed a static load flow analysis of the electric part of the system has to be carried out. The analysis is described in detail in Appendix 7. The active and reactive power of the generator is specified and the condition of the external electric grid, outside the terminals of the generator, will determined the size of the terminal voltage and its angle in relation to the stiff electric grid (with 0 elect, rad.).

Figure 4.12 shows a sketch of the electric system after the load flow analysis is performed. The impedance in the system is described by an equivalent impedance, Zequ, where the difference in voltage levels is accounted for. In nominal operation, the turbine produces an active power of Ptwbme =153.1 MW delivered to the shaft. Since the generator is modelled with only reactive losses, the active power delivered at the terminals is, Pge„ = Pnrbim- A reactive power of Qgen = 20 MVAr is specified and the terminal voltage is calculated to be Ege„ = 12.262 KV with an angle of 0.168 elect, rad.

Pum=153.1 MW P^=153.1 MW . 0^=20 MVAr

^,=12.262 KV <0.168 elect, rad.

E„=390KV 1 <0 elect rad. |

Figure 4.12 The electric part of the system after a static load flow analysis.

118

In the presented model we are able to study forced variations in the reference terminal voltage, the reference speed and the stiff electric frequency. In Figure 4.13 the angular speed and the turbine torque are shown after a drop of 0.2% in electric grid frequency. The terminal voltage and the active power of the generator are shown in Figure 4.14. In Figure 4.15 the angular speed and the active power are shown after 10% drop in the terminal voltage. Compared with simulations performed with an isolated grid, Cases 1 and 4, we can see that the system now is much more robust, with smaller transient peaks and fester response. The y-axis is in percentage change from initial condition.

-0.5

-2.520 40 60 80

Time (Seconds)

X10*

Time (Seconds)

Figure 4.13 The angular speed (left) and the turbine torque (right) after a sudden\ drop of 0.2% in the stiff electric grid frequency.

■0.015

Time (Seconds)

x10**

Time (Seconds)

Figure 4.14 The terminal voltage (left) and the active power (right) after a sudden drop of 0.2% in the stiff electric grid frequency.

119

x 10 °

Time (Seconds) Time (Seconds)

Figure 4.15 The angular speed (left) and the active power (right) after a sudden drop of 10% in the reference terminal voltage. Stiff electric grid.

In Cases 3 and 4 in the previous section, unstable behaviour under isolated load conditions are shown, caused by a badly tuned speed governor with an increased proportional gain. By connecting the isolated grid to the stiff electric grid the system will be stabilized with a very good governing performance. Figure 4.16 shows the angular speed and the opening degree and Figure 4.17 shows the terminal voltage andthe active power of the generator.

x10°

-0.5

-2.50 20 40 60 80

Time (Seconds)

xio"

Time (Seconds)

Figure 4.16 The angular speed (left) and the opening degree (right) after a sudden drop in the stiff electric grid frequency of 0.2%. The speed governor is badly tuned and the system is unstable running under isolated load conditions. The stiff electric grid stabilizes the system.

120

-0.005

•0.015

Time (Seconds)

%10*

Time (Seconds)

Figure 4.17 The terminal voltage (left) and the active power of the generator (right) after a sudden drop in stiff electric grid frequency of 0.2%. The speed governor is badly tuned and the system is unstable running under isolated load conditions. The stiff electric grid stabilizes the system.

The given data for the transmission line and the electric grid makes the interconnection from the unit to the coordinated grid very strong. The short circuit power in the grid is large and the transmission line has large capacity and small reactances compared with the unit. The stiff electric grid is therefore able to stabilize the isolated grid even when the unit is unstable, as shown above.

If the reactance in the transmission line is tremendously increased by a weak interconnection between the unit and the coordinated grid, the capacity of the transmission line becomes small Even the stable governing system operating under isolated load conditions, in Case 1 in the previous section, becomes unstable when it is connected to the electric grid. In Figure 4.18 the angular speed and the opening degree are shown after a sudden drop of 0.2% in the stiff electric grid frequency. In Figure 4.19 the terminal voltage and the active power are shown. The new line resistance is Ri = 160 Q and X\ = 1600 £2. The system is completely unstable and high-frequent noise takes over the system.

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x 10°

Time (Seconds) Time (Seconds)

Figure 4.18 The angular speed (left) and the opening degree (right) after a sudden drop in the stiff electric grid frequency of 0.2%. The transmission line between the unit and the strong coordianted grid is weak and the stable isolated system becomes unstable.

Time (Seconds) Time (Seconds)

Figure 4.19 The terminal voltage (left) and the active power of the generator (right) after a sudden drop in stiff electric grid frequency of 0.2%. The transmission line between the unit and the strong coordianted grid is weak and the stable isolated system becomes unstable.

Likewise, if the short circuit power of the stiff coordinated grid is reduced the isolated grid may be unstable when it is connected. This is shown in the last simulation example where the short circuit power is reduced tremendously and even below the rated power of the unit; &=100 MVAr. Figure 4.20 shows the unstable performance of the angular speed and the active power.

122

X10*1

Time (Seconds) Time (Seconds)

Figure 4.20 The angular speed (left) and the active power of the generator (right) after a sudden drop in stiff electric grid frequency of 0.2%. The short circuit power in the stiff coordinated grid has been reduced and the stable isolated system becomes unstable.

Concluding remarksSimulations are done in this section in order to show the possibilities and use of the two established linear models of a hydro-electric power system; isolated grid and stiff grid. The focus has been on unstable governing performance between an isolated grid and a stiff electric grid.

Simulations have shown that a stable governing system operating under isolated load conditions may become unstable when it is connected to the coordinated grid, depending on the strength of the transmission line or the coordinated grid Likewise, will an unstable isolated governing system be stabilized, if the interconnection to the coordinated grid are strong enough.

The parameters in the speed governor are always timed in isolated load condition. This is because isolated load operation is regarded as “worst case”. As simulations have shown, the stability margins may be increased or decreased when the unit is interconnected to the coordinated grid. A speed governor that automatically is capable of switching between two sets ofparameters, one tuned for isolated grid and one for interconnected operation, would in many cases improve the speed governing performance. Taking this into account it may be of great interest for both hydraulic arid electrical engineers to see the hydraulic part of the system in connection with the electrical part.

4.2 NON-LINEAR TIME DOMAIN SIMULATIONS

In this section simulations of a hydro-electric power system are performed by means of the new computer program, described in Section 3.3. Cases that can be performed in practice are treated. The purpose of the simulations is first to study the interaction between the hydraulic and the electric part of the system. In addition, these simulations are compared with simulation performed by ABB Corporate Research [49] by use of

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the well-known program, SIMPOW. In Figure 4.21 the system is shown with input data for the simulations. Simulated turbine characteristics of the actual turbine are shown earlier in Figure 3.11.

Figure 4.21 Hydro-electric power system with input data for simulations.

Input data to the system is almost similar to ones presented in the state space model in Section 4.1. Although some more details are needed concerning saturation effects and restrictions. Because of the higher order of the synchronous generator model, more input data is needed. The hydraulic system, except for the tunnel and the surge shaft, is also presented by simulations in Section 3.1.5 (Consequences of the hydraulic inertia).

The steady-state condition of the hydraulic system is equal to its nominal condition; synchronous speed and power production with the best hydraulic efficiency (b.e.p.). The steady-state condition of the electric system is determined by the given power of the turbine and a specification of the electromotive voltage Eg. The amount of reactive power, Qrk, can be governed by use of the field current in order to match the need of reactive power in the grid. The turbine power, Pnnbmc, and the produced active power, Pg, in the source, will not be affected by this; only the active power delivered on the terminals, Pk. The electromotive voltage, Eg, and the terminal voltage, Ek, will be changed though as well as the reactive power in the source, Qrp.

4.2.1 Simulations compared with SIMPOWSIMPOW is a computer program for both static load flow analysis and transient behaviour in coordinated electric power systems. The program is, for instance, in use in ABB Corporate Research, SINTEF Energy Research and Statkraft. It is very well verified against measurements and is regarded as one of the best programs for transient calculations of large electric power systems. In order to test the developed computer program, and basically the electric part, the system shown in Figure 4.21, was simulated by ABB Research Corporate by use of SIMPOW [49]. Since SIMPOW is made for analysis on the electric grid level, the hydraulic part of the system is not very

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well represented. The whole conduit system is represented by means of one pipe constant, Tw, (inelastic theory). Mass oscillations and water-hammer effects therefore cannot be simulated. Likewise, the turbine is only represented by the opening degree multiplied with the pipe constant. The turbine torque will therefore not be calculated properly. On the other hand, SIMPOW will calculate the electric part in a more sophisticated way than the model I developed, especially with a view to saturation effects in the generator. The models of the speed governor and the voltage controller that I used , were not models in the standard module library of SIMPOW. Approximated standard models were used instead.

In Figure 4.22 the initial load flow condition of the electric system, used in the simulations performed by SIMPOW, is shown. No local load is initially present. As we can see, the generator consumes reactive power from the grid (underexcitation) with the chosen electromotive voltage, Eg.

P,= 152.62 MW Qr,=-2.55 MVAr

11.95 to/<0.174° Cos$k = 0.99986 P = 0.752°

Pg= 152.91 MW Qrfl = 22.16 MVAr E0=1Z1iW<0.334°

Terminal

Put* = 152.91 MW

Figure 4.22 Steady-state load flow condition of the electric system.Input data for SIMPOW calculations.

The simulation case is a sudden connection of an ohmic local load, connected directly to the terminals, of: ASiocai = 30 MW +10 MVAr, see Figure 4.23. It was believed that this load connection would result in a minor influence on the hydraulic part. In this way the electric part could be tested.

©—GD-190 MVA T, = 7.2 s

T30 + 110 MVA

Figure 4.23 Illustration of the local load connection.

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Figures 4.24 and 4.25 respectively compare the active power of the generator, Pk, and the turbine speed, n.

I ran my simulation model with an inelastic penstock and without the tunnel and the surge shaft, in order to verify the electric part as well as possible.

^—SIMPOW calculation (ABB Corporate Research)

------SmOaticn peformed by the author

Figure 4.24 Active generator power after a sudden local load connection.The developed computer program compared with SIMPOW.

1.0006

■1 SIMPOW calculation (ABB Corporate Research)

—Sumiafion peformed by the author

1.0002

0.9994

Figure 4.25 Turbine speed after a sudden local load connection.The developed computer program compared with SIMPOW.

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The agreement is excellent, with only small deviations. The opening degree of the turbine varied in a very small scale during the simulations (maximum 1.0009 and minimum 0.9995 of nominal value) and the influence of the turbine and conduits will therefor be very small, as first assumed. The load connection is very small, seen from the capacity of strong electric grid, and the unit will only show transient oscillations in the generator active power.

Concluding remarksThe results of the performed simulation case compared with SIMPOW is excellent. The electric part is therefore well verified The developed computer program will now be used to simulate larger load connections, in order to study possible influences of the electric magnitudes caused by dynamic behaviour in the conduits.

4.2.2 Local load connectionThe origin of the next simulations are shown in the static load flow sketch in Figure 4.26. This load case is a more natural than the one, shown in Figure 4.23. With a power factor of coscf> = 0.994, the generator supplies the electric grid with both active and reactive power. An initial local load of S = 30 + jlO MVA is connected on the terminals, and is included in Z^.

P,= 152.62 MW Qr,= 17.19 MVAr E, = 12.12 KV< 0.168°

Qrs = 41.55 MVAr E, = 12.5 KV < 0.3210

Terminal

= 152.91 MW

Figure 4.26 Steady-state load flow condition of the electric system.Input data for simulations.

Figure 4.27 gives an illustration of the load condition. Initially a local load of S = 30 + jlO MVA is connected on the terminals. A sudden load connection of S = 300 + J100 MVA (broken line) then appears at the 300 KV level

£>-<£>190 MVA T,=7.2s

V ▼30+j10MVA 300 + j100 MVA

Figure 4.27 Illustration of the load connections.

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In Figure 4.28 the transient performance of the active and reactive power of the generator are shown. The reactive power will increase tremendously in order to maintain the new electric balance in the system. Then active power only shows transient oscillations and will not increase in value, since the stiff grid is maintaining the active power balance. In fact, the active power will decrease, though in a very small scale, because the ability of producing active power has been reduced. The turbine power is increasing though, but in such a low scale that the opening degree will, during the governing, be kept inside a range of 1.009% to 0.9995% of nominal value. Transient effects from the conduits can therefore hardly be seen. A permanent speed drop in the speed governor would not uploaded the unit either, since the frequency in the system still is constant (this is though a model restriction).

Although, it cannot been seen on the figures (and probably not in real life either because of the dead zone in the speed governing system), very small oscillations in the electric magnitudes, caused by the mass oscillations, are still present. This mil be considered in the next section.

-0.7

- 0.6

S 0.8

-03

Time [see.]

Figure 4.28 Active (left axis) and reactive (right axis) power of the generator after a sudden load connection.

The pressure head in front of the turbine and the turbine speed are shown in Figure 4.29. As we can see, the transient deviations are very small and the system is almost stabilized after 10 seconds. Figure 4.30 presents the water level variation in the surge shaft. The surge shaft is well sized and the oscillations are very small, in the range of ± 0.2 cm.

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1.0025

■= 1.005 0.9875 g.

- 0.9725

Figure 4.29 Pressure head (left axis) and turbine speed (right axis) after a sudden load connection.

Figure 4.30 Water level variation in the surge shaft after a sudden load connection.

Concluding remarksAs shown in the above simulations, electric load changes in a stiff electric grid, will not affect the dynamics in the conduits in large scale. But as mentioned in Section 4.1.2, the strengthening of the grid and the capacity of the transmission line, have to be taken into consideration. In order to get observable influences from the conduits in the electric system, the origin of dynamics should come from the unit itself. In the next section will therefore a study of a turbine downloading be shown.

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4.2.3 DownloadingIn this section a simulation of an unit downloading is performed. The initial condition of the system is shown in Figure 4.26. Downloading and also uploading of units are treated with care and respect on plants. Large forces are acting and the water-hammer effects can be really high if the guide vanes close too fast. The mass oscillations that occur may also cause overflow in the surge shafts (also in the cable shaft/ventilator in connection with the draft tube hatch). Air suction in the system is also a threat, depending on how the system is dimensioned. In [32] some common dimension criteria are described. Downloading is performed by use of the load setting. This is described in more details in Appendix 1 (A1.3). In this simulation we are pushing the system by loading down the unit from nominal load, PtUrbm= = 152.9 MW, to Pmrbine = 67.6 MW (44% decrease). Note that the unit is connected to the electric grid all the time. The downloading is performed by closing the guide vanes from k = 1 to k = 0.5 for 7 seconds.

= 1.06

•0.7

Mass osculations

Figure 4.31 Pressure rise in front of the turbine (left axis) and theopening degree (right axis) during the downloading.

The opening degree of the guide vanes and the pressure head in front of the turbine are illustrated in Figure 4.31. We can clearly see the water-hammer effects in the beginning and the start of the long-term mass oscillation. The maximum pressure rise is approximately 1.11 times the nominal pressure. This satisfies a common dimensioning criteria of 15% of nominal pressure. The time period of the water-hammer and the mass oscillations are respectively 1.23 and 50.6 seconds, as calculated in Section 4.1.1.

The water level rise in the surge shaft is shown in Figure 4.32. The water rises approximately 3 metres above the stationary pressure level

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3

nm#[$#e.]

Figure 4.32 Water level variation in the surge shaft during thedownloading.

Figure 4.33 presents the active and reactive power of the generator. In order to stuefy the influences from the conduits, the results are compared with an equal simulation, with an inelastic penstock instead. The tunnel and the surge shaft are not included in the simplified simulation. Figure 4.33 shows the 10 first seconds. The deviations between the curves are relatively large and are mainly caused by water-hammer effects. After approximately 2.7 seconds we can see that there is nearly 10% deviation in active power between the inelastic and the elastic curve.

A zoomed picture of the time history from 30 to 80 seconds is shown in Figure 4.34 for the turbine power, the active and the reactive power of the generator. Oscillations with the same frequency as the mass oscillations in the tunnel can clearly be seen. The turbine power and the active power are oscillating in the same phase as the mass

! oscillations but the reactive power is 180° phase displaced. The difference between theturbine power and the active power is due to ohmic losses in the generator. The amplitude of the oscillations are approximately 0.61 MW and 0.06 MVAr. The size of the oscillations will be governed by the cross section area of the surge shafts in the conduits, and the damping will be governed by the tunnel friction. In a hydropower plant with several units and with a complex conduit system with many surge shafts, a superposition of the oscillations will give a gained pressure head in front of the turbines with higher frequency. The same might happen if some of the surge shafts have undersized cross section areas. This is illustrated in the next section.

!

1Ii

if

131

09

Masbc penstock.

Bsstic penstock and tunnelSurge deft inaldedInelastic penstock.

-■0.08Bastie penstock and tunnel.

Surge shaft inalded

- 006 %

-0.04

Figure 4.33 Turbine power and active power (left axis) and reactive power (right axis) during the downloading. Early time history. Inelastic and elastic conduits are compared.

0.1235

035801234

0.12335 0.356

0.1232 s2 0.355

0.1231 •

I 0353

0.123

0.1229

Time [see.)

Figure 4.34 Turbine power and active power (left axis) and reactivepower (right axis) during the downloading. Long-term time history.

The early time history of the turbine speed is shown in Figure 4.35. The terminal voltage is shown in Figure 4.36. Deviations between the inelastic and the elastic simulation can clearly be seen. A zoomed long-term picture is given in Figure 4.37 and

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mass oscillations can clearly be seen. The terminal voltage are in the same phase as the reactive power (180°), but the turbine speed is 90° phase displaced compared with the mass oscillations. The amplitudes of the oscillations are very small, but still present.

1.0003

1.0002Bastte penstock and troeL

Surpe shaft inaided

1.0001

S’ 0.9999

0.9997

Figure 4.35 Turbine speed during the downloading. Early time history. Inelastic and elastic conduits are compared.

Bastfc penstock and turretStrge shaft toaided

o 1.014Inelastic penstock.

Figure 4.36 Terminal voltage during the downloading. Early time history. Inelastic and elastic conduits are compared.

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10170310000015

1.017021.000001

1.01701

1.0000005

1.01698

1.01696

Time [sec.]

Figure 4.37 Turbine speed (left axis) and the terminal voltage (right axis) during the downloading. Long-term time history.

4.2.4 Undersized surge shaft cross sectionThis simulation case shows the consequence of having undersized surge shaft cross sections in the hydraulic system. It is a well-known design criteria to make all free water surfaces in the system larger than the critical Thoma cross section in order to avoid unstable mass oscillations. The Thoma cross section is in this case:

As,Tboma = 0.0085 m2a513H0

= 32 m2 (4.3)

Normally., a safety margin of 1.5 is used, so the cross section area becomes, As~ 50 m2. Even the free water surface in connection with the hatch shaft at the outlet of the draft tube, has to fulfil this criteria, theoretically speaking. In practice, a large throttling will be present, resulting in a large damping effect. If the cross section is reduced below the Thoma cross section unstable behaviour will occur in the shape of standing waves. Although, this depends on the tunnel length. Long tunnels may have a safety margin of 1.3. In Figure 4.38 the water level variation in the surge shaft is shown after the cross section in the surge shaft has been reduced As = 10 m2. The frequency of the oscillation will decrease with a periodic time of 23 seconds. Though the variations are in the range of centimetres, the damping is poor. The oscillations are reflected in the active and reactive power, as shown in Figure 4.39. We can see that the oscillations is significant.

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1.5

Figure 4.38 Water level variation in the surge shaft after a suddenload connection. Undersized surge shaft cross section.

0.8065

0.8064

0.8062

r 0.8061

0.6706 §§ 0.806

0.6705

0.8057

0.8056

Figure 4.39 Active (left axis) and reactive (right axis) power of the generator after a sudden load connection. Undersized surge shaft cross section.

Concluding remarksSimulations of a downloading have been performed in Sections 4.2.3 and 4.2.4 in order to study the interaction between the conduits and the electric system. It has been shown that both the water-hammer effects and the mass oscillations -will affect the electric system. Water-hammer is reflected in the electric outputs of the generator and causes relatively large transients. This should be taken into consideration when

135

new controllers are designed. Mass oscillations will be reflected in the electric grid and it is likely that this could be the reason for many unexplained power oscillations in the electric grid. In a coordinated grid, with several units interconnected, small mass oscillations might be superposed and then gained up to be a stability problem.

Taking the new and expected operating conditions into consideration, with more rapid and large up- and downloading, it is likely that the electric system will be a lot more affected by hydraulic transients than it is in today’s situation. This emphasizes the need of analysing both parts of the power system as one unit.

Finally, in the next two sections it will be shown how the new simulation program can be used to study short-circuit and synchronizing of turbine units. Such analysis could be valuable in the design phase of new turbines in addition to the traditional stability analysis on isolated grid.

4.2.5 Uploading with line rejectionUploading of turbine units has to be done in such way that air suction from the surge shafts is avoided. If a fast uploading is performed and a line rejection occurs in a bad moment, when the water level in the surge shaft equals the stationary water level (anti­phase), the next lower surge amplitude will be larger than the first one.

In practice, the uploading process is a long-term performance based on experience and rule of thumb. In most cases, the uploading process is done manually, with large safety margins. The procedure is described in Appendix 1 (A1.3). In the near future, with more rapid up- and downloading, this process should be automatized and minimized. In this simulation example a rapid uploading with a line rejection in anti-phase is shown. A line rejection or often called a load rejection is a common way of testing the stability performance of hydropower plant. In the electric terminology, this is the same as a short-circuit of the transmission line, a disconnection or a load shedding. The unit will in this case behave as it was operating under isolated load condition.

Figure 4.40 presents the active and reactive power and the terminal voltage. Immediate after the short circuit they will drop to zero. The rotor angle will be integrated towards infinity, and will in fact have no physical meaning or interest in an isolated load condition (the unit has fallen out of synchronism). The short circuit occurs after approximately 30 seconds. In Figure 4.41 the water level variation in the surge shaft is shown. We can see that the short circuit happens just when the water level passes the stationary hydraulic line, and the next oscillations becomes very large, approximately ± 8 metres.

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Figure 4.40 Active power and terminal voltage (left axis) and reactive power(right axis) during the uploading and the disconnection.

Disconnection

Figure 4.41 The water level variation in the surge shaft during the uploading and the disconnection.

The pressure head in front of the turbine and the opening degree are shown in Figure 4.42. The pressure drop in front of the turbine is relatively large during the uploading period. The extremely rapid uploading is done in 7 seconds from idling to nominal load. After the line rejection, the speed governor is disconnected and guide vane opening is decreased. The rapid closure results in a very high water-hammer in front of the turbine.

137

16

1 4

2S 12

S 06 2

Disconnection

b

S'1

I

02

K

0---------------- ----------------1-------------- 1---------------i---------------1--------------- 1-------------- 1---------------1-------------- H0 10 20 30 40 50 60 70 80 90

Time [sec.]100

Figure 4.42 The pressure head and the opening degree during the uploading and the disconnection.

The turbine speed is shown in Figure 4.43, both as a zoomed picture of the uploading procedure and a long-term view of the disconnection. During the uploading the unit is connected to the electric grid and the variations in the speed are moderate. After the disconnection, the turbine speed will increase tremendously towards the runaway speed. In order to get the speed back to nominal value, as fast as possible, in order to synchronize the unit, the speed governor is periodically disconnected and the load setting has to be turned down to idling position The performance of the speed governor, with its normal parameter setting, will be too slow. Manual governing or implemented functions in the governor, based on simulations and experience has to be used. Probably the biggest challenge a hydraulic cybernetic engineer meets is how to get control of the speed in this situation, as fast as possible. As we can see from the simulation case in Figure 4.42, the manual closure towards idling is performed in one section (ramp). Usually, this is performed in two or three sections with different gradients. During this operation the speed will be decreased and it will going through the transient braking zone (outside the steady-state turbine characteristics). This is called “down pressing” the speed. When the hydraulic forces are controlled, the speed governor is connected again and the unit can be synchronized again. It is possible to simulate the synchronizing of the unit with the developed computer program, this is shown in the next section.

138

1.002625

Runaway Down pressing

1.00225

1.001875

1.0015 1.08 •-?

2- 1.001125 1.06 £■

1.00075 1.04 H

1.000375Dtsco/mectibn

Time [sec.]

Figure 4.43 The turbine speed during uploading (left axis, thick line) and after disconnection (right axis, thin line).

4.2.6 Short circuit with sudden synchronizingA short circuit with a sudden attempt at synchronizing is shown in this section. The initial condition of the electric system is illustrated in Figure 4.26. When a short circuit (earth fault for instance) occurs on the transmission line the circuit breaker will immediate be switched off In practice, a sudden synchronizing is attempted. If the fault that caused the short circuit is removed (only a transient defect) the synchronizing may be successful The circuit breaker will be switched on in a tenth of a second. The attempt to synchronize the unit is usually performed twice. If the fault is still present, the short circuit is defined as permanent. Synchronizing is not possible and the unit exposes a line rejection. The unit has to be downloaded to the idling position before it can be reconnected.

Figure 4.44 gives the turbine speed and the terminal voltage after a short circuit with a synchronizing (reconnection) after 0.25 seconds. As we can see, the synchronizing was successful but the variation in turbine speed is very large, and in the range of 386 rpm to 367 rpm. In this case we are pushing the limit of a breakdown and a disconnection. Normally, a frequency protection unit is installed in connection with the large power stations. Usually, they measure the frequency over a time range of 10 seconds, before they take action. So, in this case the local frequency variations may causing some “flashing lamps” in the local area The active and reactive power of the generator are shown in Figure 4.45. The transients are very large. The pressure head is shown in Figure 4.46, with a transient peak of 5% of nominal value.

139

Act

ive

and

reac

tive

outp

ut [p

.u.)

Trub

lne

spee

d [p

.u.]

1.03 2

Short circuit with sudden synchronising

Figure 4.44 The turbine speed (left axis) and the terminal voltage (right axis) after a short circuit with synchronizing after 0.25 seconds.

Short circuit with sudden synchronising

Figure 4.45 The active and reactive power of the generator after a short circuit with synchronizing after 0.25 seconds.

140

Torm

ina!

vol

tage

[p.u

,]

1.04 • ••

1 1.01

8 0.99

Short circuit with sudden synchronising

Figure 4.46 The pressure after a short circuit with synchronizing after 0.25 seconds.

It is obvious that the synchronizing is on the limit of what the system can take. If we try to synchronize the unit after 0.27 seconds after the short circuit, the hydraulic forces have taken over the governing and the synchronizing fails and the unit will be disconnected. This can be seen in Figures 4.47 and 4.48 where respectively the turbine speed and the active power are shown.

7 1.12

Short circuit

Figure 4.47 The turbine speed after a short circuit with failed synchronizing after 0.27 seconds.

%

3

Short circuit

Figure 4.48 The active power after a short circuit with Med synchronizing after 0.27 seconds.

Concluding remarksSimulations in Sections 4.2.5 and 4.2.6 have shown the possibility of performing short-circuit analysis with synchronizing attempts, by means of the new computer program. Such analysis could be valuable in the design phase or during upgrading of turbines in addition to the traditional stability analysis on isolated grid.

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5. CONCLUSIONS AND FURTHER WORK

The results of the investigation, with conclusions and recommendations for further work are given in this final section.

5.1 RESULTS OF THE INVESTIGATION

The main contribution of this work is a theoretical analysis that, due to the expected conditions in the hydropower system in the near future (larger and more rapid load variations and introduction of new technology) emphasizes the needfor analysing the whole power system as one unit.

Mathematical modelling, leading to new and improved simulation programs for both Classic and Modem dynamic analysis of a hydropower system, are successfully performed in this work. Linear and non-linear models of a hydropower system where the hydraulic part and the electric part are connected together as one unit are established. The linear model is suited for stability analysis due to small and slow load variations, in the range of ± 10%. The non-linear model is capable of simulating all kinds of variations, from up- and downloading of aggregates to load variations in the electric grid.

New rational approximations are derived for the conduits with frictional damping included. Verifications according to the Method of Characteristics show very good results.

A one-dimensional analytic model of dynamic behaviour of high-head Francis turbines [33], has been improved in order to include the hydraulic inertia in the draft tube and the runner and to derive analytic expressions for the general loss-coefficients in the model An experimental study on a low-head Francis turbine test rig is performed in order to emphasize the presence of the hydraulic inertia and test the turbine model with a turbine that has a more unpredictable turbine characteristics than the turbine tested in [33]. The results of the investigation is satisfactory and show that the improved turbine model is capable of describing low-head turbines as well, by use of a correction factor accounting for the diffusion losses.

Simulations have been performed in order to identify the consequences of hydraulic inertia in hydropower systems. The results show that this may be of great importance during the transient runaway performance of turbines that have steep characteristics. The transient speed will be higher than expected and a phase shift, reducing the stability margins, is introduced to the system.

Interactions between the hydraulic part and the electric part of a hydropower system have been investigated by means of the new simulation programs. The results of the non-linear simulations show that interaction between the conduits and turbine on the hydraulic part and the electric part is present. Water-hammer effects may cause short­term variations in the generator power and mass oscillations are reflected in the grid with an equal time period as the water level variations in the surge shafts. The variations may be seen in the active and the reactive power as well as the terminal

143

voltage, though with different phase shifts. Successful verification of the simulation model are performed by means of simulations by the well-known computer program, SIMPOW. The results from the linear simulations show that an unstable turbine unit running under isolated load conditions is stabilized by connecting it to a stiff electric grid, on the assumptions of sufficient capacity of the transmission line. On the other hand will a weak interconnection to the electric grid make even a stable turbine unit running under isolated load conditions unstable.

5.2 RECOMMENDATIONS FOR FURTHER WORK

• Throughout this study I have always concentrated on how the developed simulation models could be verified in plants. Both hydraulic and electric magnitudes, and as many of them as possible, had to be measured. The key question was how the dynamics should be excited in the system. I was convinced that a given load change in the grid, or in the local transmission line, was the way of doing it. But after a lot of investigation work and visits to plants the idea seemed to be difficult to implement. This was mainly because a given load is difficult to separate from others and the size of it is hard to determine. After the study is finished, I think that the best way of verifying the model in a plant, is performing an up- or downloading of an unit while it is connected to the electric grid. Then we got control of the sourceof the dynamics. With known short circuit performance of the outside grid and known conditions of the transmission line and rest of the electric equipment in a plant, verification can be made. Because the planning and the commissioning of this kind of measurements is a long-term project, I never had the chance to do it. However, it would be advantageous that such a verification is performed in the near future.

• In order to study oscillations and stability conditions in the electric grid the developed non-linear model of the hydro-electric system should be extended. An unit connected to a stiff electric grid is regarded to be sufficient enough to study the behaviour on the unit level. Oscillations in the electric grid between units or groups of units followed by frequency variations cannot be studied by this model Since this thesis demonstrates that hydraulic transients will affect the electric system, it is well worth trying to extend the model to at least two units.

• The advantage of installing an ASD generator has been mentioned several times in the thesis. A considerable challenge in this connection is to provide a controller unit which can adjust the system to run on the best hydraulic efficiency by finding the optimal combination of the turbine speed and the opening degree, for a given load. This can only be solved properly by using a simulation model of the total power system. In order to achieve this, cooperation across the traditional disciplinaryboundaries in engineering is strongly recommended.

• The new and prescribed operational conditions of the hydro power plants, will force units to more rapid up- and downloading. Since today these procedures are often performed manually, and are built on large safety margins and experience, it will be interesting to optimize and control this process. In power stations with several units

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connected an automatic governor, mmimimg the uploading time is very relevant. The governor must be predictable and foresee any consequences of a sudden line rejection. It is necessary to implement an exact model of the total hydro-electric system in the governor unit, in order to estimate the consequences of the governing actions. The mathematical model must also be of such a size that it fits therequirements of a real-time governor. The challenge might be in the last statement.

t

145

146

BIBLIOGRAPHY

[1] Andresen, Trond: “Monovariable systemer og signaler”, Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 1996. (in Norwegian)

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147

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[20] Haugen, Finn: “Anvendt reguleringsteknikF\ Tapir Forlag, Trondheim,Norway, 1992.

[21] Horn, Hans Erik: “Multivariable controller for hydro-electric generating units”, Norwegian Institute of Technology, Trondheim, Norway, 1977.

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[23] Jaeger, €.: “Fluid transients in hydroelectric engineering practice”, Blackie and Son, 1977.

[24] Johannesen, Arne: “Videregdende analyse av elkraftsystemer”, Department of Electrical Power Engineering, Norwegian Institute of Technology (NTH), Trondheim, Norway, 1981. (in Norwegian)

[25] Johannesen, Arne: “Egenverdiberegninger”, Norwegian Electric Power Research Institute (EFI), Trondheim, Norway, 1983. (in Norwegian)

[26] Karlsen, Trygve: “Elekroteknikk’, Tapir Forlag, Trondheim, 1983.(in Norwegian)

[27] Kemgeter, J.: ”Rohrreibungsverluste einer oszillierende turbulente stremnung in einem kreisrohr konstanten querschnitts”, Mitteilung Nr. 95, Technische Univeristat Berlin, 1980.

148

[28] Li, Xin Xin, Skarstein, 0.: “An investigation of using feedforward control to improve hydraulic turbine governing”, Norwegian Research Institute of Electric Supply (EH), Trondheim, Norway, 1985.

[29] Li, Xin Xin: “Stability analysis and mathematical modelling of hydropower systems”, Dr.Ing. thesis, Norwegian Institute of Technology (NTH), Trondheim, Norway, 1989.

[30] Li, Xin Xin, Brekke, H., Nielsen T.: “State space model formulation of turbine regualting systems”, Contribution to the IVIAHR Work Group Meeting on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, Milano, Italy, 1991

[31] Nettum, Kristian H.: “Spenningsendringer i trefaseneti”, NKI forlaget, Oslo, 1986. (in Norwegian)

[32] Nielsen, Torbjom K.: “Dynamisk dimensjonering av vannkraftverk\ SINTEF report, Trondheim, Norway, 1990. (in Norwegian)

[33] Nielsen, Torbjom K.: “Transient characteristics of high-head Francis turbines”, Dr.Ing. thesis, Norwegian Institute of Technology (NTH), Trondheim, Norway, 1990.

[34] Nielsen, Torbjom K. and Rasmussen, Finn O.: “Analytic model for dynamic simulations of Francis turbines - implemented in MOC, Contribution to the XVI International Association of Hydraulic Research (IAHR) Symposium on Hydraulic Machinery and Cavitation, Sao Paulo, Brazil, 1992.

[35] Nielsen, Torbjom K.: “Dynamic behaviour of governing turbines sharing the same electric grid", Contribution to the XVIH International Association of Hydraulic Research (IAHR) Symposium on Hydraulic Machinery and Cavitation, Valencia, Spain, 1996.

[36] Nielsen, Torbjom K: “Frekvensregulering av turbiner ved store lastendringeh”, SINTEF report, Trondheim, Norway, 1994. (In Norwegian)

[37] NORDEL: “Rekommendasjon forfrekvens, tidsawik, regulerstyrke og reserve”, draft 19/1-1996. (In Norwegian)

[38] Ongstad, Erik: Telefax, ABB Kraft, 15/8-97

[39] Oppenheim A.V., Willsky, A.S.: “Signals and Systems”, Prentice-Hall International, USA, 1983.

[40] Paynter, H.M.: “Surge and water-hammer problems”, Electrical analogies and electroncic computers Symposium, Trans. ASCE, voL 118,1953.

i

I A

149

[41] Pejovic, S. et al: ’’Forms of pump turbine characteristics in water- hammer calculations”, Winter annual meeting of the ASME, 1983.

[42] Raabe, J.: “Hydropower”, VDI Verlag, Dusseldorf, 1985.

[43] Ramos, H., Almeida, B.: “Modelling and practical analysis of the transient overspeed effect of small Francis turbines”, Contribution to the XVIII International Association of Hydraulic Research (IAHR) Symposium on Hydraulic Machinery and Cavitation, Valencia, Spain, 1996.

[44] Rasmussen, Finn O.: Internal memo, Kvasmer Energy, 1996.

[45] Reindal, Morten: ”Analyse av rusningsforlop for en lavtrykks Francis turbin”, Diploma Thesis, Department of Thermal Energy and Hydropower, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 1997.(In Norwegian)

[46] Sannes, Tor: “Undersokelse av mulige forbedringer ved innfering av foroverkoblet turbinregulator”, Norwegian Research Institute of Electric Supply (EFI), Trondheim, Norway, 1986. (in Norwegian)

[47] Siervo, F. de and Leva, F. de: “Modem trends in selecting and designing Francis turbines’’, Water Power and Dam Constructions, August 1976.

[48] Siervo, F. de and Leva, F. de: “Linear mathematical models of hydraulic turbines for frequency regulation studies”, L. Energia Elettrica n.2,1977.

[49] Sporild, Roald.: SIMPOW simulations, ABB Corporate Research, 1997.

[50] Strugstad, Svein I.: ”Francis turbine model test’, Restricted SINTEF report, Trondheim, Norway, 1988.

[51] Stuksrud, Dag Birger: “Simulering av turbinregulering i tidsplanef’, Diploma thesis, Division of Thermal Energy and Hydro Power, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 1994. (In Norwegian)

[52] Stuksrud, Dag Birger: “Simulation of turbine governing in time domain”, Contribution to the XVHI International Association of Hydraulic Research (IAHR) Symposium on Hydraulic Machinery and Cavitation, Valencia, Spain, 1996.

[53] Stuksrud, Dag Birger: “Dynamic simulation of a governing turbine connected to a strong electric grid’, Contribution to the Japan Society of Mechanical Engineering (JSME) Conference on Fluid Engineering Towards the next Century, Tokyo, Japan, 1997.

150

[54] Stuksrud, Dag Birger, Nielsen, Torbjom K., Rasmussen, Finn O.: ”Analytic model for dynamic behaviour of Francis turbines”, Contribution to the VIE International Association of Hydraulic Research (IAHR) Work Group Meeting on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, Chatou, France, 1997.

[55] Svingen, Bjecmar: “Fluid structure interaction in piping systems", Dr.Ing. thesis, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 1996.

[56] Svingen, Bjomar: “Rayleigh damping as an approximate model for transient hydraulic pipe friction”, Contribution to the VIE International Association of Hydraulic Research (IAHR) Work Group Meeting on the Behaviour of Hydraulic Machinery under Steady Oscillatory Conditions, Chatou, France, Sept. 1997

[57] Vekve, Thomas: “Styring og mating av modellturbin i Vannkraftlaboratorief’, SINTEF report, Trondheim, Norway, 1996.

[58] Vennatro, Roar, “Hydraulic losses in transient and oscillatory flow",SINTEF report, Trondheim, Norway, 1996. (In Norwegian)

[59] Weber H.W., Zimmermann, D.: “Investigation of the dynamic behaviour of a high pressure hydro power plant in the Swiss Alps during the transition from interconnected to isolated operation”, 12th Power System Computing Conference, Dresden, August 1996.

[60] White, Frank M., “Fluid Mechanics", McGraw-Hill International Editions, 1988.

[61] Wylie, E. B., Streeter, V.L.: ”Fluid Transients in Systems", Prentice-Hall, Inc., 1993.

[62] Zielke, W.: “Frequency dependent friction in transient pipe flovF, Journal of Basic Eng., March 1968.

[63] Arthun, Dag: “CFD Stromningsanalyse av et lopehjul i modellturbin ved Vannkraftlaboratorief’, Diploma Thesis, Norwegian University of Science and Technology (NTNU), Trondheim, Norway, 1996.

151

Appendix 1A DESCRIPTION OF TURBINE GOVERNING SYSTEMS

This appendix presents a description of turbine governing systems. Figure Al.l shows a schematic drawing of a common turbine governing system. The purpose of the speedgovernor is to keep the angular speed of the turbine shaft equal to a reference value. The reference value is usually equal to the synchronous frequency in the electric grid (50Hz± 0.1 Hz in Norway).

The purpose of the voltage controller is to maintain the need for reactive power in the electric grid. This is an analogy to the speed governor, where the frequency is governed in order to maintain the need for active power in the electric grid. The terminal voltage is measured and compared with its set point. Besides the governor unit, a derivative and an internal stabilization unit are added to the system. A scientific committee called the “Derivat utvalget” recommended this in the early 1970s because of problems with oscillations in the electric grid in western Norway [16]. This way of controlling a turbine unit, with two separate governors in a monovariable system, is the most common way to arrange a turbine governing system today.

QServo

Fig. Al.l Common turbine governing system.

A1.1 ANALOG/ DIGITAL SPEED GOVERNORS

Governors made before 1985 were normally of the analog type. The oldest types were pure mechanical devices with fiyball mechanisms. New types are mechanical/hydraulic types or electric/hydraulic types, with operational amplifiers and condensers [6],

I

Today almost all the new turbine governors are digital, though most governors in use are of the analog type. A digital controller is a PLS unit or a computer with an AD/DA converter where the control algorithm is a software program. A digital governor has several advantages. For instance, the amount of mechanical and electric equipment will be reduced. Problems such as lost motion and hysteresis may be avoided. The parameter settings and the control routine may easily be changed, also from a remote control site. The software algorithm may also be used in simulations of the system during upgrading and analysis of the system dynamics.

Figure A1.2 illustrates a typical analog and digital governing loop.

M=Xs)Continuous

Ansiogcpvencr physical system

Dscrebs

converter'

Figure A1.2 Example of analog (upper) and digital (lower) governing loops.

It is necessary to be aware of the time delay, caused by the DA-converter, that a digital governor introduces to the system. This delay is equal to a phase shift and may cause reduced stability margins in the system. It is therefore necessary to choose the sampling time correctly. Instead of using classic continuous Laplace transformation, discrete Z-transformation and Q-transformation [3] have to be carried out to analyse the stability and performance of the system. According to the Nyquist sampling theorem, necessary sampling time has to be twice as small as the smallest physical time constant in the system [35]. In practice, the sampling time has to be a lot smaller than this to be able to reproduce the dynamics in a continuous signal A practical rule says that the sampling time should be less than 1/5 of the minimum time constant [1]. In Figure A1.3 typical time constants in a turbine governing system are given.

n

| Typical time constants in a hydropower plant'

i

Ts=1CLJ00sTa=4.JsTw=(l9.ut3sTr=(X1...1sTo=03...1sTy=O05..4X2sTd=O01..J13s

Figure A1.3 Typical time constants in a turbine governing system.

A1.2 SPEED GOVERNING TECHNIQUES AND IMPROVMENTS

Speed governing is usually arranged by a serial compensated loop with a PID governor, see Figure A1.4. Although for high-head Francis turbines and Felton turbines a PI governor is often enough to obtain good stability margins. The influence of the derivative term (D) is often needed for low head Francis turbines, Kaplan and bulb turbines, because they have rising characteristics or positive feedback.

nref Pelca.

Generatorinertia

Turbine/ conduit system

Time delay servo/elect/mech.

Figure A1.4 Common serial compensated governing loop with PID governor.

Figure A1.5 shows a complete block diagram for a Francis turbine speed governing system [4], In order to improve the stability margins and make the governing system more robust, several techniques have been investigated. The following sections give a brief introduction to some of them.

m

Figure A1.5

Block diagram

Francis turbine governing system

FRANCIS TURBINE CHARACTERISTICS STATE SPACE VARIABLESQ„ = Linearized flow In respect of the angular speed i\ ~ Servo strokeQy = Linearized flow in respect of the guide vanes opening y = Guide vane openingK = Linearized ratio between servo strike and guide vanes opening x = Output signal from PID governor JE = Linearized efficiency in respect of the flow Q = F*ovv through the turbineE„ = Linearized efficiency in respect of the angular speed h = Pressure over the turbineh/q(s) = Transfer function between pressure/flow nrti Reference angular speed

= Electric grid self governing constant n = Angular speedPmi.ch, = Active output of the turbine Pd[CI = Active output of the generator

SPEED GOVERNOR Tn = Derivative time Td = Integral time b, = Transient speed drop bp = Permanent speed drop

TIME CONSTANTST, = Generator time constantTy = Elcct./Hydr./mech. transm. time const.T„ = Servo/carriage time constant

Al.2.1 Generator power feedforwardIn the middle of the 1980s there was a discussion among hydraulic and electrical engineers whether a feedforward loop from the load disturbance (generator power) to the PH) governor would show improved speed governing. In Figure A1.6 the block diagram in Figure A1.5 is modified with the feedforward loop sketched. TheNorwegian Electric Power Research Institute (EFI) started two projects in order to investigate possible improvements of using feedforward. The motivation for this was obvious. Since the large mass of the unit will provide a time delay from the load change to the speed deviation and from the input signal to the guide vane movement, a feedforward loop from the disturbance would speed up the system response. In [28] a system shown in Figure A1.6 was investigated. This had a strictly simplified inelastic water column and no influence from the turbine characteristics. The proportional gain in the feedforward loop, bre, was choosen to be equal to 1. The conclusion of the investigation was that feedforward from the generator power can improve system performance considerably if load variations are the only disturbance in the system. Such a system is called a single unit connected to an infinite bus or a stiff electric grid. It means that the frequency in the electric grid is constant and not dependent on the loads. But in a real system with several units connected, the load disturbance is not the only disturbance in the system. The frequency in the electric grid is a disturbance too and as a matter of feet a result of the load change in the electric grid. It was therefore necessary to investigate this case further.

In [46] a model of two units connected to a transmission line was investigated. Otherwise, the system was equal to [28]. The conclusion of this report was that a speed governor with a feedforward loop would give poorer stability margins and system response in a system with several units connected. The reason for this, is the slow and low frequent oscillations between the units that is always present in such a system. A speed governor with feedforward will react rapidly and strongly to any disturbance because the disturbance it is based on local measurements. It might therefore get in anti-phase with the other governors in the system and the stability would be poorer.

Al.2.2 Water column compensatorWater column compensators have been suggested in many different ways. A feedback signal from the servo stroke is led through a compensator and back to the angular

speed reference, see Figure A1.6. The compensator is a mathematical linear function or transfer function of second or third order. The motivation for this control technique is to improve the turbine governing stability in hydropower plant with a long penstock. The advantage of using several feedback loops, so-called cascade control, instead of a single serial compensated system is well known. Because the governor is based on an additional measurement, the governing system will be more robust to any error in the mathematical model [20]. In [8] various proposed transfer functions for the compensator, C(s), have been discussed. The conclusion is that many approaches do not always improve the stability of the complete system and in many cases the compensators will cause an unacceptably high permanent speed drop.

V

Figure Al.6 Block diagram Francis turbine governing system with a feedforward loop from the generator power, 1/bff, a water column compensator, C(s), and a pressure feedback loop, ©(s), sketched.

VI

Pres

sure

Fee

dbac

k

Al.2.3 Pressure feedbackThe principle of pressure feedback control is described in [8]. The pressure across the turbine is measured and led through a transfer function and back to the angular speed reference, see Figure A1.6. This feedback system is, according to [8], the most advanced system for improvement of the stability of hydropower plants with a long penstock. The suggested transfer function is of third order and shows a substantial increase in the stability margins for Skjik Power Plant in Norway. This power plant has an extremely long penstock of2355 metres. The Svartisen Power Plant is the only one in Norway with pressure feedback arranged in connection with the speed governor, though it is not in use. Calculations show that the feedback system works so well, that it might even make a large compressed air accumulator in the penstock superfluous. The pressure feedback technique has not achieved a breakthrough in the market yet. The consequences of a governor failure may be the explanation. From this author’s point of view the technique seems to be very promising, especially in connection with future controllers of ASD generators.

Al.2.4 Multivariable speed/voltage governorFigure A1.7 shows a schematic drawing of the principle of a multivariable turbine controller. Only one control unit is used to receive input signals from the angular speed and the terminal voltage and send power signals to the servo and the excitation voltage system. From a general control technique point of view this should be the best way of controlling a turbine governing system. This technique was studied in the 1970s [21], but it never had any practical application. Some reasons for that may be [38]:

• The bandwidth of the two governing loops are quite different. The speed governing loop is approximately 0.5 -1 rad/s and the voltage control loop is approximately 20 - 50 rad/s. The difference is so considerable that the price of connecting them to one governor may be small

• The complexity of the system will increase and it might be difficult to maintain Physically, in hydropower plants it may be as much as 100 metres between the unit and the voltage controller.

• The only thing a multivariable governor can supply is an improvement to the electric grid stability. Stability problems in the electric grid have successfully been solved by an additional damping unit in the voltage controller.

• Usually, there are different suppliers of speed governors and voltage controller.

vn

nIEk

Generator

Turbine

Electric grid

Fig. A1.7 Multivariable turbine governing system.

Al.2.5 Water level/pressure controlControl of water level or pressure in the tailrace or penstock is a common method. The purpose is not to improve the governing stability. It is rather a protection that overrules the speed governing and will react like a disturbance. Although, it will affect the block diagram for the governing system. In Figure A1.8 the principle of water level/pressure control is shown [4]. The measured water level/pressure, hm, compared with a reference value and the deviation is fed through a PHD governor, each case the type of governor setting must be studied in order to obtain stability and normally a long integrating time is needed. The power signal from the governor is compared with the load reference and transferred to the load setting, where a correction of the angular speed is transferred to its reference value.

Figure Al.8 The principle of water level/pressure control

vm

& 8"

A1.3 SPEED GOVERNING IN A COORDINATED ELECTRIC GRID

The power transmission system in Norway consists of many sub-production and transmission sites. The system is a part of the synchronous Nordel system, which also includes Sweden, Finland and Zeeland (Denmark). Each turbine unit in the system, over a certain size, is supplied with a speed governor. A load change in the electric grid will be followed by an immediate frequency change. If the need for power in the grid decreases, for instance, the amount of power in the system will be too high. In a governing system there will be an instantaneous balance between production and consumption of electric power and the system has to get rid of the excess. The only way is to upload the units in the system and the electric power excess will be converted to rotational energy. If the need for power in the grid increases the situation will be opposite, the frequency in the electric grid will decrease because the shortage of power has to be taken from the rotational energy in the units. There is therefore a strong relation between the frequency and the power (active power) in the electric grid.

If several PID-govemors are connected to the same system, the governing stability would be impossible to maintain in a satisfactory way. It would not be clearly defined which of the units will respond to the load changes and the governors would start to work against each other and perhaps get in anti-phase. It is therefore necessary to provide each governor with a permanent speed drop function that defines the load change of each unit in percentages of the frequency change in the electric grid. In a hydraulic governing terminology the governor is called a PI governor with permanent speed drop. In an electrical terminology this is called a stationary P-type governor. According to [37] the permanent speed droop for an unit, bp, may be described as follows:

4f-iWiooAC.-A

(Al.l)

Af is the frequency change in the electric grid, Pmccm is the rated power of the turbine unit, & is the synchronous frequency (50 Hz in Nordel) and APm^* is the load change the actual unit will respond with. The permanent speed drop can be adjusted for each unit, normally from 0-10% in steps of 1%. An unit with a high permanent speed drop would respond to a frequency drop with a small load increase. This way of frequency/active power governing is called primary control. The result is a permanent deviation of the electric grid frequency after a load change, which is a disadvantage but a necessity for the system. In order to re-establish the frequency to its synchronous value a secondary control is necessary. While the primary control is fast control action with the purpose to keep the instantaneous balance between production and consumption, the secondary control is a slow control action performed in several different ways [2]. It might be performed both manually and automatically and it is governed from different control stations placed around in the system. The secondary control is performed by starting/stopping units and changing the settings of some units. Figure A1.9 shows a sketch of the frequency/active power governing with two units. A load connection, AP, results in a frequency drop in the grid. The two units will respond to this according to their permanent speed drop. The result of this is a permanent

IX

deviation that has to be corrected by the secondary control. Either one of units 1 or 2 or both of them have to cover the shortage to maintain the synchronous frequency.

T

AP,

I Primary Control b.i

y

Secondary Control?

1! 1

4

ap2

Primary Control

Figure A1.9 The principle of frequency/active power governing in a coordinatedelectric grid.

Each unit, that has a speed governor connected, has the speed reference adjusted to the synchronous frequency. According to instructions from NORDEL [37], a lot of different control actions can be performed if the electric grid frequency becomes too high or too low. Figure A1.10 illustrates some of them. In practice a high frequency is considered to be less dangerous than a low frequency.

f Disconnection of | hydropower plants

Speed governing Primary control

Figure A1.10 Frequency control actions.

X

Because of hysteresis, lost motion and slackness in the speed governing system, the turbine will usually have a dead zone of approximately 0.02% (new turbines). This means that a turbine unit will not respond with a load change if the frequency varies in the range of49.99 to 50.01 Hz. For older units the dead zone may be larger.

When it comes to the voltage/reactive power control, the sequence is almost the same as for frequency/active power control The possibilities of controlling the voltage, in the secondary control phase, is more manifold. Power electronic components such as capacitor batteries, transformers and SVC (Static Var Compensator) can for instance be used.

A turbine unit runs normally with constant load reference; yref = constant. The unit changes its loading due to frequency variations in the electric grid and adjusted permanent speed drop (traditional primary control). An unit can also change its production by adjusting the load reference manually (or often automized). This way of governing production is expected to be more common in the future, because of the new operation conditions that are determined at short notice.

In practice, the new load reference is led through an in-built unit that governs the actuator directly. The power of this unit is always a linear ramp-function that controls the speed of the load change. Normally, the slope of the ramp can chosen in the range of 10-150 seconds for a 100% load change. Large load changes are often performed in equal steps with an inserted break. The sequence is based on experience and simulations.

The procedure is equal during uploading of units. First the unit is uploaded from standstill to idling. The speed governor is connected when the speed has reached 90% of nominal speed. The unit is synchronized from idling position when the speed is equal to the electric grid frequency. During downloading or load rejection a quick closing device is connected and the speed governor is overruled. Normally, a fast closure down to half load followed by a slow one is performed.

XI

Appendix 2DERIVING THE ALLIEVI EQUATIONS

The Allievi Equations are fundamental equations for one-dimensional transient and elastic flow in a pipe. They form a base for almost all methods and techniques of water hammer calculations. Figure A2.1 shows a pipe with length, L, connected to two water tanks. H is the piezometric head and z is the elevation of the pipe above the datum line. The Equation of Momentum and the Continuity Equation for the pipe flow can be derived by means of a differential element, dl, marked on the figure.

Hydraulic---

Figure A2.1 System for deriving the Allievi Equations.

The Continuity EquationIn Figure A2.2 the differential element showed in Figure 1 is enlarged. Notations used in the deriving are indicated on the figure. The velocity, c, is defined positive in opposite direction of the direction, 1, as shown in the figure.

(Mass fiow)in - (Mass flow)0m = Mass Accumulation in the element

—in­i' . v\

* ji77ZH------dl&\ JJ

fin dm „ dm (A2.1)

m = pQ = pAc and m — pV = pA-dl

xn

(A2.2)\ <3 d d) d d d

Figure A2.2 Notations for deriving the Continuity Equation.

Assuming that the pipe is fixed in both ends and the axial deflection can then be neglected;

m=0d

Equation (A2.2) can then be expressed:

dp dA d: .dp 8AAc-a*pcn*pAn’‘Ai;*pi;^

pAcfdp , dA\ d \ p AJ + pA dc _pA(dp [ dA')

d d\p A) (A2.3)

Introducing the wave speed, a:

where pis the actual pressure. By introducing the flow, Q = Ac, the following equationcomes forth:

e.»+£l®+a=0^l-lSpA d A d d p d (A2.4)

xra

The change in pipe area pr. length unit:

<34a = -jzDtaxip -

^j/Y)__ 1_ <34 _ xD tan fi

a ~~~A*~a~ a2

The pressure:

4? (an a) ,a’4a"~aJ ^

(A2.5)

dp (dHdz\?=HaL|) (A2.6)

Where — = 0 3

A ^ .and — = sin aa

Inserting Equations (A2.5) and (A2.6) into Equation (A2.4). After rearranging the following equation comes forth:

30 = gAdH QgdH (4tanyg gsina^j a a2 a a2 a V D a2

(A2.7)

The term can be neglected due to its smallness for rigid pipes. By studyinga along tunnels and pipes the term taking into account the taper, , can be

D

neglected. Since a2» gsina, the term taking the inclination of the pipe, Qgsw.cc

may usually also be neglected. Then the Continuity Equation can be expressed:

3Q AgdHa ~ a2 a

By introducing dimensionless magnitudes, h = H/Ho and q = Q/Qo, the Continuity Equation can be expressed as follows:

dj _ AgH0 ai a a2Q0 a (A2.9)

Where Ho and Q0 may for instance be the values in the state of nominal (Best efficiency point) or rated (full load) operation.

Equation of MomentumIn Figure A2.3 the differential element showed in Figure 1 is enlarged and notations used in the deriving the Equation of Momentum are indicated on the figure.

XIV

T

z

Figure A2.3 Notations for deriving the Equation of Momentum.!

Equilibrium of Forces:

V- r dc m dQ

where Am is the average cross section of the element. The mass of the element can be expressed as: m = pV = pAm •dl

There are four kinds of forces, ]T F, that influence the free body.

1. Pressure difference.

A(pA) — pA — ^pA + —^dlj = dl

2. Shear forces.

Fr = nDmT-dl = T^D+^^-dljr-dl

Neglecting higher order terms, (dl)2 = 0: Ft =7tDz-dl

(A1.10)

(A2.ll)

XVt

,4V',WFWWt

.it"

(A2.12)

3. Gravity.

Fc = -mgsin a = -pgAm sin a-dl = A + ^j-dl^dl • sin a

Neglecting higher order terms, (dl)2 = 0: Fc = —pgA sin a-dl (A2.13)

4. Taper force.

Neglecting higher order terms, (dlf = 0: FK=p—dla

Equation (A2.10) can then be written:

= AH + Fr + Fg + Fc =

(A2.14)

(A2.15)

Inserting Equations (A2.ll), (A2.12), (A2.13) and (A2.14). After rearranging and simplifications the Equation of Momentum can be caressed:

Adp n , . dQ SO SQ—A---- h ttDt — pgA sin a = —p— =-------- O—d dt d d

(A2.16)

For small Mach numbers yields: ~ 0

By introducing the pressure, according to Equation (A2.6) the following equation can be established:

dH _ 1 ( tzDt + SO d Ag\ p d

(A2.17)

In dimensionless form:

di _ Q0 tiDt i dq|d AgH0 l Q0p d) (A2.18)

The Allievi Equations are partial differential equations.

XVI

Appendix 3DERIVING THE PRESSURE/FLOW, h/q, TRANSFER FUNCTION

In Appendix 1 the Allievi Equations, the Continuity Equation and the Equation of Momentum, is derived. They are partial differential equations. In dimensionless form they can be expressed as follows:

ckj _ AgH0 3i & a2Q0 a

at _ Qq (nDr [ dq |

a AgH0 kQop a)

Laplace transformation of the equations give:

a a Qo(A3.1)

(A3.2)

Introducing the friction parameter, k, in Equation (A3.2) :

(A3.3)

where:

zDt kDt(A3.4)

Qo9P QP

Differentiation of Equation (A3.3) with respect to l gives:

(A3.5)

Inserting Equation (A3.1) into Equation (A3.5):

(A3.6)

Introducing the complex friction variable, z:

xvn

(A3.7)

where:

(A3.8)

Equation (A3.7) is an 2. order ordinary differential equation, with the solution:

(A3.9)h = a/aJ +a2e w

Differentiation of Equation (A3.9) with respect to / gives:

Combining Equations (A3.10) and (A3.3):

a AgHo

? =AgH0 s Q0a z

Z (l V"-<z2e

v y

&

(A3.10)

(A3.ll)

The pressure/flow transfer function can now be found by means of Equations (A3.9) and (A3.11):

hq

QqO z AgH0 s

r\ + /3eVal

v y (A3.12)

where /? = —a,

Introducing the Allievi Constant:Qfi2^

(A3.13)

Then:

xvra

(A3.14)

The constant, (3, has to be determined by means of the boundary conditions of the system, see Figure A3.1.

Figure A3.1 Boundary condition at the right pipe end.

If the right end of the pipe is connected to a reservoir, then h = 0. The following boundary condition comes forth:

— = 0 for l = Lq

Using the boundary condition in Equation (A3.14) the constant, (3, is found to be equal to:

The transfer function between pressure and flow then becomes:

(A3.15)

The transfer function is non-linear and transcendental

XIX

Appendix 4FRICTIONAL DAMPING MODEL BY K0NGETER/BREKKE

This Appendix presents brief resume of equations and iteration procedure of theKemgeter/Brekke model for frequency dependent friction in pipes:

Pipe:— = -2h -tanhf—;

q s \az = Vs2 +Ks

h=0

Valve:

q = -^h-Qkqy^

Pipe + valve:

gA>w? = "-Mr)'1

h = -2(^ + Qykqyrtf)

- i-1 _J/ Qn ~ qn-1

q=

1 fqni==qni'1

i

Noj Yes * Plot q,,, hn

fflo+i=con+5(0

Kengeter:

T' D>Kongeter

Brekke:

=Hi) ^ ^t|K?)+/sin(!

Q>\q\coAD

- < 0.1446535

■=(2-665-73S)w5"®Kf)+,si,(f))- ^>0'1446535

XX

Appendix 5SURGE SHAFT MODEL IN THE FREQUENCY DOMAIN

In this Appendix a linear transfer function between the water level rise in the surge shaft, Z, and the flow down the penstock and through the turbine, Q, is derived. The water level rise will give an additional long term pressure rise in front of the turbine together with the short term water hammer flotations, between the turbine and the surge shaft. In figure A5.1 the mass oscillation system is shown. The surge shaft connects the upstream tunnel and the reservoir whit the downstream penstock and turbine.

Reservoir Surge shaft

Figure A5.1 The mass oscillations system; reservoir, tunnel, surge shaft and penstock.

Since the mass oscillations have large time constants, the inelastic effects in the tunnel can be neglected. The equilibrium of forces of the tunnel gives:

(A5.1)

Where the linearized loss coefficient, k, is given by:

,c ug 2go, a; (A5.2)

The assumption of a circular cross section of the tunnel is made. Laplace transformation of Equation (A5.1) gives:

(A5.3)

XXI

The water level rise in the surge shaft can be expressed as:

dZ = &dt As

Laplace transformation gives:

Qs — AsZs

(A5.4)

(A5.5)

The continuity in the branching gives:

a=a+a (*5.6)

Combining Equations (A5.5) and (A5.6):

Q,=Q+AsZs (A5.7)

Combining Equations (A5.3) and (A5.7):

z = -(Q + AsZs) t-s + k\gd,

1 s + a.O Ass2+als+a0

where:

(A5.8)

44

By introducing dimensionless magnitudes (Qo,Ho) and substitute the frictional loss coefficients, k, the following equation can be established:

Az _ s + aAq s2+as + P

(A5.9)

where:

Where Q0 and Ho are respectively the steady state values of the flow and the pressure head in front of the turbine.

xxn

Appendix 6METHOD OF CHARACTERISTICS (MOC)

In this Appendix the Method of Characteristics is described. Boundary conditions as constant reservoir and surge shaft are treated. In order to approximate the frequency dependent damping (dynamic or unsteady-state damping) the one-dimensionally Rayleigh damping model is added to the basic Allievi water-hammer equation. In order to describe the frequency dependent damping from a more physically point of view, a two-dimensional model is required. A one-dimensional model can not handle the variations in the velocity profile, which in fact is the origin of the frequency dependent damping. The Rayleigh damping model must therefor be regarded as a rough approach. In [55] and [56] the model is presented in details and compared with other damping models in the frequency domain. One of the greatest advantages is the possibility of solving the model in both frequency- and time domain.

In Appendix A2 The Allievi Equations are derived. In Equations (A2.8) and (A2.17) are respectively the Continuity Equation and the Equation of Momentum given.

cQ AgdH d a2 d

«.JLf^£+Sd p d)

The wall shear stresses can be expressed as [61]:

fpq MT 8 A2 (A6.1)

Where /is the Darcy-Weisbach friction factor. The connection between this frictionfactor and the well-known Moody factor, X, is given by:

!

A = 4/ (A6.2)

By introducing the shear stresses and the Rayleigh damping term, the Allievi Equations the become:

oQ AgdH d a2 d

dH_ 1 8Q , fM ctQ d Ag dt 2DgA2 pgA d2

(A6.3)

(A6.4)

The dynamic friction term is a function of the Reynolds number, wall roughness and viscosity. It can be found by experiment, in the same way as the steady-state friction factor,/.

xxm

The sets of Equations (A6.3) and (A6.4) can be rearranged as follows. Proof can be found in [55] and [56].

= o

a # a, 2d Ag d pgA dd

The transient friction term can be approximated by:

lf d>Q _ Xf (8Q 8Q pgA dd pgAAt V d d,^

(SQ A Q

PgAAt \ d Alt_u

(A6.5)

(A6.6)

(A6.7)

The set of equations (A6.5) and (A6.6) can now be written:

S+g^+/!de=o

a a 2 da

& , am SO Af AQ _ Q d Ag d pgAAt Al,^

(A6.8)

(A6.9)

Where the magnitude am is a modified wave propagation speed depending on the dynamic friction term and the time step.

am = Ja2 + —f~ (A6.10)V pto

The set is now hyperbolic and can then be transformed from partial differential equations to ordinary differential equations by integration along the characteristics, see Figure A6.1.

Figure A6.1 Characteristic grid.

xxrv

II f (A6.ll)

dh AamdQ^ am f\Q\Q Xf (Ag\

dt gA dt gA 2 DA pgAAtK At)(A6.12)

After rearranging and integrating the equations, the following algebraic relations are produced:

hp~Ha +-^{Qp~Qa)+&sQp\Qa\~ rd{Qb ~ Qa) = 0 (A6.13)

Hp Hb~(Qp-Qb)-RsQp\Qb\-Rd{Qb-Qa)=° (A6.14)

5 2gDA2 ’ 'a 2AlpgA (A6.15)

The time step is given by:

At = (-(A D/p)+^D/p)2 + (2oA/)2j (A6.16)

The traditionally MOC solving technique then becomes:

to ll (A6.17)

Cp = + BQEl + Rd(q£ - Qm) (A6.18)

cm=b$-bq£+rd(qz-or,1) (A6.19)

=£+^1^1 (A6.20)

Bm = B + Rs\Q%\ (A6.21)

(A6.22)

- -}UP

XXV

(A6.23)rx _ + CMBpBr + Bu

The notations (n-1) is used instead of (t-At) and i instead ofAJ* and B. In Figure A6.2 a simulation example is shown. The pressure in front of a instantaneous closing valve is calculated. In curve A the friction losses are neglected, in curve B the steady-state friction is included and in curve C the frequency dependent Rayleigh damping term is added.

Pressure in front of valve after momentanous closure

0.7 •

Figure A6.2 Pressure curves calculated by means of MOC. A: No friction losses;B: steady-state friction; C: frequency dependent damping.

Figure A6.3 presents a comparison of experimental and calculated pressure after instantaneous valve closure [58]. The experiment is performed on a 24 metre pipe with a diameter of 0.1 metre, connected to a tank with a constant water level 3.2 metres above the ground. The wave propagation speed is measured to be 1006 m/s and the flow 0.0009 m3/s. The calculated steady-state friction is X = 0.067. Adjusted to the dynamic friction term this is found to be: Xd = 7.5E5. The thin line is the calculated pressure with steady-state friction included, the thick line also includes dynamic friction and the broken line presents the experimental results. As we can see, the curve with dynamic friction gives a more correct picture of the pressure history than the curve with only steady-state friction. Because the modified wave propagation speed, am, is dependent on the time step, there will be a phase shift between the experimental and calculated pressure. This phase shift can be seen in the figure. Work is also going on in to determine an analytic expression to estimate the dynamic friction term, Xd. Nevertheless, the present approach of including dynamic friction in one-dimensional transient pipe flow solved with the MOC gives satisfactory results with a small amount of calculations.

XXVI

20

Figure A6.3 Comparison of experimental and calculated pressure. Thinline: no friction; Thick line: frequency dependent; Broken line: experimental.

Boundary conditionsA reservoir with a constant water level in the upstream left pipe end is usually an accepted boundary condition for hydro power plants. Figure A6.4 shows that the grid point at the boundary (P) only gets contribution from the downstream characteristics. The equations become:

Cu = H£-BQ£ +±rd(q$ -er1)BM = B+Rs\Qti\

H,=H0

(A6.24)

(A6.25)

(A6.26)

(A6.27)

Figure A6.4 Reservoir as a boundary condition at the left pipe end.

xxvn

In Figure A6.5 the characteristic grid points for a surge shaft as a boundary condition between to serial connected pipe ends.

i,= nx+1

Figure A6.5 Surge shaft as a boundary condition between two pipes.

The inlet pipe (supply tunnel) and the outlet pipe (penstock) have receptively index 1 and 2. The Continuity Equation and the assumption of constant pressure between ij and I2 makes it possible to establish a connection between the two pipes.

The compatibility equations for the two pipes are:

H = HlX = Ha = CFl — QllBPl = CM2 + Q,2^m2 (A6.28)

Solved with respect to outlet branching flow, Qa'.

Cpi ~Qi\Bp\ ~Cm—<2 - □ (A6.29)

The pressure head or water level variation in the branching can be expressed as:

dH,___ 1_dt As(x) 8s

Discretisation gives:

(A6.30)

(A6.31)

Here is As(x) the cross sectional area of the surge shaft, which in feet varies with the water level Therefor, a function connecting them together has to be established. In many cases the geometry of the surge shafts can be very complicated and approaches has to be used. In simulation programs it is beneficial to avoid sudden area transitions

xxvni

in the junction because of the numerical instability they may cause. Os is the discharge in the surge shaft. The Continuity equation yields in the branching:

(A6.32)

For throttled shafts and creek intakes the pressure head in the branching has to be corrected for the loss of kinetic energy [4]:

ff"=S+flr'5re;le:l (A6.33)

Where Kt is a constant and A, is the throttled cross sectional area.

By combining Equations (A6.28), (A6.29), A6(31) and (A6.32) we obtain an expression for the inlet branching flow:

460**2(C« ~ fC') + At{CPi - CM)

+ Bpi) ■*" BM2BplAs(x)(A6.34)

XXIX

Appendix 7LOAD FLOW ANALYSISOriginal: Per-Arne Edvardsen, ABB Power Generation.Modified and translated by Dag B. Stuksrud, NTNU.

In this Appendix a load flow analysis is shown for a synchrounos generator connected to a stiff electric grid (infinite bus) and the linearized constants for a 3. order machine is derived. A load flow analysis has to be carried out before the dynamics in the system can be excited in order to determind the initial state of the system. The voltage in the stiff electric grid is spesified and its refernce angle is set to zero degrees. On the terminal of the generator either the active and reactive power (PQ-generator) or the active power and the voltage (PU-generator) have to be spesified. The load flow analysis in this Appendix is based on a PQ spesification. Magnitudes in per unit (pu) are written in small letters and real values are written in capialized letters.

Data for generator:Sgen := 190-10*

Egen := 12-103

xq :=0.8

xqsub :=0.2

Data for transformer:St := 190-10*

Uts :=390-103

Utp := 12-103

ex :=0.12

er:=0.01

Data for the transmission line given in Cl:

Rl:=6 XI :=60

xd : = 1.2 TdOsub :=0.075 xf := 12

xdtr:=0.3 TqOsub :=0.25

xdsub :=02 fs := 50

ra :=0 cosd> :=0.85

Data for the stiff electric grid: Sk :=10-109

Un :=390-103

XgridUn2

"siTRgrid :=0

Data for local load connection: Ulokal -12-103

Plokal :=0

Qlokal :=0

XXIX

The rated output and voltage of the generator are choosen as a refemce for the pu-system.

Sbase :=Sgen

Ubase :=Egen

Zbase :=Ubase2

Sbase

In pu values:

un :=Un-P^-.. ■— Uts Ubase

zt:=(er+ex-j )Utp2 1

St Zbase

zl:=(Rl+Xl-j\Uts/ Zbase

zgrid :=(Rgrid+Xgrid-j )'lfUtp\2 1 [XJtsj Zbase

Equivalent impedance and voltage for a stiff grid in pu without a local load connection:

ze :=zt+zl-t-zgrid => ze =0.017495 +0.213951i

ue :=un => ue = 1

re :=Re(ze)

xe :=Im(ze)

The load flow analysis is carried out by means of the Newton-Raphson iteration method. Elements in the admittance matrix:

Yll :=— ze

Y12ze

611 :=arg(Yll) 612 :=arg(Y12)

Y21 Y22 :=— 621 :=arg(Y21) 622 :=arg(Y22)ze ze

Spesifies the voltage ue with angle 8= 0 (electric radians):

U2spes :=ue 82spes :=0

Spesifies active og reactiv power on the terminal:Plspes :=0.805789

Qlspes :=0.105263

XXX

Guessed start values for U og 8 (U1 and U2 are in real values):

Ul:= 1.021808 81 :=0.167702 U2:=U2spes 82 :=82spes

Starting the iteration loop:

PI :=Ul-Ul-i Yll |.cos(Sl-81-911) + Ul.U2i Y12 |-cos(Sl - 82- 912) => PI =0.805787

Q1 :=U1-Ul'j Yll |-sm(81 - 81 - 911)+U1-U2-| Y12 |sin(81-82-912) => Q1 =0.105264

Deviation from spesified values:

API :=Plspes-Pl

AQ1 :=Qlspes- Q1

Establishing the Jacobi matrix:

dP1/dU1= J21 :=U2j Y12 |.cos(51-82-912) + 2.UlJ Yll i-cos(911)

dP1/dd1= J22 :=-(Ul-U2'| Y12 | sin(Sl - 82-912))

dQ/dU1= Jll :=U2j Y12 |-sin(81 - 82- 912)- 2-Ul-| Yll !-sin(911)

dQ1/dd1= J12 :=UMJ2j YI2 |-cos(81 - 52 - 012)

J :=Jll J12

J21 J22

Deviation for U1 and 81:

AU1 AQ1:=r -

A81 _AP1_

AU1 =-2200686*10 7

A81 = 4.059036*10-7

New values for U1 and 81:

Ulny:=Ul + AUl => Ulny = 1.021808

51 ny :=81 + A81 => 51ny =0.167702

The iteration loop carries on until AU1 og A81 approxematly are zero.

The generator current og voltage can the be calculated:

u:=Ul-exp(Sl-j ) => u = 1.007473 +0.170557!

XXXI

1 u—ue i =0.794722 +0.030057*ze

Calculating the angle p between the reference vector ue and the q-axis: eaf :=u+(ra+xqj )*i => eaf =0.983427 +0.806335*

p :=arg(eaf) => p =0.686772

Calculating d- og q-axis components of the voltage and current6i :=arg(i) => 61=0.037803

id :=| i i-sin(p - 6i) => id =0.480646

iq :=| i |'Cos(P —6i) => iq =0.633614

ud:=ue-sin(P)+re-id-xe-iq => ud =0.506891

uq :=ue-cos(p)-f-re-iq+-xe id => uq = 0.887217

Linearized constants for the synchrounos machine:

uq' :=uq-f-(ra+xdtr-j ) id uqO' :=| uq' |

idO :=id iqO :=iq udO :=ud uqO :=uq po :=p

iO :=| i | uO :=| u | ©N :=2it-fs

MeO := iqO -(uqO' + (xq - xdtr) -idO)

A :=re -)- (xe+xdtr) (xe+xq)

KO :=—•( iqO •( xq—xdtr) •( re-udO + (xe+xq)-uqO) + ( uqO +- xq -idO ) -(re -uqO — (xe q- xdtr) -udO)) A

Kl :=—-fiqO-(xq-xdtr)-((xeq-xq)-sin(pO)-re-cos(pO)) ... IA [+(uqO'+(xq -xdtr)-idO)•((xe+xdtr)-cos( p 0 ) -t- re -sin( p 0 ) ) J

K2 ;=iqO-[l + (xq~ xdtrXxe+xq) j + (uq0^(xq_ xdtr).id0).£i

L A A

A+(xd - xdtrH xe q- xq)

K4 :-ue^xd~ xdtr)^(xe+Xq).sin(po)_ re-cos(pO)) A

K5 re-udO-h(xe-f-xq)-uqO)A

XXXII

1

a/( xf-(xd — xdtr))

K6 :=—1—-[re-xq-udO + [re2+xe-(xeq-xq)] -uqo] uO-A

^2.-xq-((xeH-xdtr)-cos(pO)-Hre-sin(pO)) + 212£..xdtr-(re-cos(pO)- (xe+xq)-sin(pO)) uO uO

K8 :=—1— -[[re2 + xe (xe+xdtr)] -udO2 + [re2 q-xe-(xeq-xq)] -uqO2 + re-(xq- xdtr) -udO -uqo] uO-A

K9 :=----- •(((xe-hxq)-sin(pO)- re-cos(pO))-idO + ((xe+xdtr)-cos(pO)-i-re-sin(pO))-iqO)iO-A

KIO :=—!—-(idO -(re-iidO + (xe + xq)-uqO) + iqO-(re-uqO- (xe+xdtr) udO)) iO-A

Kll :=----- •(idO •(xe+xq) q- iqO-re)iO-A

K12 :=^i-((uqO-xdtr-idO)-((xe+xq)-sm(pO)- re-cos(pO))-2-udO<(xe+xdtr)-cos(pO) + re-sm(pO)) A

K13 :=_-- * .IT,.nn2[uq02<xe+. xq) + ud02-(xe+xdtr) - udO•(uqO + xdtr-idO)-re] +- id0-uq0-[re2+xe{xe-hxq)] ...A ,+-[iqO -udO -[re2 -h xe-( xe+xdtr)]]

KI4 :=—-[uqO -(xe + xq) - 2 -udO -re + idO -[re2 +- xe-( xe -t- xq)]] A

D :=ue <oN* xdtr xdsub .Td0sub^sin(p0))2 +xq xc»sub.TqOsub-(cos(pO))2

(xdtr+xe) (xq+xe)

Calculated values for block-diagram representation:

KO =-0.04557 K1 =074914 K2 = 1.287889 K3 =0363621 K4 = 1.086297 K5 = 1.568036 K6 =0.375067 K7 =-3.47214«10~3K8 =0.38151 K9 = 1.353674 KIO =0.678625 Kll =1201962

K12 =0.102545 K13 =1.876135 K14 = 1.891491 Kr = 0.96225 D =30.995409 MeO =0.7218

xxxni