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Designing a Transmission Line Using Pareto Approach

Svetlana Beryozkina, Antans Sauhats Faculty of Power and Electrical Engineering

Riga Technical University Riga, Latvia

svetlana.berjozkina@gmail.com; sauhats@eef.rtu.lv

Viktoria Neimane Vattenfall Research and Development

Vattenfall Stockholm, Sweden

viktoria.neimane@vattenfall.com

Abstract—Transmission networks are facing an investment challenge due to the limited capacity of the existing power lines. Construction of the new overhead lines is associated with several legal, technical and economic problems. This paper describes Pareto approach for choosing the optimum design of the power line by minimizing the invested capital costs and maximizing the net present value. The analysis of presented solution including assessment of technical and economic efficiency is based on the model of a real line project. The results of the analysis are discussed in the paper.

Index Terms-Design engineering, electromagnetic fields, Pareto optimization, power transmission, transmission lines.

I. INTRODUCTION Power systems are among the most complex systems

created by mankind. They include hundreds and thousands of components: boilers, turbines, generators, transformers, transmission lines, etc. The functions of the components are interdependent: the processes going on in one part of the system influence the functioning of other elements. Interdependence of the elements determines the need to account it yet at the designing stage of the new energy facilities, for example, transmission lines. Impact of high or super high voltage transmission line on remaining parts of power system is significant, as well as changes in the system may substantially influence the transmission situation. The operating conditions are continuously varying in time – new customers and power plants appear, prices grow and power consumption changes. Costly elements and components of power systems have a finite life lasing for several decades. This motivates the need to estimate the conditions that may arise in a rather distant future. Clearly, these conditions cannot be predicted exactly; therefore, it is necessary to account for numerous impacting factors and conditions. The random and uncertain nature of these factors generates the complexity of formulating and solving the task of optimal design of expensive power objects, in particular, transmission lines.

Significant increase of electric power flows, caused by notable changes in the power system structure (competitive energy markets, development of large-scale wind power

farms) as well as a larger impact of technical and legislative restrictions on the allowable load current of the existing transmission and distribution network has been observed over the last decades. There is a need to build new high voltage overhead lines (OHLs) that requires considerable invested capital and causes a significant impact on the environment.

The goals to reduce invested capital, increase the reliability of the system, decrease the impact on the environment during the design, construction and operation of OHL are natural and clear. To achieve these objectives, new advanced power transmission and OHL design technologies were developed and implemented within the framework of the Smart Grid [1]. For instance, there are available High Temperature Low Sag Conductors (HTLS), which is one of the possible solutions for upgrading and improving the existing transmission lines [2], [3]. In this way, replacement of the conductors of the traditional type with HTLS conductors, using modern insulators [4], effective towers [5], the earthing and lightning protection systems [6] and monitoring and control systems may offer several technical and economic advantages like long-term reliability, higher capacity, low sag-tension property, easy and quick design and installation, low reconstruction investments. The powerful software is developed for simplifying the designing of OHLs [7], [8].

The most widely used methods of transmission line design are based on cost-benefit analysis and maximization of the net present value (NPV) [9] or minimization of capital costs subject to several restrictions.

The approach to find a minimum cost design of the transmission lines is described in [10]. Development of the methodology that explores the sensitivity of the required present worth of revenue in order to achieve design performance at minimum cost is presented in [11]. The OHL impact on the environment is significant, which causes the complication during the transmission line route selection (it is necessary to take into account the public and activity places both for people and for animals and birds) is taken into consideration in [12]. All the above-mentioned publications are based on the deterministic methodology.

A study based on game theory criteria and application of the stochastic approach for the network development is presented in [13].

This work has been supported by the European Social Fund within the

project «Support for the implementation of doctoral studies at Riga Technical University».

This work has been supported by the Latvian National Researchprogramme “Technologies for Innovative Production and Use of EnergyResources and Provision of Low Carbon Emissions by Means of RenewableEnergy Resources, Support Measures for the Mitigation of Environmental andClimate Degradation - LATENERGI”.

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This paper deals with the OHL design method, which also is partially based on a stochastic approach and game theory criteria used for the final decision-making.

II. THEORETICAL FORMULATION OF THE OPTIMIZATION TASK FOR THE DESIGN OF THE POWER TRANSMISSION LINE

In order to formulate the tasks of transmission line design taking into account the influence of random and uncertain factors, assume that:

• Company, the owner of OHL strives to minimize invested capital IC and increase its revenues Rti (the net cash flow i.e. cash inflow - cash outflow, for each year ti of the planning period T, T=m·ti).

• Revenues Rti and invested capital IC of the company depends on the structure Σj and the parameters Пj of the OHL chosen by this company. The parameters Пj are described by a set of continuous variables (the span lengths, coordinates for construction of the towers etc.). The continuous variables are substituted with the discrete variables for simplifying the task of the optimum solution searching (with a sufficiently small discretization step for providing the required accuracy).

• As a result, taking into account that the continuous variables have been changed into discrete and that the structures Σj also are described as a set of discrete variables (the number of wires per phase, the height and parameters of the standard towers, conductor area, the mechanical and electrical parameters of the conductor etc.) we can declare that for OHL realization it is possible to choose any combination of the structure and parameters. Each such combination is called alternative. Let us assume that N alternatives Aj are considered, j=1, …, N. The freedom of choosing the structures and parameters of the company is limited. Let Sj be the set of all permissible structures and parameters for OHL and sj = Σj , Пj as chosen combination of the structure and parameters, sj ∋ Sj. The frontier of a space of permissible structures is determined by the inequalities that describe the technical and legislative restriction and limitation.

• OHL functions under the influence of the ambient environment, which is characterized by a set of random and uncertain parameters Xti (the load current in a power line, the ambient temperature, the wind speed, the humidity, ect.). In general case, Xti are changing in time. We assume that during each year ti, Xti are constant.

Due to the influence of random and uncertain factors, the revenues are also uncertain. In this case, the scenario approach in combination with probabilistic variables can be used for solving the planning task. The uncertain information will be always modeled by a number of scenarios. For each scenario SCn (n=1, …, k) we can declare that the revenues Rtij are probabilistic values. Hence, we can write the following:

( ), (1)j tijntijn tijnR R s X=

Suppose that distribution functions that have the following form may be assigned to each combination of scenario SCn and alternative Aj:

( )( , , ) (2)j tijntijn tijn tijn jnF F R s X SC A=

If the distribution function is known, the expected value of the revenues for each year ti, for each scenario SCn and alternative Aj combination can be determined using the following equation:

...( , ) ( , ) ( ( , )), (3)E R s R s dF R sΩ

⎡ ⎤⎣ ⎦Χ = Χ Χ∫ ∫

where Ω – an integration area, which is limited by the existence space of the random parameters Xti. The frontier of the space of the allowable parameters and structures Sj is limited by restrictions:

( ), 0. (4)j tiFR s X >

In the equation (3) the Lebesgue-Stieltjes integral is used [14].

The knowledge of the expected values of the revenues allows finding the equation for the determination of the net present value (NPV), which serves as main criteria of an optimization:

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( ) (5)(1 )

m

Ci i

E RNPV II t=

= − + ∑+

where IC – invested capital, which is a function of sj; I is the discount rate.

Then the task for optimal planning can be presented in the following form:

*

1

( )arg max (6)(1 )

mCjn

i i

E Rs II t=

⎛ ⎞⎜ ⎟∑⎜ ⎟⎝ ⎠

= − ++

where arg stands for “the argument for” the subject of maximization.

Solving the task (6) will result in system’s structures and parameters s*

jn that maximize NPV for the planning period T for all selected scenarios SCn and alternatives Aj. For each combination SCn and Aj the evaluation of NPV can be determined. Realizing the NPV maximization process the extended space s*

jn is selected additionally with the purpose to determine alternatives and the values of parameters, which is near to the frontier of the allowable space and beyond its limits, that can be stored. Such parameters Pri will be reviewed as the additional criteria of an optimization. Having as a result the NPV and Pri values for all combinations of scenarios and alternatives, we can get the matrix, which is formulated in Table I. After that it remains to choose the best of the alternatives. In the given set of alternatives there are even the solutions that lead to violation of the most impacting restriction.

Each of the columns of Table I may contain Pareto set; if the indexes Pri consider the second optimization criteria (note

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that in general case the number of these indexes can be numerous).

TABLE I. EXPECTED VALUES OF NET PRESENT VALUE AND RESTRICTION INDICATORS RRI

The final decision-making based on the decision set in Table I in common case is a complex task, for solving which several methods are proposed [15]. When dealing with the specific tasks, as it will be illustrated below, significant simplifications may occur.

III. THE CHOICE OF THE SCENARIOS AND ALTERNATIVES The need to construct a new transmission line can be

revealed based on the forecasts of economic development for some geographic region. If there is a prediction of new loads or of the increase in the existing loads, new generation sources, inability to supply the demand of electricity using the existing network, a decision concerning the building of a new line can be made. Due to the impact of the uncertainties, there is a need to consider several scenarios, for instance, for the load increase ratio. After selecting the power system development scenarios, the numerous alternatives are examined with the purpose of choosing the optimization parameters Sj (see part II).

The following parameters can be the subject of optimization: the nominal voltage of a line, the mechanical and electrical characteristics of a conductor (conductor diameter, cross-section, the coefficient of linear expansion, modulus of elasticity, the allowable conductor temperature as well as the load current, the reduced specific load, the linear loads, destructive loads), conductor type, quantity in a phase; tower type, geometry, height, the allowable wind, weight and clearance spans; the insulator type, the damper type; the lightning wire type, the earthing and lightning protection systems; the optimum line route.

Thus, to achieve the best technical and economical solution of transmission line design, a large number of optimization variables need to be taken into account. As mentioned before, OHL is operated under the influence of the ambient environment that is characterized by a set of random and uncertain parameters Xti. The impacting parameters which describe the OHL operating conditions are the following: climate conditions (ambient temperature, wind speed, wind direction, solar radiation, air humidity, ice thickness); the capacity of the transmission line in the future power system; the forecast electricity consumption; expansion of the existing interconnection systems; environmental impacts.

To simplify a very complex and labour-consuming procedure of revenues calculation for vast number of the scenario/ alternative combinations, the task of filtering the limited number of the competitive variants may be formulated and solved. Until recently, this task was solved using the

accrued experience, which is described in the different guidances, glossaries, descriptions of the standard cooperative solutions and rules. In the last decades, the powerful software tools are used [7]-[8], which provide the opportunity to form a significantly smaller initial set of the competitive alternatives.

Additional diminution of the number of the competitive variants may be achieved by using a simplified form of the optimization task. In this case, the task is formulated as follows:

,arg min (7)ij CS I≈

The equation (7) is much easier to use than (6). Using a simplified formulation of the problem, the obviously non- competitive variants of a project are discarded. Note that considered problem (7) does not take into account the number of important factors (for example, the power losses in a line) and it can be implemented only for reducing the number of the options, which are subject to further consideration with the use (6).

Taking into account the above-said, the solving algorithm of an optimization task can be presented as a block scheme in Fig. 1.

Figure 1. The block scheme of the solving algorithm of an optimization task

IV. THE MAIN RESTRICTIONS DURING THE OPTIMIZATION The search for the best alternatives of OHL has to take into

account several restrictions: thermal (allowable conductor temperature, load current), mechanical (conductor sag, mechanical tension), electrical (insulation levels) and environmental (the climatic conditions, the impact of the electromagnetic field) [16]. During designing of the OHL it is necessary to provide adjusted maximum capacity (allowable load current), the ground clearance as well as the clearance to the crossed objects, and the distributions of electric and magnetic fields. In respect of the fact that a large number of optimization parameters are discrete, it can be argued that for solving the optimization task formulated in part II (6), it is necessary to perform investment calculations for all possible combinations of parameters sj corresponding to the area that meet the restrictions: sj ∋ Sj.

Alternatives Scenario 1 … Scenario k A1 NPV11|| Pr11 … NPV1k|| Pr1k

……. ……. … ……. AN NPVN1|| PrN1 … NPVNn|| PNk

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In general case, if one of the existing restrictions is not fulfilled, the corresponding alternative is not considered. In contradiction to that, this study allows to take into consideration the alternatives that violate these restrictions, because the development and implementation of new technical possibilities (different monitoring systems as well as the existence of HTLS) are giving additional opportunities to avoid the modes in which the restrictions are violated.

The most significant restrictions are described below.

A. Thermal Restrictions Commonly, thermal restrictions are defined by the load

current of the overhead line and the physical characteristics of the conductor type that prescribes the permissible conductor temperature. To check the fulfillment of the restriction conditions, the IEEC 1507 thermal rating method [17] is most widely used. Here the heat-cooling balance equation of the conductor is used as well as the numerous parameters that characterize the climate conditions.

B. Mechanical Restrictions Basically, the mechanical restriction is represented by the

conductor sag, the “clearing distance” to the ground as well as to the crossed objects like crossing roads, waterways, railways, and other (low-voltage, medium voltage, high voltage) power lines or telecommunication lines and buildings.

A special software program, for example, “SAPR LEP 2011” [7], may be used for a systematic calculation of the examined conductors as well as for the placement of towers on an actual profile of the designed line model, which was based on a terrain laser scanning data.

C. Environmental Restrictions The environmental impact is, first of all, the study of the

electrical and magnetic fields, which depend on the line voltage, the line current, the height of the conductors above ground, the height above ground at the point of field measurement, and the configuration of the conductors.

All the calculations of the strengths of electric and magnetic fields have to be performed by using appropriate software, for example, [18].

V. CALCULATION OF REVENUES AND NPV The NPV calculation is a difficult and very time-

consuming task. There are difficulties that are determined by the following main factors:

• The price of OHL elements must be taken into account, which will be defined “experimentally” by conducting negotiations with the stakeholders (the compensation of the land value for the OHL route, the cost of the towers and conductors, the conditions for getting the credit etc.)

• There is a lack of common recognized methodology of revenue, which occurs as a result of the OHL construction calculation.

In this study, let us accept the following hypotheses:

• The OHL construction is defined by the necessity of grid development for providing the planned consumption growth and energy transit;

• The construction projects are realized by a coalition, which consists of two companies – C1. The owner of the existing network and the owner of new facilities C2, in particular a new power line;

• The new construction provides receiving the additional annual revenue of transmission grid, which is equivalent to the normalized percentage (generally, 7-12 % [19]) of the average cost of energy transit;

• The dividing of total additional revenue Rad and the additional revenue of each company RC1 and RC2 is based on Shapley distribution, which in the case of two players is elementary simple [20]:

1 2 2(8)ad

C CRR R= =

In this study suppose that the additional income of the transmission grid is distributed among the existing network and newly built objects in an equivalent proportion, and then we get the amount of annual income:

, (9)2adl

incs

RIRI

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

=

where ( ) , (10)an antrad E ER β +=

where IS – annual investment in a transmission grid; Il – investment in the OHL construction; Ean – the annual growth of power consumption; Eantr – transit energy growth; β – rate of the cost of energy.

VI. CASE STUDY Consider an example of using the described methodology

of optimization of the transmission line design based on a real project, which will be realized in Latvia.

In the Baltic region (Latvia, Lithuania, Estonia, the west coast of Russia and Belarus) a design and construction of a number of the major power plans, including the nuclear power plant is performed. There is a number of projects including submarine cables and overhead lines, which are already realized or will be constructed in the near future for connecting with the power systems of Scandinavia and Poland. Power system analysis for scenarios of development of the region resulted in the following alternatives for the new transmission line:

• 110 kV transmission line with a maximum current of 1200 A (optimistic scenario No. 1) or 800 A (pessimistic scenario).

• 330 kV transmission line with a maximum current of 2000 A (optimistic scenario No. 2) or 1500 A (pessimistic scenario).

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There is a 118 km long overhead line, which has to be designed for the particular climatic conditions of the region near the Baltic Sea: a wind pressure of Q = 650 Pa, the thickness of ice and frost on the conductor C = 10 mm.

The optimization task of a structure and parameters of new transmission line was formulated. After searching the optimum solutions by using of the software for OHL design, 20 alternatives were selected, formed by implementing the various combinations of examined towers and conductors (see Fig. 3).

For simplifying the posed task at the first stage instead of NPV maximization according to the V part there is the minimization of invested capital. In the final comparison of the options the time-consuming task with the NPV criteria will be used.

Based on the construction experience of the power line of the given voltage and the interviews with the tower manufacturers, the two types of towers were accepted. There are two general technical solutions for the design of a line. Firstly, the use of the standard towers (the height to the lower conductor is 20 m) with combination of the different conductor types. Secondly, the same solution as for the first, but with a difference in the use of the higher towers (the height to the lower conductor is 22 m).

The height of the towers, the span lengths, the clearances between the conductors and the ground or the crossed objects define the parameters of the electric and magnetic fields, the values of which are regulated with the purpose of decreasing the impact on the environment. For simplifying evaluation of this impact, the calculations were done; the results are presented in graphic form (see Fig. 2). Verification of the restrictions was simplified due to existence of these results in the evaluation of alternatives.

Figure 2. Electric field distribution (Emax) at Y = 1.8 m above ground in the cross-section of a 330/110 kV line for minimum “clearing distances” of the 330/110 kV conductors: 8/10 m, 9/11 m, 9.5/11.5 m, 10/12 m, 11/13 m and

12/14 m

Fig. 2 shows that with minimum “clearing distances” of 8/10 m for the 330/110 kV circuits, the strength of the electric field value Emax reaches 8.8 kV/m. In this case the value of Emax is less than 7 kV/m with minimum “clearing distances” of the 330/110 kV circuits equal to 9.5/11.5 m. However, with an

acceptable increase in the line voltage by 10%, when the circuit voltage is 363/121 kV, the electric field value Emax for minimum “clearing distances” 9.5/11.5 m will increase to 7.5 kV/m, whereas for 10/12 m it becomes equal to 7.0 kV/m.

Thus, when there is an acceptable 10% of overvoltage, the condition Emax ≤ 7 kV/m (CENELEC) [21] is fulfilled with minimum “clearing distances” of 10/12 m, and the condition Emax ≤ 5 kV/m (ICNIRP) [22] – with minimum “clearing distances” of 12.5/14.5 m.

Summing up calculation results of the environmental restrictions, it can be concluded that the strength of the electric field may not satisfy the necessary requirements. To solve the posed problem, the solution described below was proposed.

Therefore, the following groups of combinations were selected: V1 and V10; V11 and V20 mean that there are the traditional type conductors for both 110 kV and 330 kV circuits; V6 ÷ V9; V16 ÷ V19 – there are the HTLS conductors for both 110 kV and 330 kV circuits; V2 ÷ V5; V12 ÷ V15 – there are the traditional type conductors with a combination with the HTLS conductors for 110 kV and 330 kV circuits (see Fig. 3).

Figure 3. The diagram of Pareto set for the combination variants

Fig. 3 reflects the optimum solution results of two-objective optimization by implementing Pareto approach [23], where IC (p.u.) is the invested capital and E (kV/m) is the strength of the electrical field. As a result, the competitive variants are V6, V7, V9, V10, V15, V17, V19 and V20.

The resulting values of the optimization criteria are summarized in Table II (only competitive alternatives are left for further consideration). Analysis of the table reveals the following:

1. If the “classical” problem formulation, namely, maximization of the NPV, complying with all the restrictions would be used, one of the alternatives V10 would be chosen. The final decision would be taken using one of the criteria in [16].

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2. If constraint violation is allowed, one of the alternatives V17 or V19 should be chosen.

TABLE II. EXPECTED WORTH OF NET PRESENT VALUE AND RESTRICTION INDICATORS RI

The risks of appearance of conditions leading to changes in operation in its turn leading to additional economic losses were estimated in order to make the final decision. Violation of limits for the strength of electric field for the considered example for Latvian climate can arise at the maximum air temperature of +35°C and the maximal load current. The combination of these two conditions has a very low probability; for the alternative V10 it is 0,01, while for alternative V17 0,000001 (both probabilities are estimated considering the optimistic scenario). After taking into consideration the probabilities mentioned above the alternative V17 was chosen as a final decision for real-life implementation, besides, to satisfy the required legislative conditions for the electric field, a permanent monitoring and operation is needed.

VII. CONCLUSIONS The latest tendencies of development of power systems –

deregulation of the electricity market, introduction of new technologies and increase in distributed and renewable generation – considerably influence the process of transmission network planning. The new methods are required to be able to facilitate decision-making resulting in reduced invested capital and power losses, and improved reliability and power quality.

The problem of optimization of the design of transmission lines is a multi-criteria optimization task with a large number of state and decision variables. This task has to be formulated taking into consideration the possible influence of random and uncertain parameters. The solution of such problem is associated with considerable mathematical, computational and informational difficulties.

Utilization of the multi-criteria optimization partially based on the stochastic approach provides the possibility to consider even the alternatives with constraint violations, which with some additional measures can represent the most cost-effective solution.

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Alternatives Optimistic scenario 1 Pessimistic scenario 2

NPV11||E11 NPV11||E11 V10 4.7 // 6.9 5.4 / 5.0 V17 6.2 // 7.1 5.33 / 5.6 V19 6.7 // 8.0 5.7 / 7.3