Energy Conversion and Management 78 (2014) 306–315

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Energy Conversion and Management

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Improved gravitational search algorithm for parameter identificationof water turbine regulation system

0196-8904/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2013.10.060

⇑ Corresponding author. Tel.: +86 18907198929.E-mail address: yxh71@163.com (X. Yuan).

Zhihuan Chen, Xiaohui Yuan ⇑, Hao Tian, Bin JiSchool of Hydropower and Information Engineering, Huazhong University of Science and Technology, 430074 Wuhan, China

a r t i c l e i n f o

Article history:Received 10 June 2013Accepted 23 October 2013Available online 21 November 2013

Keywords:Water turbine regulation systemParameter identificationGravitational search algorithmParticle swarm optimizationChaotic mutation

a b s t r a c t

Parameter identification of water turbine regulation system (WTRS) is crucial in precise modeling hydro-power generating unit (HGU) and provides support for the adaptive control and stability analysis ofpower system. In this paper, an improved gravitational search algorithm (IGSA) is proposed and appliedto solve the identification problem for WTRS system under load and no-load running conditions. Thisnewly algorithm which is based on standard gravitational search algorithm (GSA) accelerates conver-gence speed with combination of the search strategy of particle swarm optimization and elastic-ballmethod. Chaotic mutation which is devised to stepping out the local optimal with a certain probabilityis also added into the algorithm to avoid premature. Furthermore, a new kind of model associated tothe engineering practices is built and analyzed in the simulation tests. An illustrative example for param-eter identification of WTRS is used to verify the feasibility and effectiveness of the proposed IGSA, as com-pared with standard GSA and particle swarm optimization in terms of parameter identification accuracyand convergence speed. The simulation results show that IGSA performs best for all identificationindicators.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The engineering simulation study largely depends on the typeof the model used. More accurate model leads to more accurate de-sign and application. WTRS which is used to control frequency andoutput power of hydro-turbine generator sets is one of the mostimportant parts of hydropower plant. This complicated system ismade up by water turbine speed governor, guide servomotor,water turbine and generator, in which mechano-electric dynamicsand hydrodynamics are all involved [1–8]. As the complex charac-ters of regulation system, it is difficult to extract features from ac-tual prototype turbine directly for modeling simulation andcontrol, and hence parameter identification technique which isused to get accurate simulation model and parameters throughexperiments data and priori knowledge is widely studied for WTRSsystem in the past decades. Kishor used the neural network nonlin-ear autoregressive approach on the elastic character and inelasticcharacter water turbine column respectively and successfullymodeling the pipeline [9,10]. Xiao et al. established RBF neural net-works models to identify water turbine generating unit [11] andget a good result. An improved genetic algorithm (GA) isintroduced and applied to the identification problem of hydrogeneration system model with fluid transients [12], which obtain

higher parameters accuracy. In [13,14], particle swarm optimiza-tion (PSO) is brought in and used for WTRS system model andhas been proved to be effective in handling the identificationproblem. Kou et al. [15] proposed a new approach of bacterial for-aging optimization algorithm (BFOA) and simulation results showthat the tested and the estimated outputs are in favorableagreement.

However, for the complicated characters of WTRS, which thelinear model is a non-minimum phase system, these methods usu-ally have some shortcomings in some extent. Artificial neural net-works (ANN) are easy to handle but hard to establish as thedifficult in choosing sufficient and accurate training data, suitabletraining algorithm, number of neurons in the ANN, number ofANN layers, etc. [16]. As an early proposed swarm intelligencealgorithm, some bad characters of GA and PSO such as trapped inlocal optima in possible and premature phenomenons are gradu-ally emerged in deeper research [17–21]. Bacterial foraging optimi-zation algorithm has little possibility tracking in local optima butthe rejection and attraction computation between Escherichia colibacteria is big and need a long time [22]. Therefore, a simplifiedand easily implemented approach is especially reasonable for theidentification problem of WTRS.

Recently, a newly developed evolutionary computation optimi-zation called gravitational search algorithm (GSA) [23,24] which isbased on the law of gravity and mass interactions has beenverified high quality performance in solving different optimization

Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315 307

problems. Behrang et al. applied GSA to solve the future oil de-mand forecasting problem [25] and successfully to estimate Iran’soil demand based on the structure of the Iran socio-economic con-ditions, Güvenç et al. present the effectiveness and robustness ofGSA in solving the combined economic and emission dispatchproblem under various test systems [26], Duman et al. used GSAon the optimal power flow and optimal reactive power dispatchproblems in power system [27,28], which obtained a high-qualitysolution compared to PSO.

However, like others stochastic algorithm, classical GSA alsomay suffer from premature and local convergence problem, espe-cially in large scale and complex problems. In order to improvethe performance of classical GSA, numerous variants have been re-ported. Tsai et al. combined PSO velocity and GSA acceleration intoa hybrid variant [29]. Khajehzadeh et al. developed GSA throughintroducing an adaptive maximum velocity constraint to the stan-dard GSA algorithm [30]. Li et al. proposed a kind of piecewise-based gravitational search algorithm by designing a piecewisefunction as the gravitational constant function [31]. Kumar et al.introduced a new strategy which tunes the gravitational constantusing fuzzy ‘‘IF/THEN’’ rules [32]. Han and Chang [33] used a cha-otic secure communication scheme on the GSA to minimize pre-mature convergence. Ghasemi applied fuzzy-based mechanisminto GSA to figure out multi-objective problems [34]. Mallicket al. [35] proposed a hybrid PSO–GSA based algorithm on the sta-tic state estimation problem.

Inspired by the thoughts of adding a new component or distur-bance in classical GSA [29–35], an improved GSA is proposed inthis paper, which is conceptually more concise and performs moreeffectively than some GSA variants. In the proposed IGSA, we makethree improvements: firstly, the speed of each agent is not onlybased on the law of gravity but also impacted by information ofbest particle; secondly the off-boundary agents are treated witha novel strategy, which increases diversity of the agents; finally,a chaotic mutation operator is incorporated into the searching pro-cess ensuring any feasible domain visited.

A mathematical model of WTRS associated to prototype turbineis taken into consideration. This model keeps an eye on the differ-ent values of water turbine coefficient under frequency distur-bance condition and load disturbance condition, which is moresuitable for the analysis of power system. In addition, distinguishfrom the model in [1–8], the newly model considers water turbinein three parts with engineering practice, which makes the modelmore close to the real turbine.

The rest of this paper is organized as follows: In Section 2, WTRSmodel is introduced, and the parameter identification problem forthe model is formulated. GSA algorithm is introduced and im-proved in Section 3. Section 4 illustrates the general structure ofusing IGSA approach to solve the identification problem. The com-parative experiments are designed and the results are discussed inSection 5. The conclusion is drawn in Section 6. Acknowledgmentis given in the end.

2. Model of WTRS

WTRS is a complicated system, mainly contains four parts, i.e.speed governor, servomechanism, hydraulic system, generator sys-tem. The structure of WTRS is illustrated in Fig. 1. In Refs. [1–8], thesimulation of WTRS system is either modeling in an ideal situationor overly-simplification, which is not suitable for the electricitysystem dynamic process modeling and analysis, especially whenthe load of HGU is in a fluctuated and transient process. In this sec-tion, a kind of WTRS model with respect to the engineering prac-tice has been studied and component models of WTRS areilluminated respectively.

2.1. Model of speed governor

A major of micro-regulator are PID controller in the world. ThePID controller in water turbine could be expressed as [3,8,36]:

rðsÞ ¼ 11þ bp � ki

s

� kpþ kisþ kd � s

1þ Td � s

� �� ðcðsÞ � xðsÞÞ ð1Þ

where kp is the proportional gain, ki is the integral gain, kd is thedifferential gain, s is the Laplace operator, Td is the differential timeconstant, bp is the feedback coefficient, c(s) and x(s) are the Laplacetransform of given speed c(t) and generator unit speed x(t).

2.2. Model of servomechanism

Servomechanism is the actuator of water turbine. The control-ler’s output signal changes into hydraulic signal by the mechano-electric servomechanism converter and then gradually strengthenin turn across the guide device, auxiliary servomotor and main ser-vomotor for providing enough power to operate guide vane. Thetransfer function of servomechanism could be shown as [1–8]:

yðsÞ ¼ 1Ty � sþ 1

rðsÞ ð2Þ

where y(s) is the Laplace transform of guide vane opening signaly(t), Ty is the inertia time constant of servomechanism.

2.3. Model of hydraulic system

Hydraulic system is the key component in WTRS, and it is a verycomplicated system with multi-parameters, time-varying and non-minimum phase characteristics. There is not any analytic expres-sion to describe this system until now. However, associated tothe engineering practice [36], it usually can be considered intothree parts:

(1) The fluid characteristics of penstock pipeline. Water pen-stock is taken to be incompressible if penstock is short ormedium in length, the transfer function of penstock systemis [3,8,36]:

hðsÞqðsÞ ¼ �Tw � s ð3Þ

where Tw is the water time constant, h(s) is the Laplace transform ofwater head signal h(t), q(s) is the Laplace transform of turbine flowrate signal q(t).

(2) The characteristics of hydro-turbine. In small signal perfor-mance condition, we describe the fluid character and torquecharacter of hydro-turbine as following [3,8,36]:

mtðsÞ ¼ ex � xðsÞ þ ey � yðsÞ þ eh � hðsÞqðsÞ ¼ eqx � xðsÞ þ eqy � yðsÞ þ eqh � hðsÞ

�ð4Þ

where mt(s) is the Laplace transform of generated water torquemt(t) throughout the water-hammering action, ex, ey, eh, eqx, eqy,eqh are the partial derivatives of turbine whose calculation methodsare introduced in Ref. [36].

(3) The character of turbine rotor and mechanical inertia ofhydraulic plant. It can be accounted as a part of generators’mechanical inertia.

2.4. Model of generator system

Generator system in WTRS can be considered into two parts,namely synchronous generators and load in grid. The model of syn-chronous generators is often simplified as a one-order system, inwhich HGU is considered as rigid body with a certain rotated

Generator systemHydraulic system

Speedgovernor

Servo-mechannism

Penstock

Waterturbine Generator

Load

Fig. 1. Block diagram of water turbine regulation system.

308 Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315

inertia. The transfer function of synchronous generators could beshown as [1–8]:

xðsÞmtðsÞ �mgðsÞ ¼

1Ta � sþ ðeg � exÞ ð5Þ

where mg(s) is the Laplace transform of load torque mg(t), Ta is theinertia time constant of generator, eg is the adjusting coefficient ofgenerator.

The model of load in grid also has a certain rotated inertia,which includes the rotational inertia of various electromotor andelectric drive system. The function of inertia time constant of loadis similar to the function of inertia time constant of generator,which is always recorded as Tb. Thus, Eq. (5) can be modified as fol-lows once we considered the impact of load character.

xðsÞmtðsÞ �mgðsÞ ¼

1Ta0 � sþ ðeg � exÞ

ð6Þ

where Ta0 = Ta + Tb, the value of Ta and Tb is usually measured bymeans of the simulation experiments. Based on previous experi-mental data, Tb = (0.24–0.30)Ta.

The transfer function of WTRS is shown in Fig. 2 [36]. FromFig. 2, it can be found that WTRS is a high-order, multi-inputsand multi-outputs (MIMO) system. In order to obtian the relation-ship of different inputs and outputs, state space analysis is used inthis section. To easy building the state equations, the PID controller

in Fig. 2 makes the following changes: kd�sTd�sþ1 ¼ kd

Td�kd=Td

Td�sþ1 and the

first part kdTd is incorporated into the proportional gain kp. Selecting

x(t), xd(t), xi(t), y(t), h(t) as state variables (i.e. state vector

X ¼ ½ x xd xi y h �T , the parameters in X are marked in Fig. 2)

Fig. 2. Mathematical simulation of water turbine regulation system.

and c(t), mg(t) as input variables (i.e. input vector u ¼ ½mg c �T ,the parameters in u are marked in Fig. 2), the state equations forWTRS could be deduced as:

_X ¼

a11 0 0 a14 a15a21 a22 0 0 0a31 a32 a33 0 0a41 a42 a43 a44 0a51 a52 a53 a54 a55

26666664

37777775

X þ

b11 00 b220 b320 b42

b51 b52

26666664

37777775

u ð7Þ

where

a11 ¼ ex� egTa0

; a14 ¼ eyTa0

; a15 ¼ ehTa0

a21 ¼ � kd

Td2 ; a22 ¼ � 1Td

a31 ¼ ki � bp � kpþ kdTd

� �� 1

� �; a32 ¼ ki � bp; a33 ¼ �ki � bp

a41 ¼ � kdþ kp � TdTd � Ty

; a42 ¼ � 1Ty; a43 ¼ 1

Ty; a44 ¼ � 1

Ty

a51 ¼ eqx � ðeg � exÞTa0 � eqh

þ eqyTy � eqh

� kpþ kdTd

� �; a52 ¼ eqy

eqh � Ty;

a53 ¼ � eqyeqh � Ty

a54 ¼ 1eqh� eqy

Ty� eqx � ey

Ta0

� �; a55 ¼ � 1

eqh� eqx � eh

Ta0þ 1

Tw

� �

b11 ¼ � 1Ta0

; b22 ¼ kd

Td2 ; b32 ¼ ki� ki � bp � kpþ kdTd

� �;

b42 ¼ kdþ kp � TdTy � Td

b51 ¼ eqxeqh � Ta0

; b52 ¼ � eqyeqh � Ty

� kpþ kdTd

� �

3. The improved gravitational search algorithm

3.1. Brief introduction of GSA

GSA is a newly developed stochastic search algorithm based onthe physical law of gravity and the law of motion. In this new ap-proach, a set of agents has been proposed to find optimum solutionby analogy of Newtonian laws. Agents are considered as objectsand their performance are measured by their masses, and these ob-jects attract each other by the gravity force, while this force causesa global movement of whole objects towards the objects with

Generate initial population

Evaluate the fitness for each agent

Update G, best and worst of the population

Calculate M and a for each agent

Update velocity and position

Meeting end of criterionNo

Return best solution

Yes

Fig. 3. Principle of GSA algorithm.

Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315 309

heavier masses [23]. The following describes how GSA works forthe problem to be solved.

Usually, the target problem in the real world can be turned intoa minimization or maximization mathematical optimization prob-lem, and then the GSA is used to solve this optimization problem.Without loss of generality, in this section, we take minimizationproblem as an example, which can be expressed as follows:

min f ðxÞ x 2 Rn ð8Þ

where Rn is the feasible domain in n dimensions of target problem.Assumed there are N agents, the position of the ith agent is de-

fined as follows:

Xi ¼ ðx1i ; . . . ; xd

i ; . . . ; xni Þ for i ¼ 1;2; . . . ;N ð9Þ

where xdi is the position of the ith agent in the dth dimension.

Masses are computed after calculating the fitness of agents asfollows:

miðtÞ ¼ fitiðtÞ�worstðtÞbestðtÞ�worstðtÞ

MiðtÞ ¼ miðtÞPN

j¼1mjðtÞ

8><>: ð10Þ

where Mi(t) and fiti(t) represent mass and fitness value of the ithagent at time t, and best(t) and worst(t) respectively specify thestrongest and the weakest agent with regard to their fitness route,which is defined as follows (the definition of best(t) and worst(t)for maximization optimization problem is on the contrary):

bestðtÞ ¼ minj2f1;...;Ng

fitjðtÞ

worstðtÞ ¼ maxj2f1;...;Ng

fitjðtÞ

8<: ð11Þ

According to Newton gravitation theory, the force acting on theith agent from the jth agent at time t is calculated as follows:

FdijðtÞ ¼ GðtÞ �MiðtÞ �MjðtÞ

RijðtÞ þ n� ðxd

j ðtÞ � xdi ðtÞÞ ð12Þ

where G(t) is gravitational constant at time t, Rij(t) is the Euclidiandistance between ith agent and jth agent (i.e. Rij(t) = kxi(t), xj(t)k2).n is a small constant which is set for avoiding the divisor equal tozero. (There usually uses Rij(t) instead R2

ijðtÞ in GSA because re-searches show that the performance of Rij(t) is better than R2

ijðtÞ inmost cases.)

Based on the law of motion, the agent acceleration adi ðtÞ is cal-

culated as follows:

adi ðtÞ ¼

Pj2kbest; j–i randj � Fd

ijðtÞMiðtÞ

¼X

j2kbest; j–i

randjGðtÞMjðtÞ

RijðtÞ þ n� ðxd

j ðtÞ � xdi ðtÞÞ ð13Þ

where kbest is the set of first K agents with the best fitness valueand biggest mass, which is a function of time, initialized to K0 atthe beginning and decreased with time, randj is a random numberin the interval [0,1]. (kbest is set to decrease the unnecessary calcu-lation and enhance the efficiency of optimization).

Then the next velocity of an agent is considered as a fraction of itscurrent velocity added to its acceleration. Therefore, the next veloc-ity and newly position of an agent can be computed as follows:

vdi ðt þ 1Þ ¼ randi � vd

i ðtÞ þ adi ðtÞ ð14Þ

xdi ðt þ 1Þ ¼ vd

i ðtÞ þ xdi ðtÞ ð15Þ

where vdi ðtÞ and xd

i ðtÞ are the velocity and position of ith agent attime t in the dth dimension, respectively. randi is a random numberbetween 0 and 1. It is to give a randomized feature to the search.

It must be pointed out that the gravitational constant G(t) isimportant in determining the performance of GSA and defined asfollows:

GðtÞ ¼ G0 � exp �b � tt max

� �ð16Þ

where G0 is the initial gravitational constant, b is a constant, t is thecurrent iteration, t_max is the maximum iteration.

The principle of GSA is shown in Fig. 3.

3.2. Improvements on GSA

A major of meta-heuristic optimization algorithms searchingthe best solution due to the balance of two related concepts: explo-ration and exploitation [37,38]. Exploration seeks to understandthe connectivity relationship of the search space, which is helpfulto the global optimal solution; exploration hunts for better optimalsolutions in adjacent area of the visited domain, which canstrengthen the convergence capability of local search. So an excel-lent algorithm should improve the exploration ability in the firststage and then enhance the exploitation ability in the second stagewith the iterations increasing. In this section, some strategieswhich can enhance the exploration and the exploitation ability ofalgorithms are introduced.

3.2.1. Combination with particle swarm optimizationThe reason why birds are able to find foods mainly owes to the

capability of communication and memory among the flock. PSOalgorithm is the imitation of this capability. In the standard GSA,the movement direction of each agent is determined by the totalforce that other better agents act on it and lacking of communica-tions between the agents. So in this paper, we try to improve thesearching ability of GSA by introducing the communication andmemory characteristics of PSO. The newly moving which is obedi-ent to the law of gravity and received guide of memory and socialinformation is defined as follows:

vdi ðt þ 1Þ ¼ r1 � vd

i ðtÞ þ adi ðtÞ þ c1 � r2 � ðPd

ibestðtÞ � xdi ðtÞÞ

þ c2 � r3 � ðGdbestðtÞ � xd

i ðtÞÞ ð17Þ

xdi ðt þ 1Þ ¼ xd

i ðtÞ þ vdi ðtÞ ð18Þ

310 Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315

where r1, r2 and r3 are random variables in the range [0,1], c1 and c2

are learning genes in the range [0,2], Pibest(t) is the best position thatith agent has ever suffer until time t, Gbest(t) is the past global bestposition in the agents at time t.

This new strategy which is a hybrid and universal format of GSAand PSO has been confirmed to own a faster convergence speed(i.e. strong exploitation capability) than standard PSO and GSA[3]. Besides, the special moving mechanism of GSA in Eq. (17) pro-vides a slower motion of agents in the search space and hence amore precise search [23].

3.2.2. Elastic-ball strategyIt is normal to encounter the agents whose movement positions

are beyond boundary during the evolutionary process for manystochastic algorithms. We often handle it with the followingstrategy:

Generate initial population

if xdi ðtÞ > UbðdÞ xd

i ðtÞ ¼ UbðdÞ or

if xdi ðtÞ < LbðdÞ xd

i ðtÞ ¼ LbðdÞ ð19Þ

where Ub(d) and Lb(d) are upper limit and lower limit in the dthdimension.

All off-boundary agents are gathered in the boundary after suchprocessing, which will generate a huge force compelling otheragents to move forward boundary in accordance with law of grav-ity and the uniform distribution of agents is disrupted, which isgreat harmful to the global exploration, especially when thereare local optimums around the boundary. Therefore, a new treat-ment named elastic-ball strategy is used in this paper. This novelstrategy imitates the characteristic that the elastic-ball reflectsback excessive information if there is a barrier blocking its movingdirection and described as follows:

Evaluate agents fitness and store Pbest and Gbest

Update best(t), worst(t), and M(t) in the population

Calculate gravitational constant and acceleration for each agent

Update the velocity and position for each agent

if xdi ðtÞ > UbðdÞ outside ¼ xd

i ðtÞ �UbðdÞ xdi ðtÞ ¼ UbðdÞ � outside

ð20Þ

if xdi ðtÞ < LbðdÞ outside ¼ LbðdÞ � xd

i ðtÞ xdi ðtÞ ¼ LbðdÞ þ outside

ð21Þ

After dealing with the two steps, most agents rebound to thefeasible domain. The few rest agents against the boundary willbe reset position as follows:

Meeting end of criterion

Yes

Elastic-ball strategy is invoked

Chaotic mutation is conducted for the best agent

Yes

No

Beyond boundarys

Evaluate the fitness for each agent s new position

Update Pbest and Gbest in the population

Return best solution

No

No

Fig. 4. Flowchart of the newly IGSA algorithm.

if xdi ðtÞ > UbðdÞjjxd

i ðtÞ < LbðdÞ xdi ðtÞ ¼ rand � ðUbðdÞ

�LbðdÞÞ þ LbðdÞð22Þ

This novel elastic-ball strategy has overcome the shortcomingsthat pulling the off-boundary agents back to the boundary directlymaybe mislead evolution steps and enhanced the varieties ofsearching agents at the same time, which is helpful to the globalexploration.

3.2.3. Mutation operator based on chaotic behaviorPremature phenomenon and local convergence are the com-

mon problems for many intelligence algorithms and one of theeffective approaches in the current is brought in a mutation oper-ator to overcome these unhealthy performances. In this paper, achaotic mutation operator which is able to visit all points in aspecified range without any repeat is incorporated into the im-proved algorithm. The chaotic mutation searches optimal bymeans of regularity, ergodicity and intrinsic stochastic propertiesof chaotic motion and can find out global optimum in great prob-ability. By mutating the best particle based on chaotic sequences,the current best agents will leap out local tracking and looking for

a better available solution in global. In order to ensure the muta-tion operator would not make the fitness of populations worse,only if obtained agent by mutation is better than the worst agent,it will replace the worst one in next generation. The chaotic pro-cedure is described as follows:

(1) Create a new particle Xnew ¼ ðx1new; . . . ; xd

new; . . . ; xnnewÞ and let it

be the best particle Gbest in populations as: Xnew = Gbest.(2) Convert the position of Xnew into a chaos vector e as follows:

eðdÞ0 ¼ xdnew � LbðdÞ

UbðdÞ � LbðdÞ d ¼ 1;2; . . . ;n ð23Þ

where e(d) is the element of chaos vector e in the dth dimension.

(3) Search the global area by Logistic map as follows:eðdÞ0 ¼ u � eðdÞ � ð1� eðdÞÞ ð24Þ

where e(d)0 is the Logistic map result of vector variable e(d), u is acontrol variable in which u = 4, the mapping is in full state.

(4) Convert the obtained chaos vector e(d)0 into the position asfollows:

XnewðdÞ ¼ LbðdÞ þ eðdÞ0 � ðUbðdÞ � LbðdÞÞ d ¼ 1;2; . . . ;n ð25Þ

(5) Calculate fitness of obtained new position Xnew and compareit with fitness of the worst particle Gworst in the population.If Xnew’s is better, replace Gworst by Xnew; else remap andrelocate new Logistic vector with iterative calculation of

Originalsystem

Originalsystem

Measuredoutput

xEstimated

system

Estimatedsystem

FitnessEvaluator

FitnessEvaluator

y

tm

y

x

tm

Simulatedoutput

IGSA-basedidentifier

IGSA-basedidentifier

System input

Identified parameter θ )ˆ(θIOFC

Fig. 5. Diagram of IGSA based WTRS parameter identification.

Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315 311

Eqs. (23) and (24) several times until the fitness of Xnew isbetter than the fitness of Gworst or the mapping is coveredwith every corner in the searching range.

Through chaotic mutation, the best agent has a bigger possibil-ity to leap out the current local optimal domain if there are anyother better unsearched regions (i.e. strong exploration ability).In addition, excellent characters of chaos movement such assearching feasible areas without repeat are also contributed to glo-bal exploitation.

By means of three strategies added in the algorithm, the im-proved GSA (IGSA) is summarized as the following and is illus-trated by the diagram in Fig. 4. In this work, IGSA will be used tosolve the model parameter identification problem and is appliedin parameter identification of WTRS. The performance of IGSAand effectiveness of the improvements in this section will be dem-onstrated through results in the later identification experiments.

Step 1: Initialization. Randomly initialize the agent position andvelocity.

Step 2: Fitness evolution. Calculate the fitness of agents by theirinitial position, storing current position of each agent asthe best history record position of the agent (i.e.Pbest

i ðtÞ ¼ xiðtÞ for t = 1) and position of best agent gbest(t)(which owns the biggest fitness value in the population)as the best position in global (i.e. Gbest(t) = gbest(t) fort = 1).

Step 3: Update best(t), worst(t), and Mi(t) for i = 1,2, . . . ,N.Step 4: Calculate gravitational constant in the current iteration

and acceleration for each agent.Step 5: Update agents’ velocity and position with Eqs. (17) and

(18).Step 6: Judge whether the new position of the agent is beyond

the boundary. If the new position is against the bound-ary, elastic-ball program is invoked.

Step 7: Evaluate the fitness in accordance with each agent’s newposition.

Table 1Transfer coefficients of turbine under two running condition.

Working condition Transfer coefficients in the water turbine system

ex ey

No-load �1.0567 0.9080Load �1.4673 0.7713

Step 8: Compare the obtained fitness of new position xi(t + 1)with fitness of Pbest

i ðtÞ while i changes from 1 to N. Ifxi(t + 1) has a better fitness value, replace position ofPbest

i ðtÞ by xi(t + 1).Step 9: Compare fitness of the agents with Gbest(t). If there is an

agent has a better fitness than fitness of Gbest(t), Gbest(t)will be replaced by the position of the agent.

Step 10: Chaotic mutation is conducted for the best agent.Step 11: Repeat Step 3 to Step 10 until the stop criteria reached.

4. The parameter identification strategy

For a system with known model structure but unknownparameters, the parameter identification problem can beconverted into an optimization problem. The unknownparameters vector for WTRS is usually set as a particle in swarmor a gene in chromosome and a performance function measuringhow well the model response fits the system response is built tooptimize.

4.1. Objective function based on WTRS system

The model of WTRS has been illustrated by Fig. 2. The turbinespeed x, guide vane opening y and water torque mt are observedoutput state variables. Although three outputs are selected as stateoutput variables, their contribution and importance in solving theproblem of parameter identification are different. At the same var-iation of parameters, the more significant the output of identifiedmodel deviates from that of original system, the more importantthe output in the objective function will be. And the weight for thisoutput is heavy. In this way, weights of different errors outputs aredesigned, according to the significance of deviation. The improvedobject function is defined as:

CIOFðhÞ ¼XN

k¼1

Xn

j¼1

wðjÞðzjðkÞ � zjðkÞÞ2 ð26Þ

where parameter vector h ¼ ½ kp ki kd Ty Tw Ta0 eg �, out-put of actual system z ¼ ½ x y mt �, z

_¼ ½ x

_y_

m_

t� is simulated

output of identified model, N is the number of samples, n is dimen-sion of system output vector, in this work, n = 3, the weight vectorw ¼ ½w1 w2 w3 �.

The weights are calculated according to following steps:

(1) Set value of vector parameters hi, i = 1,2, . . . ,m (m is thedimension of parameter vector h) in the WTRS system, andobtain system output zj(k), k = 1,2, . . . ,N, j = 1,2, . . . ,n.

(2) Loop A: j = 1:n.Loop B: i = 1:m.Vary the ith parameter, hnew = hi ⁄ (1 + D%), and obtain sys-tem output zjiðkÞ.Calculate wji ¼ kðzji � zjiÞk2.The jth weight wj = average(wji).End Loop B.End Loop A.

eh eqx eqy eqh

1.4191 �0.0574 0.7887 0.45711.7179 �0.4901 0.8184 0.7257

Table 2Comparison of average identification results of different methods under no-load condition.

Identified parameters hi System real value Average of identified parameters (30 trials)

PSO GSA IGSA

hi PE hi PE hi PE

kp 5.5912 6.0418 0.0806 5.9529 0.0716 5.5919 0.0007ki 1.0611 0.0749 0.9295 0.5954 0.4477 1.0596 0.0033kd 3.2800 4.1630 0.2692 3.5520 0.0921 3.2805 0.0006Ty 0.1000 0.1472 0.4724 0.1180 0.1917 0.1000 0.0012Tw 1.5000 1.4712 0.0197 1.4708 0.0283 1.5000 0.0004Ta0 12.000 17.159 0.4299 14.035 0.1762 12.005 9.0E-5eg 0.4433 0.5071 0.1613 0.4148 0.0665 0.4431 0.0016

Table 3Mean best cost and mean APE of 30 times under no-load condition.

PSO GSA IGSA

Mean best cost 86.319 20.782 0.0117Mean APE 0.3375 0.1534 0.0013

0 20 40 60 80 100

0

500

1000

1500

2000

2500

3000

3500

4000

Cos

t

Iteration

PSO GSA IGSA

Fig. 6. Comparison of average iteration process under no-load condition.

20 40 60 80 100

0

20

40

60

80

100

120

140

160

180

200

Cos

t

Iteration

PSO GSA IGSA

Fig. 7. Local magnification of average iteration process under no-load condition.

0 5 10 15 20 25 30

0

1

2

Time

Spee

d original system estimated system

0 5 10 15 20 25 30

-10

0

10

TimeTo

rque original system estimated system

0 5 10 15 20 25 300

5

10

Time

Gui

de v

ane

original system estimated system

Fig. 8. Comparison of system outputs using IGSA under no-load condition.

312 Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315

4.2. Parameter identification strategy

As shown in Fig. 5, the original and estimated systems are sup-plied with a same excitation inputs and their outputs are given asinputs to the fitness evaluator, where the fitness is calculated. The

fitness function CIOFðhÞ is then used by IGSA-based identifier toidentify the unknown parameter vector h. By minimizing of fitnessfunction through IGSA, the outputs of estimated system approxi-mates to the outputs of original system, while the unknownparameters trend to be equal with real values.

Parameter identification accuracy is measured by parameter er-ror (PE):

PE ¼ jhi � hijhi

� 100% i ¼ 1;2; . . . ;m ð27Þ

and average parameter error (APE):

APE ¼ 1m

Xm

i¼1

jhi � hijhi

� 100% ð28Þ

where hi is the parameter elements of h in original system, hi is theparameter elements of h in estimated system, m is the size of h.

5. Experiments and results analysis

In this section, the WTRS is simulated in MATLAB, and the pro-posed IGSA is applied to identify the parameters of simulated sys-tem. The model of WTRS is illustrated in Fig. 2. Seven keyparameters are chosen to be estimated in simulation experimentswhich are kp, ki, kd, Ty, Tw, Ta0 and eg.

In experiments, simulation model of WTRS are excited under twocondition, load condition and no-load condition. Under no-load con-dition, step disturbance of speed is employed to excite the system,and under load condition, step disturbance of load is employed to ex-cite the system. The parameters of simulated model are set valuesadopted in a Chinese hydroelectric station as follows:

Table 4Comparison of average identification results of different methods under load condition.

Identified parameters hi System real value Average of identified parameters (30 trials)

PSO GSA IGSA

hi PE hi PE hi PE

kp 5.5912 8.7602 0.5668 6.4076 0.1538 5.6170 0.0060ki 1.0611 1.4695 0.3938 1.1587 0.0988 1.0654 0.0052kd 3.2800 4.1421 0.3530 3.5340 0.0928 3.2827 0.0044Ty 0.1000 0.0500 0.5416 0.0950 0.1591 0.0997 0.0122Tw 1.5000 1.4294 0.0471 1.4680 0.0240 1.4981 0.0021Ta0 12.000 17.685 0.4737 13.448 0.1262 12.043 0.0046eg 0.4433 0.4312 0.0377 0.4482 0.0415 0.4458 0.0082

Table 5Mean best cost and mean APE of 30 times under load condition.

PSO GSA IGSA

Mean best cost 0.5528 0.0923 0.0017Mean APE 0.3448 0.0995 0.0061

0 20 40 60 80 100

0

5

10

15

20

25

Cos

t

Iteration

PSO GSA IGSA

Fig. 9. Comparison of average iteration process under load condition.

20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Cos

t

Iteration

PSO GSA IGSA

Fig. 10. Local magnification of average iteration process under load condition.

0 5 10 15 20 25 300

0.1

0.2

Time

Turb

ine

spee

d

original system estimated system

0 5 10 15 20 25 30-2

-1

0

TimeTu

rbin

e to

rque

original system estimated system

0 5 10 15 20 25 30-1.5

-1

-0.5

0

Time

Gui

de v

ane

original system estimated system

Fig. 11. Comparison of system outputs using IGSA under load condition.

Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315 313

The gains of PID in steady-state working condition are 5.5912,1.0611, and 3.2800 (i.e. kp = 5.5912, ki = 1.0611, kd = 3.2801). Thecharacteristic parameter of penstock in the hydroelectric stationis calculated to approximate equal to 1.5 (i.e. Tw = 1.5). The inertia

time constant and the adjusting parameter of generator and loadare equal to 12 and 0.4433 respectively through the measurement(i.e. Ta0 = 12, en = 0.4433). The major servomotor time constant isnearly approach 0.1 according to the historical data (i.e. Ty = 0.1).The feedback coefficient and the differential time constant is setto 0.04 and 0.28 according to the ever experience (i.e. bp = 0.04,Td = 0.28).

The parameters of water turbine under different conditions areshown in Table 1.

Based on the above discussion, the parameter vector (i.e.h ¼ ½ kp ki kd Ty Tw Ta0 eg �) in original system is set ash ¼ ½5:5912 1:0611 3:2801 0:1 1:5 12 0:4433 �. The simu-lation time is set to be 30 s which is long enough to make sure thesystem change to be stable from a transient process. The samplingtime is set to be 0.01 s which is fast enough to capture systemdynamic process. The outputs in vector z, which contains turbinespeed x, guide vane opening y and turbine torque mt, are all sampled.

5.1. Comparison of identification method under no-load condition

In this part of experiments, IGSA, GSA and PSO have been em-ployed to identify the parameters in the dynamic model of WTRS.A step disturbance of given speed is adopted to excite the system.The model described in Section 2 is simulated as the original sys-tem, and experiments of parameter identification are conducted.

In simulation, to perform fair comparison in fitness evaluation,population size of PSO, GSA and IGSA are all 80. The maximumgeneration is set to be 100 in the three algorithms. For GSA,G0 = 30, b = 10; For PSO, w = 0.6, c1 = c2 = 2; For IGSA, c1 = c2 = 2,G0 = 30 and b = 10. In order to overcome the randomness of threeheuristic algorithms, 30 trials are tried and average results areobtained.

314 Z. Chen et al. / Energy Conversion and Management 78 (2014) 306–315

Comparing identification accuracy of different methods underno-load, PE is used and listed in Table 2. Simulation results consid-ering best cost and APE are listed in Table 3. In Tables 2 and 3, it isseen that compared with GSA and PSO, IGSA achieve better param-eter identification accuracy. In Table 2, the mean cost and meanAPE are as small as 0.0117 and 0.0013, respectively, much smallerthan those achieved by other methods, which conforms the valid-ity of improvement measures added.

The convergence of algorithms is compared in Fig. 6, whichexhibits the average convergence of 30 times, it is seen that IGSAcould converge on the optimal quickly compared with other meth-ods. Although in the incipient evolution period, the convergence ofPSO algorithm is better, but 20 iteration later, Fig. 7 which is thelocal magnification figure of Fig. 6 exhibits IGSA has a faster con-vergence speed while the other algorithms are tracking in a localoptimal soon.

Fig. 8 shows the estimated system outputs of WTRS obtained byusing the average identified parameters with IGSA, and then com-pared with original system outputs, where guide vane opening,turbine torque and turbine speed are compared. It is obvious origi-nal curves and estimated curves are very closely, which meansparameter identification is effective and obtained higher accuracy.

5.2. Comparison of identification method under load condition

In order to verify the validity of IGSA in parameters identifica-tion of WTRS, different running conditions of WTRS is considered.In this part, the system is under load condition, and a step distur-bance of load is employed to excite the system. The parametersregarding to water turbine under load condition is shown inTable 1, the parameter vector set in original system is not changed,and parameters of PSO, GSA and IGSA are also keep unchanged.Identification experiments are repeated 30 times, average indicesof results are taken into consideration

Tables 4 and 5 show the identification accuracy achieved by dif-ferent algorithms, where average PE, APE and best cost are takeninto consideration. The results show clear that compared withPSO and GSA, IGSA performs best on all indices, which means theproposed IGSA is effective. The average convergence process ofPSO, GSA and IGSA are compared in Figs. 9 and 10, which showsthat IGSA possesses excellent ability in obtaining optimal valueof cost function compared with other algorithms. Identified out-puts using the average parameters based on IGSA are comparedwith the original system outputs in Fig. 11, which exhibits theidentified system meet original system perfect.

6. Conclusion

In this paper, the problem of parameter identification of WTRSis studied. A simulation model associated with engineering experi-ence is brought in. An improved gravitational search algorithm isproposed to the parameters identification of WTRS. Simulation re-sults are provided to validate the effectiveness of the identifiedmethod. It is shown that IGSA is capable of solving the problemof parameters identification. Comparing to GSA and PSO, IGSA per-forms the best with high accuracy and stability. Meanwhile, theidentification method is not focus on the specific input signal,which is easy to implement in the system simulation and conve-nient to transplant into other parameter identification problems.

Acknowledgment

The authors gratefully acknowledge the financial supports fromNational Natural Science Foundation of China under Grant No.51379080.

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