Electrical Power and Energy Systems 42 (2012) 257–263

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Multiobjective design of load frequency control using genetic algorithms

Fatemeh Daneshfar ⇑, Hassan BevraniDepartment of Electrical and Computer Engineering, University of Kurdistan, Sanandaj, PO Box 416, Kurdistan, Iran

a r t i c l e i n f o

Article history:Received 28 August 2009Accepted 10 April 2012

Keywords:Load frequency controlMultiobjective optimizationGenetic algorithmsWind power generation

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.04.024

⇑ Corresponding author.E-mail address: daneshfar@ieee.org (F. Daneshfar)

a b s t r a c t

Recently, several modern control theory designs like H1 have been applied to the load–frequency control(LFC) problem optimization technique. However, the importance and difficulties in the selection ofweighting functions of these approaches and the pole-zero cancellation phenomenon associated withit produces closed loop poles. In addition, the order of the H1-based controllers is as high as that ofthe plant. This gives rise to complex structure of such controllers and reduces their applicability. Alsoconventional LFC systems that use classical or trial-and-error approaches to tune the PI controller param-eters are more difficult and time-consuming to design.

In this paper the decentralized LFC synthesis is formulated as a multiobjective optimization problem(MOP) and is solved using genetic algorithms (GAs) to design well-tuned PI controllers in multi-areapower systems. The proposed control scheme has been applied to the LFC problem in a three-area powersystem network and the 10-machine New England test system respectively and shows desirableperformance.

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1. Introduction

Frequency changes in large scale power systems are a direct re-sult of the imbalance between the electrical load and the powersupplied by connected generators [1]. Therefore load–frequencycontrol is one of the important power system control problemswhich there have been considerable research works for it [1–3].Usually, the load frequency controllers used in the industry arePI type and are tuned online based on trial-and-error approaches.Also recently, several approaches based on modern control theoryhave been applied to the LFC design problem and there has beencontinuing interest in designing load–frequency controllers withbetter performance using various decentralized robust and optimalcontrol methods during the last two decades [4–10].

One of the modern control techniques which has been appliedto the LFC problem is H1 optimization technique [9,10]. However,the importance and difficulties in the selection of H1 weightingfunctions have been reported. Moreover, the pole-zero cancellationphenomenon associated with this approach produces closed looppoles whose damping is directly dependent on the open loopsystem (nominal system) [11]. On the other hand, the order ofthe H1-based controllers is as high as that of the plant. This givesrise to complex structure of such controllers and reduces theirapplicability. Then despite the potential of modern control tech-niques with different structures, power system utilities prefer the

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online tuned PI controller’s. The reasons behind that might bethe ease of online tuning and the lack of assurance of the stabilityrelated to some adaptive or variable structure techniques.

One of the optimization techniques that is used for tuning the PIcontroller parameters is genetic algorithm (GA) [12,13]. The advan-tage of the GA technique is that it is independent of the complexityof the performance index considered. It suffices to specify theobjective function and to place finite bounds on the optimizedparameters. However, in practice this approach is not capable inproblems with multiple objective functions like multi-area powersystems with more than one PI controller. In this paper, the LFCproblem in multi-area power system is formulated as a multiobjec-tive optimization problem and GA is employed to solve it. The pro-posed design approach has been applied to a three-area powersystem network and the well-known New England 10 generators,39-bus system too as case studies.

The organization of the rest of the paper is as follows, In Section2, a brief introduction to LFC and multiobjective optimizationproblem (MOP) is given. In Section 3, the problem formulationunder MOP is discussed. Simulation results are provided in Section4 and the paper is concluded in Section 5.

2. Backgrounds

2.1. Multiobjective optimization

Many real-world power system problems involve simultaneousoptimization of multiple objectives. In certain cases, objective

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functions may be optimized separately from each other and insightgained concerning the best that can be achieved in each perfor-mance dimension. However, suitable solutions to the overall prob-lem can seldom be found in this way. Instead, multiobjectiveoptimization (MO) solution seeks to optimize the components ofa vector-valued cost function. Unlike single objective optimization,the solution to this problem is not a single point, but a family ofpoints known as pareto-optimal (PO) set. Each point in this surfaceis optimal in the sense that no improvement can be achieved in onecost vector component that does not lead to degradation in at leastone of the remaining components [14].

There are following definitions related to the MO [14]:

� Inferiority. A vector u = (u1, . . . ,un) is said to be inferior tov = (v1, . . . ,vn) if and only if v is partially less than u i.e.

8i ¼ 1; . . . ;n v i^ui ^ 9i ¼ 1; . . . ;n : v i � ui:

Fig. 1. A three-control area power system.

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� Superiority. A vector u = (u1, . . . ,un) is said to be superior tov = (v1, . . . ,vn) if and only if the v is inferior to the u.� Non-inferiority. Vectors u = (u1, . . . ,un) and v = (v1, . . . ,vn) are said

to be non-inferior to one another if v is neither inferior norsuperior to u.

Usually, the aim of MOP is to determine the trade off surface,which is a set of nondominated solution points, pareto-optimalor noninferior solutions. Actually each element in the PO set con-stitutes a non-inferior solution to the MOP.

2.2. Multiobjective optimization based on genetic algorithms

In an MOP, there may not exist one solution that is best with re-spect to all objectives. In view of the fact that none of the solutionsin the nondominated set is absolutely better than any other, anyone of them is an acceptable solution [15]. The choice of one solu-tion over the other requires using an optimization technique.

Conventional optimization techniques, such as gradient andsimplex based methods, and also less conventional ones, such assimulated annealing, are difficult to extend to the true multiobjec-tive optimization case, because they were not designed with multi-ple solutions in mind. Evolutionary algorithms (EAs), however,have been recognized to be possibly well-suited to multiobjectiveoptimization since early in their development. Multiple individualscan search for multiple solutions in parallel, eventually takingadvantage of any similarities available in the family of possiblesolutions to the problem. The ability to handle complex problems,involving features such as discontinuities, multimodality, disjointfeasible spaces and noisy function evaluations, reinforces the po-tential effectiveness of EAs in multiobjective search and optimiza-tion, which is perhaps a problem area where evolutionarycomputation really distinguishes itself from its competitors [15].

One of the evolutionary computation techniques that workswell with a population of points is GA. It is expected that theycan find the Pareto-optimal front easily by maintaining a popula-tion of solutions, and search for many non-inferior solutions in par-allel. This characteristic makes GAs very attractive for solvingmultiobjective optimization problems.

2.3. Load frequency control in multi-area power systems

In an isolated power system, the LFC task is limited to restorethe system frequency to the specified nominal value. In order togeneralize the isolated LFC model for interconnected power sys-tems, the control area concept needs to be used as it is a coherentarea consisting of a group of generators and loads, where all thegenerators respond to changes in load or speed changer settings,in unison [16]. Therefore, a large-scale power system consists of

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a number of interconnected control areas. Fig. 1 shows the blockdiagram of a three-control area power system, which includes 3Gencos in each control area.

Following a load disturbance within a control area, the fre-quency of that area experiences a transient change, the feedbackmechanism comes into play and generates appropriate rise/lowersignal to make generation follow the load [16]. In the steady state,the generation is matched with the load, driving the tie-line power(DPtie) and frequency deviations (Df) to zero. The balance betweenconnected control areas is achieved by detecting the frequency andtie-line power deviations to generate area control error (ACE) sig-nal. The ACE for each control area can be expressed as a linear com-bination of tie-line power change and frequency deviation asfollow [16],

ACEi ¼ biDfi þ DPtie�i ð1Þ

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3. Problem formulation

3.1. Overview

A multi-area power system comprises areas that are intercon-nected by high-voltage transmission lines or tie-lines. The trendof frequency measured in each control area is an indicator of thetrend of the mismatch power in the interconnection and not inthe control area alone. The LFC system in each control area of aninterconnected power system should control the interchangepower with the other control areas as well as its local frequency[16]. According to above discussions, the main objectives for theLFC problem in a multi-area power system can be expressed asfollow.

If the disturbance magnitude is greater than the availablepower reserve (supplementary control) i.e. PC < PL, the frequencydeviation and tie line power changes do not converge to zero insteady state [16]. Therefore, the main goal of the LFC system in amulti-area power system is to converge each area’s ACE signal tozero in steady state in the presence of load disturbance, and themultiobjective problem is reduced to optimize the PI controllersparameters, such that the ACE signals converge to zero in encoun-tering the load disturbance too.

According to the above explanation, to have some degree of rel-ative stability in all areas of a multi-area power system, the param-eters of the PI controllers may be selected so as to minimize theobjective function (2).

Fig. 2. The GA model.

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sss

ObjFnci ¼XK

t¼0

jACEi;t j ð2Þ

which ObjFnci is the objective function of power area i, k is equalwith the simulation time (s) and |ACEi,t| is the absolute value ofACE signal of area i at time t.

To minimize the mentioned objective function in a multi-areapower system with n areas, a vector-valued cost function (3) hasbeen defined, which the multiobjective optimization solution seeksto optimize it by determining a set of nondominated solutionpoints which the choice of one solution over the other is done byGA.

½ObjFnc1; . . . ;ObjFnci; . . . ;ObjFncn� ð3Þ

To implement the above solution, a simulation study is pro-vided in the Optimization Toolbox of MATLAB software. In thissimulation the following design criteria have been considered forsimplicity through the process of applying GA to the MOP,

1. The population and individual representation has to be able todeclare all candidate solutions.

2. The LFC problem includes specific constraints, then it has to beguaranteed that any produced individual by the cross over andmutation presents a valid candidate solution, otherwise it hasto be repaired to a feasible one.

3. Fitness function of the GA problem based on the vector objec-tive function (3), is as follow,

½ObjFnc1; . . . ;ObjFnci; . . . ;ObjFncn� ¼ FitnessFunctionð. . .Þ: ð4Þutho

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3.2. GA Operators

Here GA is a technique used to tune PI controller parametersbased on the optimum values gained for the vector-valued costfunction (3).

The basic line of the algorithm is derived from a steady state ge-netic algorithm, where only one replacement occurs per genera-tion. Also when defining initial population of GA individuals anddesigning the crossover and mutation operators, the need for therepair function is necessary to make a high correlation betweenparents and offspring. Finally to reach a good combination of indi-viduals, GA operators are defined as follow [15],

3.2.1. InitializationEq. (5) shows the GA individual vector whose elements present

PI controller parameters.

½ðK1; P1Þ; . . . ; ðKi; PiÞ; . . . ; ðKn; PnÞ� ð5Þ

which (Ki, Pi) are PI parameters related to area i.Initial solutions to the above individuals are generated using a

uniform random number of PI controller parameters between[�1, 1] (since the most PI controller parameters are between [�1,1], the initial population is spread along the search space [�1,1]).

Also an m � n matrix presents the whole GA population, (mrows correspond to m individuals and n columns present individ-ual elements). To initialize individuals at random, we start withan empty matrix, and fill it with PI parameters generated using auniform random number.

3.2.2. Selection, mutation and crossoverIn each generation phase, individuals (PI parameters) are ap-

plied to the specified multi-area model and the model is simulatedfor appropriate seconds. After the simulation terminated differentcontrol area ACE signals will produce the individual fitness accord-ing to (4). Then two different individuals have been selected basedon the roulette wheel selection. The crossover and mutation oper-ators are then applied. The crossover is applied on both selectedindividuals, generating two childes. The mutation is applied uni-formly on the best individual. The best resulting individual is inte-grated into the population, replacing the worst ranked individualin the population. The above procedure has run for many genera-tions until gain to the minimum value of the fitness functions(Fig. 2 presents the model of the applied GA algorithm).

In some cases mutation operator will change the individual insuch a way that we cannot guarantee the individual to be still legal.Then, the repair function has to be applied and guarantee that eachindividual parameters are randomly generated in [�1, 1] and alsothere is not any repetitive individual.na

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4. Application results

To illustrate the effectiveness of the proposed control strategy,and to compare the results with the recent applied linear robustcontrol techniques [9,10], a three-control area power system isconsidered as the first test system that its parameters are givenin [9,10] (see Fig. 1).

Also as the second test case study the proposed optimizationmethod is examined on the well-known New England 10 genera-tors, 39-bus system too, that the simulation parameters for thegenerators, loads, lines, and transformers are given in [17].

4.1. Three-control area test system

In this section, the performance of the closed-loop system usingthe linear robust PI controllers [9,10] compared to the well tunedPI controllers with multiobjective optimization for the various pos-sible load disturbances,

Case 1: As the first test case, the following large load distur-bances (step increase in demand) are applied to three areas:

DPd1 ¼ 100 MW; DPd2 ¼ 80 MW; DPd3 ¼ 50 MW

The frequency deviation, area control error and control actionsignal (DPC) of the closed-loop system are shown in Fig. 3.

Case 2: Consider larger demands by areas 2 and 3, i.e.

DPd1 ¼ 100 MW; DPd2 ¼ 100 MW; DPd3 ¼ 100 MW

The closed-loop responses for each control area are shown inFig. 4.

According to Figs. 3 and 4, by using the proposed method, thearea control error and frequency deviation of all areas are properlydriven back to zero, as well as robust controllers. Also theconvergence speed of frequency deviation and ACE signal to itsfinal values are good, they attain to the steady state more rapidly

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Fig. 3. System responses in case 1, (a) area 1, (b) area 2, and (c) area 3, (Solid line: proposed method, dotted line: robust PI controller [9], dashed line: robust PI controller[10]).

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than signals in [9,10]; however the most frequency deviation is oc-curred at 2 s which load disturbances are occurred in it.

For more investigation the average of |ACEi| over 1 min isused as a performance index, and the results for the proposed

controller design are listed in Table 1. It is cleared, in comparisonof complex and high order robust controllers, the closed-loopperformance of the current simple design is significantlyimproved.

Fig. 4. System responses in case 2, (a) area 1, (b) area 2, and (c) area 3, (Solid line: proposed method, dotted line: robust PI controller [9], dashed line: robust PI controller[10]).

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4.2. New England test system

To demonstrate the effectiveness of the proposed control de-sign, extra simulations were carried out. In these simulations, the

proposed method was applied to the well-known New England10 generators, 39-bus system. It has three areas and 10 generators,19 loads, 34 transmission lines, and 12 transformers (more expla-nations on this system are given in [17]). There are 198.96 MW of

Table 1Performance evaluation.

|ACE1|avg (pu) |ACE2|avg (pu) |ACE3|avg (pu)

Case 1MO-approach 0.0104 0.0071 0.0063Robust-approach [9] 0.0122 0.0096 0.0056Robust-approach [10] 0.0147 0.0103 0.0057

Case 2MO-approach 0.0103 0.0087 0.0114Robust-approach [9] 0.0104 0.0102 0.0103Robust-approach [10] 0.0147 0.0129 0.0113

Fig. 5. Area-1 responses.

Fig. 6. Area-2 responses.

Fig. 7. Area-3 responses.

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conventional generation and 265.25 MW load in Area 1. In Area 2,there are 232.83 MW of conventional generation, and 232.83 MWload. In Area 3, there are 160.05 MW of conventional generation,and 124.78 MW of load. Also in the present work, it is assumedthat only one generator in each area is responsible for the AGC task.

As a test scenario, the following load disturbances (step increasein demand) are applied to three areas: In Area 1, 3.8% of total areaload at bus 8, 4.3% of total area load at bus 3 in Area 2, and 6.4% oftotal area load at bus 16 in Area 3 have been simultaneously in-creased in a step form. The frequency deviation and ACE signalsof the closed-loop system are shown in Figs. 5–7.

As shown in the simulation results, using the proposed method,the area control error and frequency deviation of all areas are prop-erly driven close to zero.

5. Conclusion

In this paper a new method for LFC design, using the MOP ap-proach has been proposed. It is formulated to optimize a compositeset of objective functions comprising the PI controllers parametersby GA. The design strategy is very simple and includes enough flex-ibility to set the desired level of performance. It was applied to athree-control area power system and the New England test powersystem respectively. Simulation results demonstrated the effec-tiveness of the methodology.

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