A Review of the Use of Genetic Algorithms in Economic Load Dispatch

W. Warsono PLN Indonesia, University of

Abertay Dundee, UK, 0604722@abertay.ac.uk

Dr. C.S. Ozveren University of Abertay

Dundee, UK, c.s.ozveren@abertay.ac.uk

Dr. David J KingUniversity of Abertay

Dundee, UK D.king@abertay.ac.uk

Prof. D.BradleyUniversity of Abertay

Dundee, UK D.Bradley@abertay.ac.uk

Abstract- This paper presents a review of many previous

papers on the use of genetic algorithms (GA) for solving the problem of economic load dispatch (ELD) for power systems. The paper will cover several topics, i.e. a brief description of the GA method, the various power system models and topologies solved by the GA method, various GA techniques used for ELD, and hybrids of GA with other techniques for solving ELD.

I. INTRODUCTION

Economic load dispatch (ELD) is one of the major issues in power system operation. It is defined as a process of allocating the output of generators to satisfy electrical demand in a power system in the most economic way considering all constraints [1]. The complexity of the ELD problem depends upon many factors, such as the size of the system, system constraints, and generator characteristics.

Several techniques have been introduced to solve the optimisation of ELD, which can be divided into conventional and stochastic methods. Conventional methods use a deterministic approach, such as the LaGrange multiplier, Linear Programming (LP) and Dynamic Programming (DP) [2]. These methods have limitations or drawbacks when coping with more complex problems. The LaGrange Multiplier and LP are unable to solve problems with non-linear and non-smooth characteristics. The DP method has a problem with dimensionality because its storage requirements and execution time increase dramatically when the number of generators is increased and higher accuracy is needed [3].

Recent techniques have been developed using stochastic approaches for solving optimisation problems. Examples are an Adaptive Hopfield Neural Network [4], the Simulated Annealing method [5], and Genetic Algorithms (GA), amongst others. These new methods offer alternative techniques which attempt to overcome the drawbacks of conventional methods.

GA is a global search algorithm based on biological concepts which mimic the mechanics of nature and natural genetics. Along with Evolutionary Programming, Evolutionary Strategy, and Genetic Programming, GA is a part of a wider concept called Evolutionary Computation (EC). Meanwhile, EC, along with Adaptive-Neural Networks (ANN), Fuzzy Systems, amongst others, are classified as Artificial Intelligence (AI) techniques [6].

There are several different properties and advantages of GA compared to traditional methods such as those mentioned

by Chipperfield [7], such as: (1). GA seeks a number of candidate solutions in parallel, instead of a starting from a single point; (2). GA uses probabilistic transition rules using GA operators rather than deterministic ones; (3). GA may use an encoding of the parameter set instead of the parameter itself; (4). GA does not require derivative information or other auxiliary knowledge, except objective or fitness functions.

GA is capable of finding the global optimum and of coping with various difficulties, such as non-linearity, non-smoothness, discontinuity, and non-convex characteristics (Santos [8] and Sheble and Brittig [9])

The paper will focus on the use of GA methods for solving ELD. The discussion will be divided into the following four subjects: i) the various power system architectures that can be solved using GA, the various GA techniques used and hybrids of GA with other techniques.

II. POWER SYSTEM ARCHITECTURES SOLVED BY GA

The GA method has been used for solving various power system architectures in terms of size, generation characteristics, system constraints, or objective functions by many authors. This shows the flexibility and capability of the GA method to solve ELD.

Amongst the first work, Sheble and Brittig [9] examined GA to satisfy typical smooth quadratic functions for three thermal generators that can also be solved using the classical LaGrange technique. They used the fact that GA can provide similar results with the classical solution to validate the effectiveness of GA. More complex systems were studied by Walters and Sheble [10], Bakitzis et al [9] and Sewtohul [11]. They used GA for solving ELD for systems with valve point discontinuities, which occur in steam generators when a valve in the turbine starts to open and produces a rippling effect on the unit curve. In traditional methods these discontinuities are ignored and smoothed, therefore introducing inaccuracy in the results. All papers show that GA can cope well with this problem. Bakirtiz etal [3] show that GA takes significantly less time compared to dynamic programming (DP) when there is a large number of units, but the probability of getting an optimal solution is less than that for DP.

In another study by Zhang et al [19] a 24 hour Dynamic Economic Dispatch is performed for a system with valve point loading which considers ramp rate constraints as well. They use a novel GA called real-coded GA with quasi-simplex techniques, which provide convincing results.

In a recent study, Chiang [12] reports the use of GA for another complex ELD problem that deals with valve point loading and prohibited operating zones (POZ). The unit with POZ separates the decision space into disjoint subsets, and producing a non convex cost function. The proposed GA method is compared with many other methods, such as Simulated Annealing (SA), hybrid stochastic search, particle swarm optimisation (PSO), hybrid evolutionary programming and sequential quadratic programming, and Evolutionary Strategy. Using 3 simulation examples, he asserts that his proposed GA method has many merits, such as being straightforward, easy to implement, and more effective.

Hong and Li [13] study the effectiveness of using GA for a system consisting of multiple cogenerators and multiple buyers in a deregulated market. They successfully use GA in both an IEEE 30-bus system and an IEEE 118-bus system.

Hosseini and Kheradmandi [14] use a GA method in a deregulated power system which considers transmission costs and ramping rate constraints. They successfully test their GA method both on a 10-unit system and an IEEE 30-bus system

Abido [15] proposed a novel approach based on GA for solving ELD which considers environmental objectives. The problem is formulated into a multiobjective optimisation problem with competing fuel cost objective and emission cost minimisation. He uses an IEEE 30-bus system test with several constraints including valve loading effects and transmission loading restrictions. His proposed GA method provides a representative and manageable Pareto optimal set.

Hong and Li [16] report on using GA for short-term scheduling of an autonomous system containing diesel generators, wind power, solar photovoltaics and batteries. The result is compared to Simulated Annealing (SA) for the same problem and provides a solution that requires fewer iterations and takes less time.

Chen and Chang [17] used GA for a large-scale system in Taiwan Power System which contains 40 units, taking into account transmission losses, ramp rate limits and prohibited zones as well. They report the robustness and powerfulness of GA compared to Lambda Iteration Methods for solving this problem.

III. VARIOUS GA TECHNIQUES USED IN ELD

Although GA methods have the same basic principles there are a wide-range of techniques that can be used to look for the most effective and efficient solutions. Many authors have used different techniques in the application of the ELD problem to seek the most effective technique for solving various problems.

A. Encoding/decoding techniques Among the early work is a paper by Walters and Sheble

[10]. In this paper, they encode generator output values into binary strings and investigate two types of binary encoding. The first method simply stacks each value in series with each other. Therefore this type is called series encoding by Sheble and Brittig [9]. The second encoding method is more

complicated than the first, where the assigned gene structures are embedded throughout the string. So this encoding is called embedded encoding by Sheble and Brittig [9]. Both encoding methods are illustrated in Figure 1. It was shown that the embedded encoding method is more complicated.

The authors have demonstrated that series encoding tends to outperform embedded encoding. Therefore, series encoding is more commonly used, as shown in Bakirtiz [3], Sewtohul [11], Chiang [12], Hong and Li [13], Hosseini and Kheradmandi [14], and others.

Unit1 Unit 2 Unit 3 | XXXXXXX | YYYYYYY | ZZZZZZZ |

(a). Series encoding Unit1 Unit 2 Unit 3

XYZ | XYZ | XYZ | XYZ | XYZ | XYZ | XYZ |

1st bit 2nd bit (b). Embedded encoding

Figure 1. Encoding illustration

A unique encoding method is proposed by Chen and Chang [17]. Instead of encoding generator outputs as is usual, they use equal system incremental cost (λ) encoding. Thus, the chromosome consists of one variable, and the length of the string is only related to its resolution. They report the speediness, robustness and effectiveness of their method for a large-scale system, as the length of the string will be entirely independent of the number of generators. However, if the incremental cost functions (λ) of the generators can not be found from the generator output function, this method will not work effectively.

Another unique encoding method is offered by Kumaran and Mouly [18]. Unlike regular encoding schemes in which each string represents a particular value, in their technique each string represents a region of the search space. In this technique, each generator’s operational output is divided into 2 halves. The upper-half region is encoded by 1, and the other is 0. The number of bits in a string then represents the number of generators. Therefore, the size of population needed in this method will be much smaller than common GA methods. But it is not clear how to calculate the fitness function of each generator, because each chromosome has a lot of possible values rather than a specific value.

Besides binary coded GA, some work has been done based on real coded GA (RCGA) for different ELD problems with satisfactory results. Chiang [12], Zhang et al [19], Wong and Wong [25], Abido [15], and Das and Patvardhan [27] use RCGA to solve valve-point loading, however Chiang [12] combines the problem with POZ. Abido [15] and Das and Patvardhan [27] use it in a multiobjective optimisation problem. From their work, it is shown that RCGA is an effective technique for various scenarios and has the capability of being combined with other methods.

B. Objective and Constraint Function Handling An objective function in GA is transformed into a fitness

function. As for constraint functions, if possible they are satisfied in the population construction, such as the minimum and maximum operating limit. The highest encoding value represents the maximum operation limit, and the lowest value for minimum limit. If this technique is impossible or ineffective, the other common technique is to handle the constraint function by including it in the fitness function, along with the objective function. Hence, the fitness function will represent two purposes at the same time, i.e. optimising the objective value and satisfying the constraints. A simple example is a fitness function formulated by Kumaran and Mouly [18], which sums all of the function representing cost, load balance and loss objectives. In this technique each part of the fitness function is equally weighted.

A more complex technique is used by Walters and Sheble [10], where each component of the fitness function is weighted by scaling factors and scaling power factors.

Chen and Chang [17] only use the constraint function deviation as the only component of the fitness function, since their method searches for an appropriate system incremental cost (λ) which satisfies the load balance constraint.

Another similar technique for handling constraints is by implementing penalty factors added to the fitness function (Sheble and Brittig [9]). The aim of penalty factors is to prevent the solution variables from violating constraints.

Many papers use penalty factors for solving constraint problems, such as the papers by Hong and Li [13], Chiang [12], Ma etal [20], Chiang etal [21], and Nanda and Narayan [22]. Hosseini and Kheradmandi [14] do not use penalty factors, but they set the objective function to a specific large value if the solutions do not satisfy the constraints. Other wise, they do not change the objective function value.

There are two basic techniques for solving multiobjective ELD problems. The first is to convert it into one objective function, which usually gives a best solution. Using this technique, Ma etal [20] convert the emission objective into a cost function and Kumaran and Mouly [18] convert the minimisation of losses and the cost objective, along with the load balance constraint, into an index value and then all index values are summed into a single fitness function. The second technique is to use a specific multiobjective method. In this technique, all objective functions are in competition and a search algorithm is used to find a Pareto-optimal solution. In Pareto optimality, a solution cannot be improved upon without adversely affecting the other objectives. Therefore, the result will be a set of optimal solutions that can be presented in a trade-off curve among all objectives. The second approach is used by Abido [15] and Yalcinoz [24].

C. GA operators Typically, GA uses crossover and mutation as operators for

producing individuals for the subsequent generations; therefore all authors use these operators in their papers. The probability of crossover (Pc) is usually high, whereas the

probability of mutation (Pm) is always very low. These probabilities reflect what happens in nature, where probability crossover is high and low probability of mutation by external factors. The values for Pc and Pm are chosen so as to find a suitable balance between fast convergence and increasing the diversity of the population. For example, Pc = 0.9 and Pm = 0.0001 (Hong and Li [13]); Pc = 0.8 and Pm = 0.1 (Chen and Chang [17]), Pc = 0.95 and Pm = 0.0011 (Ma et al [20]), and Pc = 0.75 and Pm = 0.1 (Sheble and Brittig [9]). It is not mentioned how they chose those values. As for mutation rate (Pm), the values range widely from 0.0001 to 0.1, and it is also not clear why this is so.

The elitism technique is clearly described in some papers, such as Yalcinoz and Altun [24], Das and Patvardhan [27], Hong and Li [13], Ongsakul and Ruangpayoongsak [26], Nanda and Narayanan [22], and Tippayachai etal [32], but in many others details are not mentioned. The number of individuals chosen for elitism is also rarely mentioned in the papers. A study by Sheble and Brittig [9] shows that the use of the elitism technique tends to increase the GA performance by accelerating convergence time in the calculations.

Another GA operator used in some papers (Chiang [12], and Chiang etal [21]) is migration. This operator is applied to increase the diversity of the population after a pre-specified generation by generating newly diverse individuals of a small part of the population in the space search. This operator is used in addition to help the algorithm to escape from local extreme value traps.

D. The Size of Population and the Number of Generations The size of population and the number of generations used

in the papers vary widely depending on techniques used, as well as the size and complexity of system modeled. Walters and Sheble [10] as well as Sheble and Brittig [9] utilise 100 chromosomes and 100 generations for a small system with 3 generator units, on the other hand Chen and Chang [17] only use 16 chromosomes and around 20 generations for a large system with 40 generators. For his multiobjective optimisation, Abido [15] selected the size of population and the number of generations as 200 and 500 respectively. Nanda and Narayanan [22] investigate three different population sizes (10, 15, and 20) and three different numbers of generations (5, 10, and 15) for the same systems and assert that in this case the population size of 10 with 15 generations provides an optimum solution. In a unique piece of work by Wong and Wong [25] only two chromosomes are used in each generation, but they produce 40 chromosomes from crossover.

IV. HYBRIDS OF GA WITH OTHER METHODS USED IN ELD

Besides the simplicity of the procedure, GA methods can be improved and easily combined with other methods creating a hybrid GA. The aim of a hybrid GA is to utilise the advantages of other techniques and then, it is hoped, increase the effectiveness of the GA. Some authors investigate the use of Hybrid GA for solving ELD problems. These methods

include Simulated Annealing (SA), Tabu Search (TS), Fuzzy System, and Artificial Neural Networks (ANN).

In an early work a hybrid GA was developed by Wong and Wong [25] who investigated a hybrid of GA and Simulated Annealing, called Genetic Annealing Algorithm (GAA). They developed two types of GAA, i.e. GAA and GAA2. In GAA, the inclusion of SA aims at eliminating premature convergence caused by crossover and attempts to avoid the destructive effects of mutation. This is done by implanted SA in the crossover and mutation processes. In GAA2, the main purpose is to reduce the memory requirement, by developing a very small population size with only two individuals. However, in each generation, 40 feasible and valid chromosomes are produced by performing a maximum of 160 crossover operations. Compared to other GA and SA based methods the simulations show that GAA and GAA2 outperform in terms of the cost objective, and for GAA2, in the execution time as well.

Das and Patvadaran [27] utilise SA in the selection process of a GA and for initialising a Pareto-optimal set in a multiobjective ELD problem. The results of the simulation are also compared with other methods, incuding GAA and GAA2 by Wong and Wong [25], and they assert that their method outperforms the other methods.

A different combination technique proposed by Ongsakul and Ruangpayoongsak [26] is a Genetic Algorithm based on a Simulated Annealing solution (GA-SA). Their algorithm is relatively simple, where both SA and GA are used in sequence. The results are compared with some other methods, including dynamic programming (DP), SA, merit order loading, and local search. The GA-SA outperforms the other methods in the quality of solutions, leading to substantial fuel cost savings especially for a large number of generators.

Integrating GA and a Tabu Search (TS) technique is done by Sudhakaran and Slochanal [29] for the system with combined heat and power (CHP). A TS is characterised by the capability to avoid local optima traps by memorising a short set of recent solutions. TS classify certain moves (trial solutions) as forbidden or “Tabu”, and therefore it prevents cyclical moves. In their proposed method TS is deployed to generate new members (up to 25%) for the next population. They report the superiority of this method compared to conventional methods and other non-hybrid GAs.

Fung et al [28] performed a more complex study. They investigate the application of hybrid GA with SA and with both SA and TS using a single computer and nine parallel computers on a Local Area Network. SA is integrated into the GA procedure during the crossover and mutation process while TS is used afterward to evaluate the results of the previous process. The simulation, which is tested using a 13 generator system, shows that the integration of both SA and TS into the GA method outperforms the integration of SA and GA only, but the execution time of the first hybrid method is much longer than the second one. Unfortunately, they do not take advantage of the availability of a number of networked computers to extend the use for a large scale system.

Kumarappan and Mohan [30] proposed a neuro-hybrid GA method for solving ELD, which consists of three methods, i.e Artificial Neural Network (ANN), TS and GA. As in the other cases they use TS for generating new members to improve the population diversity. As for the ANN, it is used in the final process where a back propagation ANN training algorithm is applied to the final set of results. Training data are produced from the hybrid GA calculation. The number and the range of input-output variables can be extended easily by identifying the input - output relationship using an ANN technique. They report the consistency of the results and the superiority of their performance compare to conventional GA for simulations using a 66 bus system in terms of the cost objective, convergence characteristics and computing speed. Unfortunately, they do not show any comparisons of their computing times.

The hybrid of GA with fuzzy logic controller (FCGA) is studied by Wang et al [31]. They use a fuzzy logic controller in the crossover and mutation processes to improve their results by dynamically modifying the crossover and mutation rate during the process.

Chiang [12] and Chiang etal [21] explore the potential of improving the RCGA method by implanting an Improved Evolutionary Director (IEDO) and Multiplier Updating (MU). IEDO is deployed in the selection process before performing GA operators, and MU is used to overcome the weaknesses of using penalty parameters. If the penalty parameter value is too large, it may lead to an unstable condition. On the other hand, if the penalty value is too small, it may not work effectively as a constraint objective. MU overcomes this problem and small penalty parameter values can be used. They report that their method outperforms the conventional GA in terms of cost objective and computational speed as shown in their simulation using 15, 30, 60 and 90 generators.

Zhang et al [19] improve RCGA by adding a quasi-simplex technique, which is a modification of a variant of the direct search methods (DSM), into the process of producing each new generation, along with crossover and mutation. Furthermore, instead of a randomly-generated initial population in conventional GA, they introduce three specific ways for generating the initial population in order to speed up the overall process. The first is by using the proportional rule, which chooses a generator mix output proportional to the maximum capacity of each generator and a number of random values around these quantities. The second is by using the minimum cost rule, which chooses a number of values based on each generator’s average cost. The third is by using a linear programming approach. The results of the simulation provide convincing results with small standard deviation. Unfortunately, its performance is not compared with other methods.

In order to provide a better Pareto optimal set in a mutiobjective ELD problem, Abido [15] employs hierarchical clustering and a fuzzy base mechanism into the GA procedure. The hierarchical clustering is used to reduce the number of Pareto optimal sets, without destroying the trade-

off characteristics between objectives. A fuzzy-based mechanism is applied at the end to find out the best compromise solution.

V. CONCLUSIONS

Due to its attractive properties, the GA has become very popular for use in various power system applications, including ELD. Many papers on the use of GA for solving the ELD problem have been reviewed. ELD problems of varied complexity have been investigated in the literature using GA with satisfactory results. The various types of complexities of the ELD problems, such as the problems with non-smoothness, discontinuity, non-linearity and non-convex characteristics; the size of system; the type of constraints; and the type of objective functions, have been studied. It is demonstrated in many papers that GA methods are able to cope with such diverse problems.

The GA method provides a wide-range of techniques that can be exploited in order to look for the most effective and efficient way for a specific problem. The techniques utilised in the papers are varied in terms of encoding/decoding techniques; objective and constraint handling, GA operators used, and the size of population and the number of generations. The effective formulation of the ELD problem in terms of GA methods plays an important role in producing an efficient procedure.

The other interesting point is that the GA method can be easily improved and combined with other methods, such as the Simplex Technique from the Direct Search method, SA, TS, Fuzzy systems, ANN, amongst others, forming a hybrid method. By using a hybrid method, the advantages or superior properties of other methods can be utilised to increase the capability and effectiveness of the GA. However, the hybrid GA algorithms become more complicated than those of simple GA methods.

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