Energetic Operation Planning Using Genetic
Algorithms
Patricia Teixeira Leite, Adriano Alber de França Mendes Carneiro, Member, IEEE, and
André Carlos de Ponce Leon Ferreira de Carvalho
Abstract—This paper investigates the application of genetic al-gorithms to optimize large, nonlinear complex systems, particu-larly the optimization of the operation planning of hydrothermal power systems. Several of the current studies to solve this kind of problem are based on nonlinear programming. This approach presents some deficiencies, such as difficult convergence, oversim-plification of the original problem or difficulties related to the ob-jective function approximation. Aiming to find more efficient solu-tions for this class of problems, this paper proposes and investigates the use of a genetic approach. The characteristics of the GAs such as simplicity, parallelism, and generality, can provide an effective solution to these problems. The paper presents an adaptation of the technique and an actual application on the optimization of the op-eration planning for a cascade system composed by interconnected hydroelectric plants.
Index Terms—Genetic algorithms (GAs), hydrothermal systems, operation planning, optimization.
I. INTRODUCTION
I
N a hydrothermal power system (HPS), the available water resources for electrical generation are represented by the inflows to the hydroelectric power plants (HPPs) and the water stored in their reservoirs. Thus, the available resources at each stage of the operation planning horizon depend on the previous use of the water, which establishes a dynamic relationship among the operation decisions taken along the whole horizon [1]. The main issue in energetic operation planning is: what is the most appropriate way to manage the water resources in the present without compromising their availability in the future? The authors believe that the use of artificial intelligence based techniques can provide a valuable contribution to answer this question.The operation of an HPS is characterized as a nonseparable problem, with an equally nonseparable objective function. This is more evident when the system has several HPPs along the same river. In this case, the inflow of a downstream reservoir is composed of the water discharge from the upstream reservoirs. Even when HPPs are located in different rivers, the operation planning still persists as nonseparable, as HPPs are linked by transmission lines. Furthermore, the objective function adopted on the optimization of the Hydrothermal Power Systems Opera-tion Planning (HPSOP) is nonlinear and nonconvex. The
nonlin-Manuscript received November 28, 2000; revised August 8, 2001. P. T. Leite and A. A. F. M. Carneiro are with the Electrical Engineering Department, University of São Paolo, São Carlos, Brazil (e-mail: pa-tricia@sel.sc.usp.br).
A. C. P. L. F. Carvalho is with the Department of Computing Science, Uni-versity of São Paolo, São Carlos, Brazil.
Publisher Item Identifier S 0885-8950(02)01071-4.
earity comes from the thermal subsystem operational cost func-tion and from the hydraulic generafunc-tion funcfunc-tion. The noncon-vexity may be shown by the eigenvalues of a Hessian matrix [2].
The main goal of the optimal operation planning of an HPS is to determine an operation strategy for each power plant. This strategy has to minimize the expected value of the operative cost along the planning horizon and assures that the energy market will be supplied according to reliability rates. The operation cost includes costs from the operation of the thermal units, purchase of energy from neighboring systems and penalties for failure in the load supply.
In hydro-dominated HPSs, optimal operation planning is con-cerned with the possible replacement of the generation from thermal units by generation from hydro units. This concern is due to the almost null operative cost of hydroelectric power plants [3], [4].
Due to the planning complexity and the different aspects that this problem has to address, it may require a chain of models to be solved. This chain of models is divided into long-term, midterm and short-term horizons, where energetic and electric operation aspects are considered. In the Brazilian System, the energetic aspect of the operation is developed during the long-term and midterm horizons.
The long-term planning is performed in a five-year horizon. Due to the reservoir inflow randomness, it is solved using ag-gregation techniques and stochastic dynamic programming [5]. In the mid-term planning, which takes two years, the genera-tion target of the aggregated system is distributed among the HPPs. This planning may be solved using nonlinear program-ming techniques in a deterministic context. The short-term plan-ning is concerned with daily or hourly decisions, where the elec-trical constraints of the system are taken into account [1], [6].
The main focus of this study is the mid-term horizon plan-ning, although the proposed technique may also be applied to other horizons. The traditional algorithms employed to solve the mid-term horizon planning problem, with individual represen-tation of each HPP, have a few inherent difficulties, such as the need for derivative calculus and convergence problems. Addi-tionally, the shape of the objective function, due to its noncon-vexity, introduces numerical problems and difficulties to avoid local minima.
To overcome these difficulties, this paper proposes the use of Artificial Intelligence techniques in the mid-term planning op-timization process. As such, it demonstrates how genetic algo-rithms (GAs) can be applied to determine the optimal operation of real HPPs from the Brazilian System.
174 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
This article is organized as follows. The next section presents the mathematical formulation of the HPSs optimal operation. Section III introduces GAs and describes how they are applied in the optimization of the HPSOP. Section IV shows a real ap-plication of the methodology, comparing the results achieved by using GAs against those achieved by using nonlinear pro-gramming techniques. Finally, Section V presents comments and conclusions.
II. HPS OPERATIONPLANNING
The optimal operation of a HPS, with individual representa-tion to HPPs and deterministic inflows, can be formulated as the following mathematical programming problem [1].
A HPS has as operation cost at a stage ( ), , which is de-fine by the generation cost of the nonhydraulic complementary subsystem. This nonhydraulic generation, , represents the difference between all the demand and all the hydraulic generation , including, if necessary, load shortage. Equa-tions (1) and (2) show how the operation cost can be de-fined.
(1) if
otherwise (2)
As the nonhydro sources must be employed in an economical order according to icreasing marginal costs, the operational cost function is a convex increasing function of nonhydro gen-eration [7].
A general formulation of the energetic operation planning of HPS may be estabilished as the minimization of the operation cost, , along the whole planning horizon [1, ], which is given by:
(3) The total hydroelectric generation at the stage , , is given by the sum of all HPPs generations:
(4) where:
number of hydro plants;
hydroelectric generation function of hydro plant ; water stored in reservoir ;
water released from turbines of hydro plant ; spillage from reservoir .
The hydroelectric generation function of the -th HPP de-pends on the net head, the water released from its turbines and its spilling. This is illustrated in (5).
(5) where
constant that takes into account gravity accelera-tion, water density, turbine-generatior efficiency and conversion factors;
(a)
(b)
(c)
Fig. 1. (a) Uniform crossover, (b) weighted average crossover, and (c) mutation.
total water released from reservoir at the stage
, given by ;
upstream level as a function of storage; downstream level as a function of release; losses in the head caused by the water flow. The dynamic water balance equations, for each reservoir, are given by:
(6)
where: is the incremental water inflow into reservoir ; is the index of immediately upstream reservoirs.
Equation (6) does not consider time lag between hydro plants since, in general, this is not necessary for the operation planning in monthly steps. However, if necessary, it can be considered without any difficulty.
The storage volume and water released by the HPPs are lim-ited by several operation constraints, such as flood control and sailing. The (7) and (8) present the constraints adopted in this work.
(7) (8)
Fig. 2. Proposed algorithm.
The solution of (1)–(8), for a given initial condition , , constitutes the individualized hydrothermal sched-uling. Its solution can be carried out by Nonlinear Programming Techniques such as, for example, the nonlinear network flow al-gorithm (NNFA) especially developed for hydrothermal sched-uling [8].
This article proposes the use of GAs for this optimization problem. The costs associated with NNFA and the GAs based approach are compared. The next Section briefly introduces GAs.
III. OPTIMALOPERATION BYGENETICALGORITHMS
GAs are a search and optimization technique based on ge-netics. Proposed by John Holland in 1976 [9], they have been successfully applied in a growing number of power engineering problems, such as economic dispatch [10], hydrothermal coor-dination [11], reactive power planning [12], power plant control [13] and generation expansion planning [14].
Every search or optimization problem has a set of compo-nents, such as a search space, where possible solutions for a given problem can be found and a cost or evaluation function, which measures the fitness or quality of each particular solution [9], [15].
In opposition to traditional optimization methods, GAs work with a population of solution candidates in parallel. As a result, they can simultaneously search different areas of the solutions space. According to Goldberg [15], GAs differ from traditional methods of search and optimization in four main aspects:
• GAs work with a code of a group of solution parameters and not with their own solution parameters;
• GAs work with several possible solutions and not with a single solution;
• GAs use cost information or reward functions instead of derivative or other auxiliary knowledge;
• GAs use probabilistic transition rules, instead of determin-istic rules.
Each individual in the population is called a chromosome. A chromosome is composed of genes, which are the features present in the solution provided by the chromosome.
The GAs initial population is usually formed by a random group of individuals, which can be seen as first guesses to solve the problem. The processing goes through a number of itera-tions, named generations. For each generation, a new popula-tion is derived from the previous populapopula-tion. For such, the cur-rent population is evaluated and for each individual a score is given, reflecting the quality of the solution it represents. A per-centage of the most capable individuals is selected for a repro-duction phase, with the others being discarded. In the reproduc-tion phase, operators based on evolureproduc-tionary theory are applied to the selected individuals, generating a new population. Sev-eral generations may be required before a suitable solution is reached.
The next section presents the approach proposed in this paper for the optimization of the HPSOP problem using GAs. A. Proposed Approach
To optimize the HPS operation through GAs, it is neces-sary to represent the original system in an appropriate way, en-coding the individuals and adapting the reproduction operations. Most of the implementations of GAs encode the individuals by sequences of binary values. For a better representation of the HPSOP problem, real values are used in this paper. Each of these values, which is a gene, represents the percentage of water stored in a given reservoir in a certain month.
In order to select the best individuals in the current popula-tion, an evaluation function calculates the operative costs in the
176 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
Fig. 3. Part of the Brazilian hydroelectric power system.
Fig. 4. A typical individual.
planning period for each individual in the population. The in-dividuals are then ranked according to their costs and the indi-viduals with lower costs have a higher chance of being selected using a roulette wheel.
Three different methods to define the initial population are used in the experiments:
• Random volumes; • Full volumes;
• Volumes produced by NNFA.
To guarantee that the best individuals are not lost from one generation to another, an elitism operator is employed. This op-erator guarantees that the best individuals automaticaly pass to the next generation. Three other genetic operators are also used: a modified uniform crossover, weighted average crossover and mutation.
Applied according to a crossover rate, the crossover operator randomly selects pairs of chromosomes, which are called parents. By exchanging a group of genes from each pair, one offspring is produced. For the modified uniform crossover, a random mask, based in the fittness of the parents, is used to define which genes from each parent are copied to the offspring. Fig. 1(a) shows an example of the modified uniform crossover. The uniform crossover was used to keep the values of the parents’ genes. The average crossover was employed to allow searches in regions near to those defined by the values of the parents’ genes. It can be noted that, with weight values 1 and 0, the average crossover behaves like the uniform crossover.
For the weighted average crossover, the value of each gene in the offspring is the weighted average of the correspondent
TABLE I HYDROPLANTSDATA
TABLE II OPERATIONCOST
genes in the parents. Fig. 1(b) illustrates the weighted average crossover.
The mutation operator randomly changes the value of ran-domly chosen genes of ranran-domly selected chromosomes. The number of genes that are subjected to mutation is defined by a mutation rate, show in Fig. 1(c).
Finally, a group of individuals, similar to other individuals in the same population, may have a small change in their genes’ values. This operation aims to increase the diversity of the pop-ulation and explore the neighborhood of high rank individuals. New individuals with improved cost substitute for the original individuals in the population. Fig. 2 shows the whole algorithm. The processing stops when either the algorithm converges or a maximum number of generations has been reached.
IV. EXPERIMENTS
In order to evaluate the performance of the proposed ap-proaches work on the operation planning problem, they are applied to a HPSOP problem. The system investigated is com-posed of 7 large hydroelectric plants from the Brazilian system. These plants are: Emborcação, Itumbiara, São Simão, located in Paranaiba River, Furnas, Marimbondo, Água Vermelha, located in Grande River and Ilha Solteira, located in the Paraná River, as shown in Fig. 3. Several tests were carried out using different operative situations.
In the experiments performed, each individual represents the volume of a particular reservoir during a planning period. The main goal is to minimize the operation costs associated with this period, in order to assist the requested demand within a reliability limit.
(a)
(b)
Fig. 5. Optimal storage trajectory (a) [NNFA] and (b) [NNFA-Gas].
Fig. 4 illustrates an example of a typical individual using a planning period of one year. The first gene represents the per-centage of the volume in plant A in the first month, which is equal to 100% of the useful volume. The second gene repre-sents a useful volume of 93% for the same plant in the second month and so forth. The tool user may define the initial and final values.
The tests adopted the LTA (Long Term Average) inflows, with initial and final volumes in 100% of the storage capacity, a planing horizon of 24 months and a demand varying along the time. The average demand was assumed to be 80% of the in-stalled capacity.
The discount rate employed was of 10% a year. May, the be-ginning of the dry period, was adopted as the initial month and April, the end of the higher inflows period, was considered the final month.
Table I presents installed capacity, volume and turbine of each plant.
To evaluate the potential of using GAs, simulation experi-ments were performed with 1, 3 and 7 hydroplants. For each group of hydroplants, four sets of experiments were performed. 1) The first set of experiments defined the volumes for each plant using the NNFA algorithm [1]. Using these
vol-(a)
(b)
Fig. 6. (a) Thermal generation [Isolated hydroelectric plant] and (b) thermal generation [Isolated hydroelectric plant].
umes, the costs are calculated by (1)–(8). In these simula-tions, all reservoirs had an initial volume equal to 100%; 2) The second set of experiments used GAs. In these experi-ments, the initial volume for each reservoir was randomly defined with values from 0% to 100%;
3) The third set of experiments also used GAs. But now, similar to the experiments with NNFA, the initial volume of each reservoir was equal to 100%;
4) Finally, in the fourth set of experiments, the volumes de-fined by NNFA were used as the initial population by GAs.
Each experiment using GAs had a population of 20 individ-uals. Table II presents the operation costs achieved by these four sets of experiments using three groups of hydroplants. The experiments with one plant used the Emborcação Hydroelec-tric. The experiments with three hydroplants used Emborcação, Itumbiara and São Simão power plants.
The labels NNFA, GAs (random), GAs (100%) and NNFA-GAs, refer to the results achieved by the Nonlinear Net-work Flow Algorithm, GAs with a random initial population, GAs with a full capacity initial population and GAs with the initial population defined by NNFA, respectively. Each result represents the average of three runs.
It is important to point out that NNFA uses a unique tion and, through methods of traditional optimization, this solu-tion is improved. In experiments with GAs, each individual is a
178 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, FEBRUARY 2002
(a)
(b)
Fig. 7. (a) Thermal generation [three hydroelectric plants] and (b) thermal generation [three hydroelectric plants].
possible solution for the problem. As a result, experiments with GAs require a longer running time. However, the running times obtained in the experiments with GAs do not represent an ob-stacle to their application for mid-term planning.
According to the results achieved, the combination of the Nonlinear Network Flow Algorithm with GAs presented the best performance for all these groups. The GAs with initial vol-umes of 100% also reduced the costs achieved by the nonlinear network flow algorithm in these groups. Although the perfor-mances obtained by all four approaches are very close, it must be observed that a reduction of 1% in the operation costs repre-sents a large economy for the system.
The tests with the seven hydroelectric plants are presented in more detail in Fig. 5(a) to (b). The cascade behavior of both the NNFA and NNFA-GAs approaches can be seen. According to these figures, the curves obtained by the NNFA-GAs approach are smoother, suggesting that NNFA-GAs are able to capture with more accuracy the system behavior.
Fig. 5(a) and (b) also show that the hydroelectric plants of Emborcação and Itumbiara, the upstream plants, regulated the water flow, oscillating their volumes and reaching the maximum volumes at the end of the planning period. On the other hand, São Simão, the downstream reservoir, presented maximum ef-ficiency during the whole period [1], staying at the maximum level, probably due to the head effect.
(a)
(b)
Fig. 8. (a) Thermal generation [seven hydroelectric plants] and (b) thermal generation [seven hydroelectric plants].
Still, according to Fig. 5(a) and (b), only the Furnas plant os-cillated its volume in order to regulate the system. The Marim-bondo, Água Vermelha and Ilha Solteira plants stayed at the same maximum level. The results indicate that GAs captured these different characteristics of the plants without difficulty.
Since the operation costs are given by a polynomial function, there is a tendency to consider that a more uniform complemen-tary thermal generation along the time implies a lower operation cost. Figs. 6(a)–8(b) present the thermal generation when one, three and seven plants are used.
The operational cost function has been considered as a quadratic function. This approximation do not essentially change the conclusions achieved, based on comparisons. Moreover, any other function may be easily adopted on the proposed approach without loss of generality.
It should be noted that, with the adopted quadratic function, the marginal operation cost is given, except for the units, by the amount of the thermal generation showed in Figs. 6(a)–8(a).
It can be observed in Figs. 6(a)–8(b), the trajectories pro-duced by NNFA-GAs are usually smoother than those propro-duced by NNFA, GAs (random) and GAs (100%), indicating a more stable planning. Figs. 6(a), 7(a), and 8(a) present the curves pro-duced by the four sets of experiments.
In order to have a clearer view of the improvement achieved by the NNFA-GA technique over the original NNFA technique,
Figs. 6(b), 7(b) and 8(b) show a zoom of the curves obtained by the NNFA and NNFA-GAs approaches.
Note that in these figures the apparent large oscillations of the curves are due only to the used scale.
The results achieved suggest that GAs can provide efficient computational tools to support HPSOP. They can reduce the overall cost, deal with the complexity of the system model and assist the data analysis.
V. CONCLUSION
From the results obtained, it can be observed that the use of GA, reduced the costs achieved by NNFA in the optimization of HPSOP.
This proposed technique allows an individualized represen-tation of the hydroplants, is relatively easy to implement and presents a processing time compatible with the application pro-posed. Due to its intrinsic characteristic, this method overcomes several difficulties presented by the classical techniques. In this paper, the GAs based methods were applied to one, three and seven large hydroelectrics in cascade, showing their potential to deal with plants presenting different behaviors.
It can be argued that conventional search techniques may reach an efficient solution in a shorter computation time; how-ever, these methods have difficulties with the problem complex-ities previously discussed.
One of the main advantages of using GAs, is that, for each additional problem constraint, only the equation used to define each individual performance needs to be modified. Thus, the im-plementation of the technique does not become more complex. New tests are being performed with larger systems and sev-eral alternative operative conditions. The results achieved so far indicate that the proposed approaches can be an efficient alter-native or complementary technique for the planning of an HPS operation.
ACKNOWLEDGMENT
The authors would like to thank the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and the Department of Computing and Information Science, University of Guelph, Canada.They would also like to thank Dr. S. Massago, Federal University of São Carlos, São Carlos, Brazil, and Dr. M. Wineberg, Department Computing and Information Science, University of Guelph, for their valuable comments.
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Patricia Texeira Leite received the B.Sc. degree in electrical engineering from
the Federal University of Mato Grosso, Cuiabá, and the M.Sc. degree in power systems from the Electrical Engineering Department, São Carlos Engineering School, University of São Paulo, São Paulo, Brazil, where she is currently pur-suing the Ph.D. in electrical engineering.
Her M.S. thesis focused on the application of artificial intelligence techniques in the planning of hydroelectric power system operation.
Adriano Alber de França Mendes Carneiro received the B.Sc. degree in
elec-trical engineering from the Catholic University of São Paulo, São Paulo, Brazil, the M.Sc. degree in power systems from the Federal School of Engineering of Itajubá, Itajubá, Brazil, and the Ph.D. degree in operation planning of power systems from the State University of Campinas, Brazil.
He is currently an Assistant Professor with the Electrical Engineering De-partment, São Carlos Engineering School, University of São Paulo, São Paulo, Brazil.
André Carlos Ponce Leon Ferreira de Carvalho received the B.Sc degree in
computer science and the M.Sc. degree in informatics from Federal University of Pernambuco, Pernambuco, Brazil, and the Ph.D. degree in electronic engi-neering from the University of Kent, Canterbury, U.K.
He is currently an Associate Professor with the Department of Computer Sci-ence, University of São Paulo, São Paulo, Brazil.
