Application of Cultural Algorithm to Generation Scheduling of Hydrothermal Systems

http://www.paper.edu.cn

Energy Conversion and Management 47 (2006) 2192–2201

Application of cultural algorithm to generation scheduling of hydrothermal systems

Xiaohui Yuan a,*, Yanbin Yuan b

a Department of Hydropower Engineering, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan, Hubei 430074, PR China b Department of Environment Engineering, Wuhan University of Technology, Wuhan 430071, PR China

Received 30 June 2005; accepted 12 December 2005 Available online 21 February 2006

Abstract

The daily generation scheduling of hydrothermal power systems plays an important role in the operation of electric power systems for economics and security, which is a large scale dynamic non-linear constrained optimization problem. It is difficult to solve using traditional optimization methods. This paper proposes a new cultural algorithm to solve the optimal daily generation scheduling of hydrothermal power systems. The approach takes the water transport delay time between connected reservoirs into consideration and can conveniently deal with the complicated hydraulic coupling simul- taneously. An example is used to verify the correctness and effectiveness of the proposed cultural algorithm, comparing with both the Lagrange method and the genetic algorithm method. The simulation results demonstrate that the proposed algorithm has rapid convergence speed and higher solution precision. Thus, an effective method is provided to solve the optimal daily generation scheduling of hydrothermal systems. Ó 2005 Elsevier Ltd. All rights reserved.

Keywords: Cultural algorithm; Hydrothermal systems; Daily generation scheduling

1. Introduction

The short term daily generation scheduling of hydrothermal power systems has been one of the most important and challenging optimization problems in the economic operation and control of power systems. Along with the increase in exploitation and utilization of hydraulic resources, more and more cascaded hydropower plants are established. The innate operational ﬂexibility of hydropower plants and coordinating compensatory functions between cascaded hydropower plants make them play a very important role in the operation of the power system for economics and security. Cascaded hydropower plants are related to each other in both power and hydraulic aspects. A certain delay in time of the water used in generation or spilling in upstream

* Corresponding author. Tel.: +86 2762994000; fax: +86 2767884228. E-mail address: [email protected] (X. Yuan).

X. Yuan, Y. Yuan / Energy Conversion and Management 47 (2006) 2192–2201 2193

plants will affect the generation and spilling of downstream plants, while the regulation role of reservoirs and the constraint of the released water ability of downstream plants will make an impact on the water planning of the upstream ones in turn. Therefore, the optimal daily dispatch of hydrothermal power systems is a large scale, dynamic and constrained non-linear programming problem with time delay and is very difficult to solve. Scholars have proposed many methods to solve this problem in the past decades, including calculus of vari- ations [1], maximum value principle [2], functional analysis [3], dynamic programming [4–6], network flow and linear programming [7,8], non-linear programming [9], Lagrange relaxation method [10,11], artificial neural networks [12,13] genetic algorithm and evolutionary programming [14–16] and chaotic optimization [17], but these methods have one or more drawbacks, such as dimensionality difficulty, inability to handle a non-linear cost function, premature convergence and being trapped into local optimum or taking too much computation time. Therefore, improving the current optimization method and exploring new methods to solve the optimal daily generation scheduling of hydrothermal power systems has important real life significance. In recent years, a new optimization method known as the cultural algorithm (CA) proposed by Dr. R. Reynolds in 1995 has become a candidate for many optimization applications due to its flexibility and efficiency. The cultural algorithm is a technique that incorporates domain knowledge obtained during the evolutionary process so as to make the search process more efficient. It has been successfully applied to solve optimization problems and promises to overcome some shortcomings of the above optimization methods. This paper presents a new cultural algorithm to solve the daily generation scheduling of hydrothermal power systems. Simulation results demonstrate the feasibility and effectiveness of the proposed method in terms of convergence speed and solution precision compared with both the Lagrange method and the genetic algorithm.

2. Problem formulations

The optimal daily generation scheduling problem of hydrothermal power systems can be stated as ﬁnding the water release from each reservoir and the corresponding unit thermal generation over all the planning time intervals so as to minimize the total cost of fuel for thermal generation while satisfying diverse hydraulic, thermal and load balance constraints. Typically, the total planning period is one day and the time interval is 1 h.

2.1. Notation

To formulate the problem and its solution mathematically, the following notation used in this paper is ﬁrst introduced:

F composite fuel cost function t fiðP siÞ fuel cost of ith thermal plant Ns number of thermal plants Nh number of hydro plants ai, bi, ci thermal generation coeﬃcients of ith plant t P D system load demand at time interval t t P sj power generation of thermal plant j at time interval t t P hi power generation of hydro plant i at time interval t min P si minimum power generation of thermal plant i max P si maximum power generation of thermal plant i min P hi minimum power generation of hydro plant i max P hi maximum power generation of hydro plant i t V i water volume of reservoir i at end of time interval t min V i minimum water volume of reservoir i max V i maximum water volume of reservoir i t Qi water discharge of hydro plant i at time interval t min Qi minimum water discharge of hydro plant i 中国科技论文在线 http://www.paper.edu.cn

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max Qi maximum water discharge of hydro plant i begin V i initial storage volume of reservoir i end V i final storage volume of reservoir i at end of dispatching horizon si water travel time from plant i 1 to plant i t Si water spillage of hydro plant i at time interval t t Ii natural inflow into reservoir i at time interval t C1i, C2i, C3i, C4i, C5i, C6i power generation coefficients at ith hydropower plant T total time horizon t time index, t =1,2,...,T

2.2. Objective function and constraints

The objective function considered is to minimize the summation of the fuel cost for all the thermal plants over the complete dispatch period. The unit fuel cost is generally assumed to be a quadratic function of thermal generation power. So, the optimal daily generation scheduling of hydrothermal power systems can be mathematically formulated as an optimization objective function as follows:

XT XN s XT XN s no t t t 2 min F ¼ fiðP siÞ¼ ai þ biP si þ ciðP siÞ ð1Þ t¼1 i¼1 t¼1 i¼1 subject to the following constraints:

2.2.1. System load balance

XN h XN s t t t P hi þ P sj ¼ P D t ¼ 1; 2; ...; T ð2Þ i¼1 j¼1

2.2.2. Thermal plant power generation limits

min 6 t 6 max P si P si P si i ¼ 1; 2; ...; N s; t ¼ 1; 2; ...; T ð3Þ

2.2.3. Hydro plant power generation limits

min 6 t 6 max P hi P hi P hi i ¼ 1; 2; ...; N h; t ¼ 1; 2; ...; T ð4Þ

2.2.4. Hydro plant discharge limits min 6 t 6 max Qi Qi Qi i ¼ 1; 2; ...; N h; t ¼ 1; 2; ...; T ð5Þ

2.2.5. Reservoir storage volumes limits

min 6 t 6 max V i V i V i i ¼ 1; 2; ...; N h; t ¼ 1; 2; ...; T ð6Þ

2.2.6. Initial and terminal reservoir storage volumes

0 begin T end V i ¼ V i V i ¼ V i i ¼ 1; 2; ...; N h ð7Þ

2.2.7. Water dynamic balance equation with travel time

t t1 t tsi tsi t t V i ¼ V i þ Ii þ Qi1 þ Si1 Qi Si i ¼ 1; 2; ...; N h; t ¼ 1; 2; ...; T ð8Þ 中国科技论文在线 http://www.paper.edu.cn

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2.3. Hydropower generation characteristics

The power generated from a hydro plant is related to the reservoir characteristics as well as the water discharge. A number of models have been used to represent this relationship. In general, the hydro generator power output is a function of the water discharge through the turbine and the net head. Since the net head is a function of the volume of stored water, hydropower generation can be written in terms of turbine discharge rate and storage, and a frequently used expression is t t 2 t 2 t t t t P hi ¼ C1i ðV iÞ þ C2i ðQiÞ þ C3i V i Qi þ C4i V i þ C5i Qi þ C6i ð9Þ

3. Solution technique—cultural algorithm

3.1. Notions of cultural algorithm

Some social researchers have suggested that culture might be symbolically encoded and transmitted within and between populations as another inheritance mechanism. According to this idea, Reynolds [18] proposed a cultural algorithm model in which cultural evolution is seen as an inheritance process that operates at two levels: the micro-evolutionary and the macro-evolutionary levels. At the micro-evolutionary level, individuals are described in terms of behavioral traits (which could be socially acceptable or unacceptable). These behavioral traits are passed from generation to generation using several socially motivated operators. At the macro- evolutionary level, individuals are able to generate ‘‘mappa’’, or generalized descriptions of their experiences. Individual mappa can be merged and modified to form ‘‘group mappa’’ using a set of operators. Both levels share a communication link. As seen in Fig. 1, cultural algorithms operate on two spaces. First, they operate on the population space, in which a set of individuals (called population) is adopted. The second space is the belief space, where the knowledge acquired by the individuals, along the evolutionary process is stored. To unify both spaces, a communication protocol is established such that it dictates rules regarding the type of information to be exchanged between these two spaces. For example, to update the belief space, the individual experiences of a select set of individuals are incorporated. This select group of individuals is obtained with the function acceptance, which is applied to the entire population. On the other hand, the operators that modify the population and the selection operator are modified by the function influence. This function acts in such a way that the individuals resulting from the applications of the operators tend to approach the desirable behavior while staying away from undesirable behaviors. Such desirable and undesirable behaviors are defined in terms of the information stored in the belief space. These two functions are used to establish the communication between the population space and the belief space. The pseudo-code of a cultural algorithm is shown as follows:

Generate the initial population Initialize the belief space

Belief Space Adjust

Acceptance Influence Communication Function Function Protocol

Inherit Performance Selection Function Population Space

Fig. 1. Spaces used by a cultural algorithm. 中国科技论文在线 http://www.paper.edu.cn

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Evaluate the initial population Repeat Update the belief space (with the individuals accepted) Apply the variation operators (under the inﬂuence of the belief space) Evaluate each child Perform selection While the end condition is not satisﬁed

3.2. Cultural algorithm for constrained optimization [19,20]

Consider that the optimal daily generation scheduling of hydrothermal power systems in Section 2 can be converted into the following constrained optimization problem: min F ¼ f ðQÞ s.t. 6 ð10Þ giðQÞ 0

hjðQÞ¼0 T where Q =[Q1,Q2,...,Qn] is a vector of the n discharge variables of the optimization problem;

n ¼ T N h

i ¼ 1; 2; ...; ð4 T N h þ 2 T Þ

j ¼ 1; 2; ...; N h When using a cultural algorithm to solve non-linear constrained optimization problems (10), the acceptable beliefs can be seen as constraints that direct the population at the micro-evolutionary level. Therefore, the key problem is how to represent and store the knowledge about the constraints in the belief space of the cultural algorithm. In this paper, we adopt an n-dimensional, regional based schema called a belief cell to support the acquisition, storage and integration of knowledge about the non-linear constraints in the cultural algorithm [19]. The belief space can contain a set of these schemata, which can be used to guide the search in a direct way by pruning the infeasible regions and promoting the promising regions. Meanwhile, the population component of the cultural algorithm will be adopting evolutionary programming (EP). The exploration of the EP population will serve to modify these belief cell schemata, which are used to guide the production of new individuals in the population component in turn.

3.2.1. Constraint knowledge and belief cells We can find that problem constraints cut the domain space into smaller different characters regular sub- regions. Some areas are feasible, which satisfy all the constraints, while others are infeasible. Information about the feasibility of the solutions is included in the belief space. Taking advantage of the intervals of good solutions that are stored in the normative portion of the belief space, belief cells are created. In fact, these belief cells are a subdivision of the intervals of good solutions, such that the feasibility of the cells can be deter- mined. When the intervals of the variables are modified, the cells are also modified. There exist four types of cells (see Fig. 2): feasible, infeasible, semi-feasible and unknown, that is, some cells are feasible since they are totally in feasible regions, some cells are infeasible since they are totally in infeasible regions, some cells are semi-feasible because they span both feasible and infeasible areas and others are unknown (if that region has not been explored yet). Consequently, the cells can be used to represent and store certain patterns of constraint knowledge of the domain space. The influence that the belief space has on the generation of off spring consists of moving individuals that lie on infeasible cells towards feasible cells. Simply speaking, the belief cells act like a navigation map that can guide the optimization search in a direct way by pruning the infeasible areas and promoting the promising areas. Fig. 2 is used to illustrate the divided belief cell. The part at the left illustrates the feasible region of a problem. In this case, the lines represent the boundary between the feasible and infeasible regions. The right part 中国科技论文在线 http://www.paper.edu.cn

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Fig. 2. Illustration of belief-cells.

illustrates the representation of the constraints part of the belief space for the search space of the same problem.

3.2.2. Initialization of the belief space The lower and upper boundaries of the promising intervals for each variable are stored in the normative part of the belief space together with the fitness for each extreme of the interval. For the minimization problem, the initial fitnesses are set to positive infinite values. Regarding the constraints of the problem, the interval given in the normative part is subdivided into s sub- intervals such that a portion of the search space is divided into hypercubes. The following information for each hypercube is stored: number of feasible individuals within that cell, number of infeasible individuals within that cell and the type of region. The type of region depends on the feasibility of the individuals within it. Four types are defined:

(1) if feasible individuals = 0 and infeasible individuals = 0, then cell type = unknown (2) if feasible individuals > 0 and infeasible individuals = 0, then cell type = feasible (3) if feasible individuals = 0 and infeasible individuals > 0, then cell type = infeasible (4) if feasible individuals > 0 and infeasible individuals > 0, then cell type = semi-feasible

3.2.3. Updating the belief space The update of the constraints part consists only of adding any new individuals that fall into each region to the counter of feasible individuals. The update of the normative part is more complex. When the interval of each variable is updated, the cells or hypercubes of the restrictions part are changed, and the counters of the feasible and infeasible individuals are reinitialized. So, this update is done taking into consideration only a portion of the population, which is selected by the function accept (), taking the percentage of the total population size given by the user beforehand.

3.2.4. Inﬂuence of beliefs in the mutation operator According to the following rules, mutation takes place for each variable of each individual for the inﬂuence of the belief space [20].

(1) If the variable is outside the interval given by the normative part of the constraints, then it attempts to move within such interval through the use of a random variable. 中国科技论文在线 http://www.paper.edu.cn

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(2) If the variable is within a feasible, a semi-feasible or an unknown hypercube, a perturbation is done to try to place it within the same hypercube or very close to it. (3) If the variable is in an infeasible cell, we try to move it ﬁrst to the closest semi-feasible cell. If none is found, we try to move it to the feasible or unknown closest cell. If that does not work either, then we move it to a random position within the interval deﬁned by the normative part.

3.2.5. Selection Tournament selection is performed considering the entire population according to the following rules:

(1) If both individuals are feasible or both are infeasible, then the individual with the best ﬁtness value wins. (2) Otherwise, the feasible individual always wins.

When the tournament selection ﬁnishes, the p (population size) individuals with the larger number of vic- tories are selected to form the next generation individuals.

3.3. Cultural algorithm implementation

The procedures of the cultural algorithm for a solution of a constrained optimization problem is described as follows:

(1) Initialize p population individual solutions using a uniform distribution random method. (2) Assess the performance score of each individual by the given objective function. (3) Initialize the belief space. (4) Generate p new off spring solutions using a variation operator as modified by the influence function. (5) Assess the performance score of each off spring solution by the given objective function. (6) For each individual, select c competitors at random from the population of size 2p. Conduct pair-wise competitions between the individual and the competitors. (7) Select p solutions that have the greatest number of wins to form the next generation individuals. (8) Update the belief space by accepting individuals using the acceptance function. (9) Repeat to (4) until the termination condition is achieved.

4. Numerical examples

In order to verify the correctness and eﬀectiveness of the proposed cultural algorithm, a simulation calculation has been applied on the optimal daily generation scheduling of hydrothermal power systems, which con- sist of cascaded hydropower plants with two reservoirs and a number of thermal units represented by a single equivalent thermal plant. The scheduling period is 24 h with 1 h time intervals. The hydro generation data and load demand data are given in Tables 1–3.InTable 2, the units of storage are 103 m3 while the units of the water discharge rate are 103 m3/h. The water travel time from the upstream hydropower plant to the downstream hydropower plant is 3 h. The composite thermal plant fuel cost coeﬃcients a, b and c are taken as 0, 10.0 and 1.0, respectively. With the data given above, the proposed cultural algorithm, coded in Visual C++ 6.0, was applied to solve the daily optimal generation scheduling of the cascaded hydropower system. The parameter values chosen in the calculation are: the population size is 120, the maximal iterative generations number is 1500. Under the

Table 1 Hydro generation coeﬃcients

Plant C1 C2 C3 C4 C5 C6 1 0.001 0.1 0.01 0.4 4.0 30 2 0.001 0.1 0.01 0.3 3.0 30 中国科技论文在线 http://www.paper.edu.cn

X. Yuan, Y. Yuan / Energy Conversion and Management 47 (2006) 2192–2201 2199

Table 2 Characteristics of hydro plants

Plant Vmin Vmax V0 VT Qmin Qmax 1 95 135 100 115 5 15 2 155 185 170 170 5 15

Table 3 Load demand Time 1 2 3 456789101112 Load 90 90 95 95 100 100 105 105 110 110 115 115 Time 13 14 15 16 17 18 19 20 21 22 23 24 Load 120 120 115 115 110 110 105 105 100 100 90 90

Plant 1 Plant 2 60

20 generation(MW) 10

0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324 Hour

Fig. 3. Hourly hydro plants power generations.

Plant 1 Plant 2

) 14 3 12 10 8 Water 6 4

discharge (X10000m 2 0 1 32 4 5 6 7 98 101112131415161718192021222324 Hour

Fig. 4. Hourly hydro plant discharge trajectories.

chosen parameters, starting with the random initial population, the ﬁnal hourly hydropower plant power out- puts and the water discharge results are shown in Figs. 3 and 4, respectively, while the total fuel cost of the thermal plant is 37678.9$. To validate the results obtained with the proposed algorithm, the same problem was solved using the genetic algorithm and the Lagrange method. A comparison of the total fuel cost of the 中国科技论文在线 http://www.paper.edu.cn

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Table 4 Comparison of total fuel cost of the thermal plant with other methods Method Total thermal plant cost ($) Genetic algorithm (GA) 42027.8 Lagrange method (LM) 37697.8 Proposed cultural algorithm (CA) 37678.9

56000

52000

48000

Genetic Algorithm

Cost($) 44000

40000 Cultural Algorithm

36000 0 300 600 900 1200 1500 generations

Fig. 5. Variation of cost with the number of generations.

thermal plant with that of the other methods, the genetic algorithm and Lagrange method, are shown in Table 4. The total fuel cost of the thermal plant is 42027.8$ using the genetic algorithm, while it is 37697.8$ using the Lagrange method. The variation relationship of the thermal plant fuel cost with the evolutionary generation number is shown in Fig. 5. From Fig. 5, it is clear that the results obtained using the cultural algorithm are much better than the corresponding values computed from the genetic algorithm in terms of convergence speed and solution precision. At the same time, the simulation results from Fig. 5 and Table 4 demonstrate that the proposed cultural algorithm has rapid convergence speed and higher solution precision for solving the optimal daily generation scheduling of the cascaded hydropower system.

5. Conclusions

In the daily generation scheduling problem of hydrothermal power systems, the complexity introduced by the cascade nature of the hydraulic network, the scheduling time linkage, non-linear relationships in the problem variables and the water transport delay time, has made this problem very diﬃcult to solve using traditional optimization methods. This paper presents a new cultural algorithm to solve the optimal daily generation scheduling of hydrothermal systems. Not only can complicated hydraulic coupling be dealt with conveniently but also non-linear relationships in the problem variables and the water transport delay time are all taken into account. Finally, the proposed cultural algorithm is applied to solve a hydrothermal systems economic operation scheduling problem. It is clear from the simulation results that the proposed cultural algorithm can avoid premature phenomena in the evolutionary process and obtain a better quality solution with higher precision and quick convergence speed, so it provides a new eﬀective method for the solution of the optimal daily generation scheduling of hydrothermal systems, yet it is simple as well as being easy to implement.

Acknowledgement

The authors would like to thank the National Natural Science Foundation of China (Under supported No. 50409010,50309013,40572166,50539140). 中国科技论文在线 http://www.paper.edu.cn

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