RESEARCH ARTICLE
A Chaotic Quantum Behaved Particle Swarm Optimization Algorithm for Short-term Hydrothermal Scheduling
Chen Gonggui^{*, 1}, Huang Shanwai^{1}, Sun Zhi^{2}
Article Information
Identifiers and Pagination:
Year: 2017Volume: 11
First Page: 23
Last Page: 37
Publisher Id: TOEEJ-11-23
DOI: 10.2174/1874129001711010023
Article History:
Received Date: 05/05/2016Revision Received Date: 03/11/2016
Acceptance Date: 29/11/2016
Electronic publication date: 25/01/2017
Collection year: 2017
open-access license: This is an open access article licensed under the terms of the Creative Commons Attribution-Non-Commercial 4.0 International Public License (CC BY-NC 4.0) (https://creativecommons.org/licenses/by-nc/4.0/legalcode), which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
Abstract
This study proposes a novel chaotic quantum-behaved particle swarm optimization (CQPSO) algorithm for solving short-term hydrothermal scheduling problem with a set of equality and inequality constraints. In the proposed method, chaotic local search technique is employed to enhance the local search capability and convergence rate of the algorithm. In addition, a novel constraint handling strategy is presented to deal with the complicated equality constrains and then ensures the feasibility and effectiveness of solution. A system including four hydro plants coupled hydraulically and three thermal plants has been tested by the proposed algorithm. The results are compared with particle swarm optimization (PSO), quantum-behaved particle swarm optimization (QPSO) and other population-based artificial intelligence algorithms considered. Comparison results reveal that the proposed method can cope with short-term hydrothermal scheduling problem and outperforms other evolutionary methods in the literature.
1. INTRODUCTION
The short-term hydrothermal scheduling (SHTS) as a significant and constrained optimization problem plays a vital role in power system. The complex and nonlinear peculiarities of SHTS problem make finding the efficient global optimal solution a huge challenge. The objective of SHTS is the determination of power generations among hydro plants and thermal plants with the result that the fuel cost of thermal plants is minimized over a schedule horizon of one day when meeting various hydraulic and electrical operational constraints. Usually, the constraints include system load balance, initial and terminal reservoir storage volume limits as well as water dynamic balance as the equality constraints and power limits of thermal plants and hydro plants, reservoir storage volume limits as well as discharge limits of hydro plants as the inequality constraints.
In the past few decades, many methods are implemented for solving the SHTS problem such as dynamic programming (DP) [1], linear programming (LP) [2] and Lagrange relaxation (LR) [3]. DP algorithm can actually tackle a quite general class of dynamic optimization problems, including the ones with nonlinear constraints. It has been widely used to solve short-term hydrothermal scheduling problem. However, the disadvantage of DP is obvious with the growth of computational and dimensional requirements in a larger system. The linear programming method is aimed at linearizing the hydro power generation depending on water discharge so as to ignore the head change effect and reduce the accuracy of the solution. The basic idea of Lagrange relaxation method is to relax demand and reserve requirements using Lagrange multipliers. LR method is efficient in dealing with large-scale problems, however, it is easy to generate dual optimal solution which rarely satisfies the power balance and reserve constraints. Additionally, the convergence and accuracy of LR depend on the Lagrange multipliers updating methods. In general, those traditional methods have lost the superiority when faced with the complicated nonlinear constraints and the non-convex short-term hydrothermal scheduling problem.
Other than the above methods, many artificial intelligence algorithms have been successfully applied to overcome the drawbacks of traditional algorithms in many areas including short-term hydrothermal scheduling problem [4, 5]. Typical algorithms such as evolutionary programming (EP) [6], genetic algorithm (GA) [7], differential evolution (DE) [8, 9] clonal selection (CS) [10] and particle swarm optimization (PSO) [11] have obtained good effect. However, those algorithms are easy to trap into the local optimum and sensitive to initial point which may debase the solution quality as well as effectiveness. The main disadvantage of PSO algorithm maybe is that, it does not guarantee to be global convergent, and sensitive to initial point although it converges fast. Compared with PSO, quantum-behaved PSO has lesser parameters to control and better search capability. However, the conventional QPSO algorithm still suffers slow convergence for complex and large-scale SHTS problems. Hence, in this paper, a chaotic local search technique is employed to enhance local search capability in exploring the global best solution. The chaotic optimization method takes advantage of the universality, randomicity, sensitivity dependence on initial conditions and it is more likely to acquire the global optimum solution. Thus, the proposed chaotic quantum behaved particle swarm optimization (CQPSO) algorithm is implemented to solve short-term hydrothermal scheduling problem in a four hydro plants and three thermal plants system. The simulation results show that the proposed method is able to obtain higher quality solutions.
This paper is organized as follows. Section 2 describes the mathematical formulation of SHTS problem. Section 3 introduces the PSO and QPSO briefly. Section 4 proposes a chaotic quantum behaved particle swarm optimization algorithm for solving SHTS problem. Section 5 presents the simulation experiments and results. Finally, the conclusions are provided in section 6.
2. PROBLEM FORMULATION
The objective of the SHTS problem is to minimize the total cost of thermal plant as much as possible while making full use of hydro resource. Generally, the scheduling period and the scheduling time interval are set to 24h and 1h respectively. The objective function and related equality and inequality constraints can be simulated as follows.
2.1. Objective Function
The objective function of the problem is formulated as follows:
(1) |
Taking the valve-point effects into consideration, the fuel cost function can be expressed as the sum of a quadratic function and a sinusoidal function as follows:
(2) |
where F is the total fuel cost; f_{i}(P_{si,t}) is fuel cost of the ith thermal plant at time interval t; P_{si,t} is the generation of the ith thermal plant at time interval t; a_{si}, b_{si} and c_{si} are cost coefficients of the ith thermal plants; d_{si}, e_{si} are value-point effects coefficients of the ith thermal plants; N_{s} is the number of thermal plants; T is the number of intervals over a scheduling horizon.
2.2. Constraints
2.2.1. System Load Balance
(3) |
where N_{h} is the number of hydro plants; P_{hj,t} is the generation of the jth hydro plant at time interval t; P_{D,t} is the load demand at time interval t; P_{L,t} is the power loss at time interval t, which can be calculated by Kron’s formula [6]:
(4) |
where B, B_{0}, B_{00} are power loss coefficients. The power generation of hydro plants is represented as a function of reservoir storage volume and water discharge as:
(5) |
where V_{j,t} is reservoir storage volume of the jth hydro plant at time interval t; Q_{j,t} is water discharge of the jth hydro plant at time interval t; C_{1}_{j}, C_{2}_{j}, C_{3}_{j}, C_{4}_{j}, C_{5}_{j} and C_{6}_{j} represent hydro power generation coefficients.
2.2.2. Output Power Constraints
(6) |
where P_{si,}_{min} and P_{si,}_{max} are the minimum and maximum power generation of the ith thermal plant; P_{hj,}_{min} and P_{hj,}_{max} are the minimum and maximum power generation of the jth hydro plant;
2.2.3. Thermal Unit Ramp Rate Limits
(7) |
where UR_{i} and DR_{i} are ramp-up and ramp-down rate limits of the ith thermal unit respectively.
2.2.4. Reservoir Storage Volume Limits
(8) |
where V_{j,}_{min} and V_{j,}_{max} are the minimum and maximum reservoir storage volume limits of the jth hydro plant.
2.2.5. Water Discharge Limits
(9) |
where Q_{j,}_{min} and Q_{j,}_{max} are the minimum and maximum water discharge limits of the jth hydro plant.
2.2.6. Initial and Terminal Reservoir Storage Volumes Limits
(10) |
where V_{j,}_{B} and V_{j,}_{E} are the initial and terminal reservoir storage volumes limits of the jth hydro plant.
2.2.7. Water Dynamic Balance
(11) |
where I_{j,t}, S_{j,t} are the nature inflow and water spillage of the jth hydro plant at time interval t; N_{j} is number of upstream plants directly connected with hydro plant j; τ_{hj} is the time delay from the upstream hydro plant h to plant j.
3. OVERVIEW OF QUANTUM BEHAVED PARTICLE SWARM OPTIMIZATION
3.1. Particle Swarm Optimization
Particle swarm optimization (PSO) algorithm was put forward by Eberhart and Kennedy in 1995. It is a population based stochastic algorithm to find an optimum solution of a problem [12]. The algorithm is different from evolutionary algorithms; however it is much simpler since it has no use for selection. In PSO, each candidate solution named as “particle” flies around the solution space and lands on the optimal position. All the particles are evolved by competition and cooperation according to fitness functions. Each particle has a memory and keeps track of its own personal best solution (P_{best}) and the global best solution (G_{best}).
Assume that there are N particles in a D-dimensional space, the position and velocity vectors particle can be represented as x_{i} = (x_{i}_{1}, x_{i}_{2}… x_{iD}) and v_{i} = (v_{i}_{1}, v_{i}_{2}… v_{iD}) where i = 1, 2… N. The updating formulas of position and velocity of the ith particle can be described as follows:
(12) |
(13) |
where w is velocity inertia weight; r_{1} and r_{2} are two random numbers from the interval [0, 1]; c_{1} and c_{2} are the cognitive and social parameters; k is the current iteration; P_{best} stands for the best solution of the all swarm founded at time k and G_{best} represents the best solution until time k.
3.2. Quantum Behaved Particle Swarm Optimization
Though PSO algorithm is characterized by fast convergence, but it has no guarantee to be global convergence. In order to solve this problem, QPSO, as a variant of PSO, was proposed by Sun et al. [13] in 2004, when they were inspired by quantum mechanics and fundamental theory of particle swarm. In QPSO, quantum theory is applied in the searching process. Because of the uncertainty principle of quantum mechanics, the position and velocity of a particle cannot be determined synchronously in quantum world. New state of each particle is determined by wave function ψ(x,t) [14]. In literature [15], Clerc and Kennedy analyze the trajectory of each particle in PSO and assume that each particle can converge to its local attractor which can guarantee the global convergence. The local attractor is defined as follows:
(14) |
where ϕ = c_{1}r_{1}/ (c_{1}r_{1} + c_{2}r_{2}); r_{1} and r_{2} are values generated according to a uniform in range [0, 1]; c_{1} and c_{2} are the cognitive and social parameters. According to the Monte Carlo method, the particles update their positions by the following iterative equation:
(15) |
where β is a design parameter called contraction-expansion coefficient; u and rd are probability distribution random numbers in the interval [0, 1]. M_{best} is the mean of the P_{best} position of all particles and it can be formulated as:
(16) |
The steps of QPSO are depicted as follows from Coelho [16, 17].
- Step 1: Initialize randomly the initial particles in the feasible range using a uniform probability distribution function.
- Step 2: Evaluate the fitness value of each particle.
- Step 3: Compare the fitness of each particle with P_{best} value. If current fitness value is better than P_{best} then set current fitness value to P_{best}.
- Step 4: Compare P_{best} values with current G_{best} value. If P_{best} values are better than G_{best}, replace G_{best} with current P_{best}.
- Step 5: Calculate the M_{best} using Eq.(16).
- Step 6: Update the position of the particles according to Eq.(15).
- Step 7: Repeat Step 2 to Step 7 until termination criteria is met.
4. CHAOTIC QUANTUM BEHAVED PARTICLE SWARM OPTIMIZATION FOR SOLVING SHTS
Chaos is a deterministic, random-like mathematical phenomenon which takes place in nonlinear systems and strongly affected by the initial conditions [18]. This kind of unpredictability of random behavior is also helpful in dealing with SHTS problem. Thus, chaos was widely utilized in order to generate high quality solutions.
4.1. Logistic Map
Logistic map is a kind of one dimensional chaotic system which is firstly introduced by Robert May [19]. It demonstrates that how complex behavior arises from a simple deterministic system without need of any random sequence. In our study, Logistic map is coupled with QPSO to enhance the global convergence rate of QPSO and the logistic map can be expressed by:
(17) |
where α is a control parameter between 0.0 and 4.0; z_{0 } is the initial condition of {0.25, 0.50, 0.75} for fear of a regular sequence. When α = 4.0, a chaotic sequence is generated.
4.2. Chaotic Local Search
In QPSO algorithm, when the solution cannot be improved through a certain iteration times, chaotic local search is considered to generate a new particle which helps to find a new solution. Chaotic local search technique is employed to enhance local search capability in exploring the global best solution. The process of chaotic local search can be described as follows:
- Set k_{c}= 0, where k_{c} is the iteration count of chaotic local search. Initialize randomly z_{0} in the feasible range;
- Calculate the the fitness value of current particle. Compare the fitness of each particle with P_{best} value. If current fitness value equals to P_{best} then k_{c} = k_{c} +1, otherwise set k_{c} = 0;
- If k_{c} = k_{c}_{max}, where k_{c}_{max} is the maximum iteration count of chaotic local search. Chaotic local search is used in QPSO algorithm, and set k_{c} = 0. The updating formulas of position of the current particle can be described as follows:
(18) |
where x_{i} is the position of the ith particle; z_{k} is the chaotic sequence generated by Eq.(17); r is a metabolic search radius which decides the range of searching space can be formulated as:
(19) |
where r_{max} and r_{min} are maximum value and minimum value of r respectively; k_{max} is the maximum iteration and k is the current iteration. In our study, r_{max} is set to 0.95 and r_{min} is set to 0.5.
4.3. Initialization
The initial population is generated in a feasible region which consists of water release of N_{h} hydro plants and the power generations of N_{s} thermal plants in T intervals over a schedule horizon of one day. Each randomly generated element covers the entire search space and is initialized as:
(20) |
where μ_{1} and μ_{2} are probability distribution random numbers in the interval [0, 1]. Hence, an individual can be expressed by an array as follows:
(21) |
4.4. Constraints Handling
Though the initial population is generated in a valid region, it may not satisfy all the equality and inequality constrains synchronously. In many cases, penalty function has been used to handle constraints and obtained good effect. However, the weakness of penalty function is obvious that the quality of solutions is closely related to the choice of penalty parameters. Inspired by [20], a new method is introduced about handling the equality constraints in this paper. The equality and inequality constraints handling strategy is planned as follows.
4.4.1. Inequality Constraints Handling
Refer to the formulas in section 2, the inequality constraints consist of water discharge limits in Eq.(9), reservoir storage volume limits in Eq.(8) as well as output power constraint in Eq.(6). Taking no account of prohibited discharge zones, the handling strategy of water discharge limits is as follows:
(22) |
As the same with water discharge limits strategy, the handling method of reservoir storage volume limits can be applied as follows:
(23) |
Refer to the output power constraint of thermal unit, these variables are kept in a feasible range due to impose of
(24) |
4.4.2. Equality Constraints Handling
There are two equality constraints of water dynamic balance and system load balance to be resolved though they are more complicated than inequality constraints. In order to simplify the water dynamic balance constraint, the water spillages are neglected and a novel reservoir volume handling strategy can be found in Fig. (1).
The system load balance constraints handling strategy executes after the water dynamic balance procedure. Balanced water discharge Q_{j,t} is updated according to Fig. (1), and V_{j, t} can be calculated by Eq.(11). It is obvious that all the needed variables in Eq.(3) are ascertained and the change of the state variables of thermal plants has no effect on the constraints handling for hydro plants. Thus, the proposed system load balance handling strategy can be found in Fig. (2).
4.5. Selection Operation
Generally speaking, the proposed constraints handling strategy takes a long time in the early iterations, but it can also reduce the running time as the target value (total fuel cost F) becoming smaller. In addition, all of the modified particles in each generation will never violate the constraints. This kind of method by parting constraints handling and objective function simplified section operation largely when compared with penalty function methods and three simple feasibility-based selection comparison rules adopted in [21]. The section operation of global best solution (G_{best}) is formulated as:
(25) |
Fig. (1). Pseudo codes of reservoir volume handling strategy. |
The steps of CQPSO are depicted as follows:
- Initialize randomly the initial particles in the feasible range according to Eq.(20), set iteration number k = 0, judge whether the particles are violate the constraints, and then handle constraints follow with the Figs. (1) and (2).
- Evaluate the fitness value of each particle, and update P_{best} and G_{best}.
- Calculate the M_{best} using Eq.(16), update the position of the particles according to Eq.(15).
- Chaotic local search scheme is implemented to generate a new particles and modify the offspring according to Eq.(18).
- Calculate particle fitness again, if the current particle fitness is better than P_{best}, then replace P_{best} with current fitness; If the current global optimal value is superior to global optimal, then replace G_{best} with the current global optimal.
- If the iteration number k equals to the maximum iteration number k_{max}, break the procedure and output the optimal solution of SHTS; otherwise, k = k+1 and go back to step 3.
5. SIMULATION EXPERIMENTS
In order to verify the effectiveness of proposed CQPSO algorithm, it has been tested on four hydro plants coupled hydraulically and three thermal plants system. In addition, the traditional PSO and QPSO algorithm are utilized for comparison. Both algorithms are coded by MATLAB R2014a programming language and run on a 2.93 GHz PC with 2 GB of RAM.
The detail data of four hydro plants and three thermal plants system can be found in [8]. The problem is solved by CQPSO and the population size (N_{p}) and the maximum iteration number (k_{max}) are set 50 and 1500, respectively. The scheduling period is divided into 24 intervals of one day. Here prohibited operating zones of hydro plants are not considered. There are two cases taken into consideration. It is necessary to point out that all of the follow case will never violate the constraints because of the proposed equality constraints handling strategy.
Fig. (2). Pseudo codes of system load balance handling strategy. |
Case 1: Value-point Effects is Considered
In this case, the value-point effects are considered and the transmission losses are neglected. To run the program 20 times, the optimal fuel cost and the average CPU time of proposed CQPSO algorithm and other artificial intelligence algorithms, including MHDE [8], CSA [10] and QOTLBO [22] are given in Table 1. The symbol ‘-’ means the respective value cannot be obtained according the original paper. Obviously CQPSO is superior for solving the SHTS problem of this test system by obtaining the optimal fuel cost with simulation time of 154.6s. The result comparison in the table has indicated that the proposed CQPSO algorithm can obtain solutions of better quality and higher robustness than the other methods. Its simulation time is good enough though some of other algorithms previously proposed have less time than the CQPSO. The comparison of the convergence characteristics is depicted in Fig. (3). It is observed that the searching ability and convergence rate are improved in the proposed CQPSO algorithm. The best schedule result of optimal hydro discharges and the optimal thermal generation obtained by the CQPSO algorithm are shown in Table 2. Based on the above optimal result, the optimal reservoir storage volume and optimal hydro generation can be calculated by formula (11) and (5) respectively. The hourly reservoir storage volumes of four hydro plants are shown in Fig. (4). It can be seen from this figure that the volumes satisfy their initial and final volume constraints and the bound constraints. The total generation of each schedule interval and the total power demand are shown in Fig. (5). It can be found that the optimal result will not violate all of the system constraints.
Fig. (3). Convergence characteristics for case 1. |
Method | Minimum cost ($) | Average cost ($) | Maximum cost ($) | CPU time (s) |
---|---|---|---|---|
MHDE [8] | 41856.50 | - | - | 31 |
CSA [10] | 42440.574 | - | - | 109.12 |
QOTLBO [22] | 42187.49 | 42193.46 | 42202.75 | 21.6 |
PSO | 42886.613 | 43474.174 | 43893.142 | 70.4 |
QPSO | 41910.958 | 42077.327 | 42290.026 | 110.7 |
CQPSO | 40989.820 | 41220.048 | 41343.252 | 51.5 |
Hour | Hydro discharge (10^{4} m^{3}) | Thermal power (MW) | |||||
---|---|---|---|---|---|---|---|
Q_{1,}_{t} | Q_{2,}_{t} | Q_{3,}_{t} | Q_{4,}_{t} | P_{s}_{1,}_{t} | P_{s}_{1,}_{t} | P_{s}_{1,}_{t} | |
1 | 10.2178 | 9.3298 | 20.2613 | 8.8759 | 36.5463 | 126.0370 | 229.7972 |
2 | 9.1368 | 7.1820 | 19.2544 | 7.3848 | 109.8551 | 211.7697 | 140.1393 |
3 | 9.6459 | 6.5576 | 19.4342 | 8.1820 | 24.4956 | 125.4209 | 232.2091 |
4 | 9.6995 | 7.7044 | 29.9999 | 8.0582 | 102.8291 | 40.2673 | 229.5763 |
5 | 8.9220 | 7.1976 | 23.1773 | 8.7948 | 102.7000 | 124.9082 | 139.7599 |
6 | 8.7120 | 9.2769 | 17.3405 | 8.0073 | 102.6736 | 124.9079 | 229.5235 |
7 | 8.6709 | 6.1667 | 14.8253 | 7.4806 | 175.0000 | 209.8180 | 230.1476 |
8 | 9.1580 | 8.6202 | 14.6216 | 16.8001 | 20.0453 | 209.8169 | 319.8594 |
9 | 9.9018 | 7.3033 | 15.4067 | 15.9997 | 103.4103 | 209.8239 | 319.4229 |
10 | 8.1948 | 7.9247 | 15.9186 | 15.1351 | 102.7299 | 209.8158 | 319.2797 |
11 | 8.5710 | 10.0305 | 15.8516 | 15.6576 | 103.8099 | 209.8162 | 319.2794 |
12 | 10.1380 | 6.1447 | 19.9330 | 16.7290 | 174.9988 | 209.8158 | 319.2790 |
13 | 7.5895 | 11.0777 | 14.9540 | 17.4826 | 102.6748 | 209.8171 | 319.2794 |
14 | 7.9344 | 6.1315 | 28.7997 | 17.0121 | 102.6736 | 209.8163 | 319.2794 |
15 | 7.1356 | 7.7380 | 15.0530 | 18.2715 | 102.6735 | 124.9080 | 319.2790 |
16 | 5.8203 | 7.8071 | 15.9904 | 14.8778 | 102.6735 | 294.7237 | 229.5195 |
17 | 5.0016 | 6.6501 | 19.5098 | 16.4194 | 102.6734 | 209.8158 | 319.2793 |
18 | 5.8185 | 13.2088 | 13.0580 | 17.7778 | 102.6735 | 209.8158 | 319.2794 |
19 | 5.0112 | 7.5769 | 14.9019 | 15.8503 | 102.6735 | 209.8158 | 319.2792 |
20 | 10.0542 | 10.1393 | 15.3120 | 19.5066 | 102.6735 | 209.8158 | 229.5196 |
21 | 5.8088 | 6.8073 | 14.8748 | 17.4938 | 102.6709 | 124.9079 | 229.5196 |
22 | 9.0025 | 7.6751 | 10.1147 | 18.6095 | 20.0055 | 124.9080 | 229.5196 |
23 | 5.3654 | 9.9044 | 12.4956 | 19.9970 | 20.0000 | 124.9070 | 229.5193 |
24 | 9.4895 | 13.8452 | 11.9320 | 20.0000 | 21.0904 | 128.6517 | 139.7697 |
Fig. (4). Optimal hourly reservoir storage volumes for case 1. |
Case 2: Value-point Effects, Transmission Losses and Ramp-rate Limits are Considered
In this case, value-point effects, transmission losses and ramp-rate limits are considered. To run the program 20 times, the optimal fuel cost and the average CPU time of proposed CQPSO algorithm compared with MHDE [8] and SPPSO [23] are given in Table 3. The best, average and worst total cost of thermal plant found by CQPSO are 41785.665$, 41972.366$ and 42098.316$ respectively. It is obvious that the proposed CQPSO method has a higher performance than QPSO and other method. Fig. (6) shows the convergence of PSO, QPSO and CQPSO for the trial run that produced the minimum cost solution. The optimal hydro discharges, the optimal thermal generation, and the total transmission losses obtained by CQPSO accompany with the system power demand are demonstrated in Table 4. The hourly reservoir storage volumes of four hydro plants are shown in Fig. (7). The Optimal hourly power generation, transmission losses and load demand are shown in Fig. (8). It is important to note that all control and state variables remained within their permissible limits.
Fig. (5). Optimal power generation and load demand for case 1. |
Method | Minimum cost ($) | Average cost ($) | Maximum cost ($) | CPU time (s) |
---|---|---|---|---|
MHDE [8] | 42679.87 | - | - | 40 |
SPPSO [11] | 42740.23 | 43622.14 | 44346.97 | 32.7 |
PSO | 44431.089 | 44711.496 | 45158.586 | 151.1 |
QPSO | 42375.926 | 42971.683 | 43389.563 | 134.5 |
CQPSO | 41785.665 | 41972.366 | 42098.316 | 64.5 |
Hour | Hydro discharge (10^{4} m^{3}) | Thermal power (MW) | Power loss(MW) | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Q_{1,}_{t} | Q_{2,}_{t} | Q_{3,}_{t} | Q_{4,}_{t} | P_{s}_{1,}_{t} | P_{s}_{1,}_{t} | P_{s}_{1,}_{t} | ||||
1 | 8.2245 | 7.9191 | 29.6237 | 9.1369 | 102.6595 | 124.7263 | 229.5104 | 10.5015 | ||
2 | 10.1855 | 10.7757 | 19.9313 | 7.4709 | 102.6561 | 209.6966 | 139.7385 | 7.4810 | ||
3 | 7.2637 | 7.1659 | 29.7072 | 6.7772 | 102.6509 | 124.9049 | 229.4984 | 10.4278 | ||
4 | 10.2128 | 6.2541 | 19.3666 | 6.9397 | 102.6533 | 124.8233 | 139.7579 | 6.1761 | ||
5 | 10.6099 | 9.3396 | 15.8075 | 11.5044 | 20.0000 | 124.3139 | 139.6987 | 4.4665 | ||
6 | 5.8312 | 9.2429 | 16.2382 | 9.9546 | 102.6176 | 209.8158 | 139.7594 | 7.4303 | ||
7 | 12.1564 | 7.0913 | 18.2705 | 13.4959 | 102.6735 | 124.9079 | 319.2793 | 16.9544 | ||
8 | 8.2605 | 6.4284 | 16.9478 | 11.9905 | 102.6114 | 209.8155 | 319.2792 | 17.9783 | ||
9 | 8.9624 | 6.4645 | 14.2832 | 10.0717 | 102.6727 | 294.7237 | 319.2764 | 19.5343 | ||
10 | 7.4292 | 10.3183 | 20.7605 | 19.6889 | 102.6700 | 209.7698 | 319.2793 | 18.3792 | ||
11 | 11.5798 | 9.1713 | 14.4692 | 17.2266 | 102.6603 | 294.7166 | 229.5196 | 13.6742 | ||
12 | 7.7718 | 6.0132 | 19.6893 | 10.0783 | 102.6733 | 294.7235 | 409.0384 | 28.0507 | ||
13 | 6.7269 | 6.8159 | 15.6891 | 18.6106 | 137.1163 | 209.8157 | 319.2794 | 19.9164 | ||
14 | 9.3596 | 8.8527 | 15.6208 | 19.2345 | 102.6714 | 124.9079 | 319.1813 | 17.2822 | ||
15 | 10.0835 | 8.2008 | 18.7433 | 18.0866 | 102.5806 | 124.9079 | 319.2793 | 17.2309 | ||
16 | 5.3518 | 9.7178 | 11.4705 | 14.6726 | 102.6734 | 294.7233 | 229.5075 | 13.4458 | ||
17 | 5.0855 | 8.3598 | 20.3461 | 16.7670 | 102.6219 | 209.7787 | 319.2793 | 18.1973 | ||
18 | 8.7348 | 12.9170 | 17.0700 | 19.8221 | 102.6735 | 209.8022 | 319.2794 | 18.4002 | ||
19 | 6.9129 | 6.4645 | 13.1685 | 17.5592 | 102.6735 | 209.6555 | 319.2785 | 18.1637 | ||
20 | 5.7935 | 9.0423 | 13.8961 | 14.6919 | 102.6735 | 294.7237 | 229.5196 | 13.3989 | ||
21 | 6.8254 | 7.5790 | 13.8209 | 17.7822 | 102.0285 | 209.8150 | 139.3596 | 7.9770 | ||
22 | 8.1673 | 10.8562 | 15.1082 | 19.0893 | 20.0000 | 122.8864 | 229.5054 | 9.2802 | ||
23 | 8.3810 | 10.2764 | 13.1953 | 19.1190 | 20.0000 | 124.5950 | 224.2423 | 8.9311 | ||
24 | 5.0901 | 6.7331 | 14.7397 | 18.1287 | 20.0000 | 124.4392 | 229.5119 | 8.9268 |
Fig. (6). Convergence characteristics for case 2. |
Fig. (7). Optimal hourly reservoir storage volumes for case 2. |
Fig. (8). Optimal power generation and load demand for case 2. |
CONCLUSION
In this paper, a chaotic quantum-behaved particle swarm optimization (CQPSO) algorithm has been proposed to solve the short-term hydrothermal scheduling problem with a set of equality and inequality constrains. In CQPSO, chaotic local search technique is employed to enhance local search capability and convergence rate in exploring the global best solution. Additionally, a novel equality constrains handling strategy ensures all control and state variables in each generation will never violate the constraints. Finally, a four hydro plants and three thermal plants system has been applied to verify the effectiveness and feasibility of the proposed method. Taken the value-point effects and transmission losses into consideration, the simulation results show that CQPSO can obtain the better feasible fuel cost than all the population-based artificial intelligence algorithms considered.
CONFLICT OF INTEREST
The authors confirm that this article content has no conflict of interest.
ACKNOWLEDGEMENTS
The authors would like to thank the editors and the reviewers for their constructive comments. This work was supported by Chongqing University Innovation Team under Grant KJTD201312 and the National Natural Science Foundation of China (Nos.51207064 and 61463014).