- Original research
- Open Access
Opposition-based differential evolution for hydrothermal power system
- Jagat Kishore Pattanaik^{1}Email authorView ORCID ID profile,
- Mousumi Basu^{1} and
- Deba Prasad Dash^{2}
https://doi.org/10.1186/s41601-017-0033-5
© The Author(s) 2017
- Received: 26 May 2016
- Accepted: 16 January 2017
- Published: 8 February 2017
Abstract
This paper presents opposition-based differential evolution to determine the optimal hourly schedule of power generation in a hydrothermal system. Differential evolution (DE) is a population-based stochastic parallel search evolutionary algorithm. Opposition-based differential evolution has been used here to improve the effectiveness and quality of the solution. The proposed opposition-based differential evolution (ODE) employs opposition-based learning (OBL) for population initialization and also for generation jumping. The effectiveness of the proposed method has been verified on two test problems, two fixed head hydrothermal test systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading. The results of the proposed approach are compared with those obtained by other evolutionary methods. It is found that the proposed opposition-based differential evolution based approach is able to provide better solution.
Keywords
- Differential evolution
- Opposition-based differential evolution
- Hydrothermal system
- Fixed head
- Variable head
Introduction
Optimal scheduling of power plant generation is of great importance to electric utility systems. Because of insignificant marginal cost of hydroelectric power, the problem of minimizing the operational cost of hydrothermal system essentially reduces to that of minimizing the fuel cost of thermal plants under the various constraints on the hydraulic, thermal and power system network.
The hydrothermal scheduling problem has been the subject of investigation for several decades. Several classical methods such as Newton’s method [1], mixed integer programming [2, 3], dynamic programming (DP) [4], etc. have been widely used to solve hydrothermal scheduling problem. Among these methods, DP appears to be the most popular. However, major disadvantages of DP method are computational and dimensional requirements which grow drastically with increasing system size and planning horizon.
Recently, stochastic search algorithms such as simulated annealing (SA) [5], evolutionary programming (EP) [6], genetic algorithm (GA) [7, 8], evolutionary programming technique [9], differential evolution (DE) [10–12], particle swarm optimization [13], artificial immune system [14], clonal selection algorithm [15] and teaching learning based optimization [16] have been successfully used to solve hydrothermal scheduling problem.
Since the mid 1990s, many techniques originated from Darwin’s natural evolution theory have emerged. These techniques are usually termed by “evolutionary computation methods” including evolutionary algorithms (EAs), swarm intelligence and artificial immune system. Differential evolution (DE) [17–20], a relatively new member in the family of evolutionary algorithms, first proposed over 1995–1997 by Storn and Price at Berkeley is a novel approach to numerical optimization. It is a population-based stochastic parallel search evolutionary algorithm which is very simple yet powerful. The main advantages of DE are its capability of solving optimization problems which require minimization process with nonlinear, non-differentiable and multi-modal objective functions.
The basic concept of opposition-based learning (OBL) [21–23] was originally introduced by Tizhoosh. The main idea behind OBL is for finding a better candidate solution and the simultaneous consideration of an estimate and its corresponding opposite estimate (i.e., guess and opposite guess) which is closer to the global optimum. OBL was first utilized to improve learning and back propagation in neural networks by Ventresca and Tizhoosh [24], and since then, it has been applied to many EAs, such as differential evolution [25], particle swarm optimization [26] and ant colony optimization [27].
Opposition-based harmony search algorithm [28] has been applied to solve combined economic and emission dispatch problems. In [29] oppositional real coded chemical reaction optimization has been used for solving economic dispatch problems. Opposition-based gravitational search algorithm [30] has been applied for solving reactive power dispatch problem.
This paper proposes opposition-based differential evolution (ODE) for optimal scheduling of generation in a hydrothermal system. This paper considers fixed head as well as variable head hydrothermal system. In case of fixed head hydro plants, water discharge rate curves are modeled as a quadratic function of the hydropower generation and thermal units with nonsmooth fuel cost function. Here, scheduling period is divided into a number of subintervals each having a constant load demand. In case of variable head hydrothermal system, multi-reservoir cascaded hydro plants having prohibited operating zones and thermal units with valve point loading are used. The proposed method is validated by applying it to two test problems, two fixed head hydrothermal test systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading. The test results are compared with those obtained by other evolutionary methods reported in the literature. From numerical results, it is found that the proposed ODE based approach provides better solution.
Problem formulation
Fixed head hydrothermal system
Fixed head hydrothermal scheduling problem with Ν _{ h } hydro units and Ν _{ s } thermal units over M time subintervals is described as follows:
Objective function
Constraints
Determination of generation level of slack generator
The loading of the slack generator (i.e. Ν _{ s } th) can then be found by solving equation (9) using standard algebraic method.
Variable head hydrothermal system
The variable head hydrothermal scheduling problem is aimed to minimize the fuel cost of thermal plants, while making use of the availability of hydro power as much as possible. The objective function and associated constraints of the hydrothermal scheduling problem are formulated as follows.
Objective function
Constraints
(i) Power balance constraints:
(iii) Hydraulic network constraints
The hydraulic operational constraints comprise the water balance equations for each hydro unit as well as the bounds on reservoir storage and release targets. These bounds are determined by the physical reservoir and plant limitations as well as the multipurpose requirements of the hydro system. These constraints include:
Description of opposition-based differential evolution
A brief description of differential evolution
Differential Evolution (DE) is a type of evolutionary algorithm originally proposed by Price and Storn [19] for optimization problems over a continuous domain. DE is exceptionally simple, significantly faster and robust. The basic idea of DE is to adapt the search during the evolutionary process. At the start of the evolution, the perturbations are large since parent populations are far away from each other. As the evolutionary process matures, the population converges to a small region and the perturbations adaptively become small. As a result, the evolutionary algorithm performs a global exploratory search during the early stages of the evolutionary process and local exploitation during the mature stage of the search. In DE the fittest of an offspring competes one-to-one with that of corresponding parent which is different from other evolutionary algorithms. This one-to-one competition gives rise to faster convergence rate. Price and Storn gave the working principle of DE with simple strategy in [19]. Later on, they suggested ten different strategies of DE [18]. Strategy-7 (DE/rad/1/bin) is the most successful and widely used strategy. The key parameters of control in DE are population size (Ν _{ Ρ }), scaling factor (F) and crossover rate (C _{ R }). The optimization process in DE is carried out with three basic operations: mutation, crossover and selection. The DE algorithm is described as follows:
Initialization
where i =1,…., Ν _{ Ρ } and j =1,…., D; \( {X}_j^{\min } \) and \( {X}_j^{\max } \) are the lower and upper bounds of the j th decision variable; \( U\left({\mathrm{X}}_j^{\min },{\mathrm{X}}_j^{\max}\right) \) denotes a uniform random variable ranging over \( \left[{\mathrm{X}}_j^{\min },{\mathrm{X}}_j^{\max}\right] \). \( {X}_{ij}^0 \) is the initial j th variable of i th population. All the vectors should satisfy the constraints. Evaluate the value of the cost function \( f\left({\mathrm{X}}_i^0\right) \) of each vector.
Mutation
where \( X{}_a{}^k \), \( {X}_b^k \) and \( {X}_c^k \) are selected randomly from Ν _{ Ρ } vectors at k th iteration and a ≠ b ≠ c ≠ i. The scaling factor (F), in the range 0 < F ≤ 1.2, controls the amount of perturbation added to the parent vector. The noisy vectors should satisfy the constraint.
Crossover
where ρ is an uniformly distributed random number within [0, 1]. The crossover constant (C _{ R }), in the range 0 ≤ C _{ R } ≤ 1, controls the diversity of the population and aids the algorithm to escape from local optima.
Selection
The process is repeated until the maximum number of iterations or no improvement is seen in the best individual after many iterations.
Opposition-based learning
Opposition-based learning (OBL) was developed by Tizhoosh to improve candidate solution by considering current population as well as its opposite population at the same time.
Evolutionary optimization methods start with some initial population and try to improve them toward some optimal solution. The process of searching terminates when some predefined criteria are satisfied. The process is started with random guesses in the absence of a priori information about the solution. The process can be improved by starting with a closer i.e. fitter solution by simultaneously checking the opposite solution. By doing this, the fitter one (guess or opposite guess) may be chosen as an initial solution. According to the theory of probability, 50% of the time, a guess is further from the solution than its opposite guess. Therefore, process starts with the closer of the two guesses. The same approach can be applied not only to the initial solution but also continuously to each solution in the current population.
Definition of opposite number
Similarly, this definition can be extended to higher dimensions [21] as stated in the next sub-section.
Definition of opposite point
By employing the definition of opposite point, the opposition-based optimization is defined in the following sub-section.
Opposition-based optimization
Let X = (x _{1}, x _{2},...., x _{ n }) be a point in n - dimensional space i.e. a candidate solution. Assume f = (•) is a fitness function which is used to measure the candidate’s fitness. According to the definition of the opposite point, \( \overline{X}=\left(\overline{x_1},\overline{x_2},.....,\overline{x_n}\right) \) is the opposite of X = (x _{1}, x _{2},...., x _{ n }). Now, if \( f\left(\overline{X}\right)< f(X) \) (for a minimization problem), then point X can be replaced with \( \overline{X} \); otherwise, the process is continued with X. Hence, the point and its opposite point are evaluated simultaneously in order to continue with the fitter one.
Opposition-based differential evolution
In the present work, the concept of the opposition-based learning [21] is incorporated in differential evolution. The original DE is chosen as a parent algorithm and the opposition-based ideas are embedded in DE.
Simulation results
Two test problems, two fixed head hydrothermal systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading are investigated. The computational results have been used to compare the performance of the proposed ODE method with that of other evolutionary methods. The proposed ODE algorithm and DE algorithm used in this paper are implemented by using MATLAB 7.0 on a PC (Pentium-IV, 80 GB, 3.0 GHz).
Simple examples
Example 1: Consider the maximization problem [31]
where − 3.0 ≤ x _{1} ≤ 12.1 and 4.1 ≤ x _{2} ≤ 5.8
This function is multimodal. The problem is solved by using ODE.
Best optimum value, the variables corresponding to the best optimum value, average value, worst value and average CPU time for example 1
Method | x ^{∗} | f(x ^{∗}) | Average value | Worst value | CPU time (sec) |
---|---|---|---|---|---|
ODE | [12.1000, 5.7227] | 38.9377 | 38.9377 | 38.9377 | 0.0473 |
DE | [12.1000, 5.7228] | 38.9375 | 38.9373 | 38.9371 | 0.0469 |
Example 2: Consider the minimization problem [31]
where − 10 ≤ x _{1} ≤ 10 and − 10 ≤ x _{2} ≤ 10
This function has 760 local minima, 18 of which are global minima with −186.73. The problem is solved by using ODE. Here, the population size (Ν _{ Ρ }), scaling factor (F), crossover constant (C _{ R }) and maximum iteration number have been selected 10, 0.3, 1 and 100 respectively for the example under consideration.
To validate the proposed ODE based approach, the same example is solved by using DE.
Best optimum value, the variables corresponding to the best optimum value, average value, worst value and average CPU time for example 2
Method | x ^{∗} | f(x ^{∗}) | Average value | Worst value | CPU time (sec) |
---|---|---|---|---|---|
ODE | [5.4830, 4.8581] | −186.7309 | −186.7309 | −186.7309 | 0.0625 |
DE | [−7.7084, −7.0834] | −186.7308 | −186.7307 | −186.7303 | 0.0781 |
Figure 4 depicts the nature of convergence obtained from ODE and DE for example 2.
Case study of fixed head hydrothermal system
Test system 1
This system consists of two hydro plants and two thermal plants whose characteristics and load demands are given in Tables 13, 14 and 15 respectively in Appendix 1. Transmission loss formula coefficients are also given in the Appendix 1. Hydro plant data is taken from [32].
The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _{ Ρ }), scaling factor (F), crossover rate (C _{ R }) and the maximum iteration number (Ν _{max}) have been selected as 100, 1.0, 1.0 and 100 respectively for the test system under consideration.
Results obtained from ODE of test system 1 of fixed head hydrothermal system
Sub-interval | Ρ _{ h1} (MW) | Ρ _{ h2} (MW) | Ρ _{ s1} (MW) | Ρ _{ s2} (MW) |
---|---|---|---|---|
1 | 244.5860 | 90.7689 | 179.4953 | 424.9773 |
2 | 307.3581 | 163.3383 | 228.7850 | 570.1572 |
3 | 285.4852 | 139.2931 | 211.2739 | 522.5895 |
Results obtained from DE for test system 1 of fixed head hydrothermal system
Sub-interval | Ρ _{ h1} (MW) | Ρ _{ h2} (MW) | Ρ _{ s1} (MW) | Ρ _{ s2} (MW) |
---|---|---|---|---|
1 | 240.3807 | 85.6583 | 206.3934 | 407.6673 |
2 | 310.1176 | 167.5754 | 206.3934 | 585.2895 |
3 | 286.6845 | 139.7912 | 206.3934 | 525.7479 |
Test system 2
This system comprises of two hydro plants and four thermal plants whose characteristics and load demands are given in Tables 16, 17 and 18 respectively in Appendix 2. Transmission loss formula coefficients are also given in the Appendix 2.
Results obtained from ODE of test system 2 of fixed head hydrothermal system
Sub-interval | Ρ _{ h1} (MW) | Ρ _{ h2} (MW) | Ρ _{ s1} (MW) | Ρ _{ s2} (MW) | Ρ _{ s3} (MW) | Ρ _{ s4} (MW) |
---|---|---|---|---|---|---|
1 | 172.6478 | 317.8272 | 93.6207 | 174.7438 | 109.2596 | 50.3779 |
2 | 243.8370 | 411.3216 | 124.8716 | 174.6929 | 123.6025 | 50.1150 |
3 | 209.7780 | 351.8750 | 116.1764 | 174.7282 | 120.3243 | 50.0519 |
4 | 249.8641 | 499.8741 | 124.8642 | 174.9127 | 222.4536 | 68.0992 |
Results obtained from DE of test system 2 of fixed head hydrothermal system
Sub-interval | Ρ _{ h1} (MW) | Ρ _{ h2} (MW) | Ρ _{ s1} (MW) | Ρ _{ s2} (MW) | Ρ _{ s3} (MW) | Ρ _{ s4} (MW) |
---|---|---|---|---|---|---|
1 | 184.4627 | 303.6346 | 88.3611 | 174.7233 | 116.2664 | 50.9170 |
2 | 241.0344 | 419.5791 | 117.4402 | 174.8712 | 124.7407 | 50.9397 |
3 | 201.9931 | 357.2371 | 123.3403 | 173.9739 | 115.3547 | 51.0280 |
4 | 249.3076 | 499.1428 | 124.0676 | 174.7184 | 221.4260 | 71.3501 |
Case study of variable head hydrothermal system
Three variable head hydrothermal test systems are considered to inspect and verify the proposed ODE method.
Test system 1
This test system considers a multi-chain cascade of four reservoir hydro plants and an equivalent thermal plant. The entire scheduling period is 1 day and divided into 24 intervals. Here, two cases are considered.
Case 1: Here fuel cost is considered as a quadratic function of the power from the composite thermal plant. The detailed parameters for this case come from [7].
The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _{ Ρ }), scaling factor (F), crossover constant (C _{ R }) and maximum iteration number have been selected 100, 1, 1 and 300 respectively for this case.
Comparison of performance for case 1 of Test System 1 of variable head hydrothermal system
Case 2: Here prohibited operating zones of hydro plants and valve point loading of thermal generator are considered. The detailed parameters for this case come from [9].
The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _{ Ρ }), scaling factor (F), crossover constant (C _{ R }) and maximum iteration number have been selected 100, 1, 1 and 400 respectively for this case.
Comparison of performance for case 2 of test system 1 of variable head hydrothermal system
Test system 2
This system considers a multi-chain cascade of four reservoir hydro plants and three thermal plants. The entire scheduling period is 1 day and divided into 24 intervals. The effect of valve point loading is considered. Transmission loss is also considered. The detailed parameters for this case are taken from [10].
The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _{ Ρ }), scaling factor (F), crossover constant (C _{ R }) and maximum iteration number have been selected 100, 1, 1 and 300 respectively for this case.
Test system 3
This system considers a multi-chain cascade of four reservoir hydro plants and ten thermal plants. The entire scheduling period is 1 day and divided into 24 intervals. The effect of valve point loading is taken into account. Here transmission loss is not considered. The detailed data for this system is taken from [12].
The problem is solved by using both the proposed ODE and DE. Here, the population size (Ν _{ Ρ }), scaling factor (F), crossover constant (C _{ R }) and maximum iteration number have been selected 100, 1, 1 and 900 respectively for this case.
Comparison of performance for test system 3 of variable head hydrothermal system
Techniques | ODE | DE | DE [12] |
---|---|---|---|
Best cost ($) | 170452.35 | 170915.57 | 170964.15 |
Average cost ($) | 170459.78 | 170924.41 | - |
Worst cost ($) | 170468.52 | 170935.28 | - |
CPU time (s) | 472.51 | 459.92 | - |
It is observed from in Tables 20, 22, 24 and 26 respectively in Appendix 3 that the third hydro unit has no output during some time interval. This is because of the fact that output from a particular hydro unit during a specified time interval depends on the availability of water, reservoir storage volume limit, water transport delay between cascaded reservoirs and on the system configuration as a whole. Depending on the system configuration and constraints for the present problem, this has happened in case of the third hydro unit.
Conclusion
In this paper, opposition-based differential evolution is demonstrated and presented to solve the hydrothermal scheduling problem. The proposed opposition-based differential evolution method has been successfully applied to two test problems, two fixed head hydrothermal test systems and three hydrothermal multi-reservoir cascaded hydroelectric test systems having prohibited operating zones and thermal units with valve point loading. The results have been compared with those obtained by other evolutionary algorithms reported in the literature. It is seen from the comparisons that the proposed opposition-based differential evolution method performs better than other evolutionary algorithms in the literature.
Nomenclature
a _{ si }, b _{ si }, c _{ si }, d _{ si }, e _{ si }: cost curve coefficients of i th thermal unit
Ρ _{ sim }: power output of i th thermal generator during subinterval m
\( {P}_{si}^{\min } \), \( {P}_{si}^{\max } \): lower and upper generation limits for i th thermal unit
t _{ m }: duration of subinterval m.
Ρ _{ hjm }: power output of j th hydro unit during subinterval m
Ρ _{ Dm }: load demand during subinterval m
Ρ _{ Lm }: transmission loss during subinterval m
Β _{ lr }: loss formula coefficients.
a _{0hj }, a _{1hj }, and a _{2hj }: coefficients for water discharge rate function of j th hydro generator
W _{ hj }: prespecified volume of water available for generation by j th hydro unit during the scheduling period.
\( {P}_{hj}^{\min } \), \( {P}_{hj}^{\max } \): lower and upper generation limits for j th hydro unit
Ρ _{ sit }: output power of i th thermal unit at time t
Ρ _{ Dt }: load demand at time t
Ρ _{ Lt }: transmission loss at time t
Ρ _{ hjt }: output power of j th hydro unit at time t
C _{1j }, C _{2j }, C _{3j }, C _{4j }, C _{5j }, C _{6j }: power generation coefficients of j th hydro unit
Q _{ hjt }: water discharge rate of j th reservoir at time t
V _{ hjt }: storage volume of j th reservoir at time t
\( {Q}_{hj}^{\min } \), \( {Q}_{hj}^{\max } \): minimum and maximum water discharge rate of j th reservoir
\( {Q}_{hj, k}^L \), \( {Q}_{hj, k}^U \): lower and upper bounds of k th prohibited zones of hydro unit j
\( {V}_{hj}^{\min } \), \( {V}_{hj}^{\max } \): minimum and maximum storage volume of j th reservoir
Ι _{ hjt }: inflow rate of j th reservoir at time t
R _{ uj }: number of upstream units directly above j th hydro plant
S _{ hjt }: spillage of j th reservoir at time t
τ _{ lj }: water transport delay from reservoir l to j
t, Τ: time index and scheduling period
Ν _{ s }: number of thermal generating units
Ν _{ h }: number of hydro generating units
n _{ j }: number of prohibited zones for hydro unit j
k: index of prohibited zones of a hydro unit
Declarations
Authors’ contributions
JKP makes substantial contributions to conception, design, acquisition of data, analysis and interpretation of data. JKP drafted the article and revising it thoroughly for preparation of the manuscript for the esteemed journal. Also he did the simulation part by using different test data for two different test systems. As a corresponding author he takes the primary responsibility for communication of the journal during the manuscript submission, peer review, publication process, and typically ensures that all the journal’s administrative requirements, such as providing details of authorship. JKP will be available throughout the submission and peer review process to respond to editorial queries in a timely manner. Also he will be available after publication to respond to critiques of the work and cooperate with any requests from the journal for data or additional information should be answered about the paper arise after publication. JKP also agrees to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. MB participated in the peer review process of the manuscript and involved in the test data preparation. She reviewed the manuscript thoroughly. She also involved in preparation for testing of the test systems. DPD participated in the peer review process of the manuscript and to compare the performance of the proposed method with that of other evolutionary methods. He also involved in review the manuscript. All authors read and approved the final manuscript.
Authors’ information
Jagat Kishore Pattanaik has received the Master’s degree in electrical engineering in 2011 from Jadavpur University, Kolkata, India. He is currently working towards the Ph.D degree in the Department of Power Engineering, Jadavpur University, Kolkata, India. His research interest are Power system and optimization techniques.
Mousumi Basu has received the Ph.D. degree from Jadavpur University, Kolkata, India. He is currently working as an Associate Professor in the Department of Power Engineering, Jadavpur University, Kolkata, India. His research interest are Power system and soft computing technique.
Deba Prasad Dash has received the Ph.D. degree from Jadavpur University, Kolkata, India. He is currently working as a Professor in the Department of Electrical Engineering, Odisha Engineering College, Bhubaneswar, Odisha, India. His research interest are Power system Operation & Control and soft computing technique.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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