Elsevier

Energy Conversion and Management

Volume 96, 15 May 2015, Pages 568-578
Energy Conversion and Management

An extended NSGA-III for solution multi-objective hydro-thermal-wind scheduling considering wind power cost

https://doi.org/10.1016/j.enconman.2015.03.009Get rights and content

Highlights

  • Multi-objective hydro-thermal-wind scheduling model (MO-HTWS) is establish.

  • The extra cost in MO-HTWS problem caused by wind uncertainty is considered.

  • An extended NSGA-III is proposed to solve MO-HTWS problem.

  • Constraint handling strategies are presented to modify infeasible solutions.

  • The feasibility and effectiveness of the proposed method is verified by example.

Abstract

Due to the characteristics of clean and renewable, wind power is significant to economic and environmental operation of electric power system so that it attracts more and more attention from researchers. This paper integrates wind power with hydrothermal scheduling to establish multi-objective economic emission hydro-thermal-wind scheduling problem (MO-HTWS) model with considering wind uncertain cost. To solve MO-HTWS problem with various complicated constraints, this paper extends NSGA-III by introducing the dominance relationship criterion based on constraint violation to select new generation. Moreover, the constraint handling strategy which repairs the infeasible solutions by modifying the decision variables in feasible zone according to the violation amount is proposed. Finally, a daily scheduling example of hydro-thermal-wind system is used to test the ability of NSGA-III for solving MO-HTWS problem. It is concluded from the superior quality and good distribution of the Pareto optimal solutions that, NSGA-III can offer an efficient alternative for optimizing MO-HTWS problem.

Introduction

In recent years, as one of the most important optimization problem in electrical power system, short-term hydrothermal scheduling (SHTS) has attracted a large amount of attention. SHTS problem devotes to seeking for the optimal operation plan including the water discharge of each hydro plant and power load dispatch of thermal units during the scheduling period to minimize the operating expense of power system. Many researchers have made contributions to developing optimization methods for solving SHTS problem, such as linear programming (LP) [1], nonlinear programming (NLP) [2], [3], Lagrange relaxation LR [4], dynamic programming (DP) [5], genetic algorithm (GA) [6], particle swarm optimization (PSO) [7], [8], [9], differential evolution (DE) [10], [11], [12], simulated annealing (SA) [13], cultural algorithm (CA) [14], chemical reaction optimization algorithm (CRO) [15]. Although the methods above have obtained some achievement in solving hydrothermal scheduling problem, all of them only considered the single objective of minimizing the operation cost of power system.

With the progress of economy and society, the economic benefit is not the only goal of solving practical engineering problems. The SO2 and NOx released by thermal plants would pollute atmospheres and the large amount of CO2 from burning coal can cause global warming. Thus it is significant to regard minimizing pollution emission as another objective of SHTS problem. Many scholars have modeled the multi-objective economy emission hydrothermal scheduling problem and utilized different techniques to solve it, such as quadratic approximation based DE [16], improved quantum-behaved PSO [17] and self-organizing hierarchical PSO [18]. Although these approaches above have taken both fuel cost and emission into consideration, the scheduling problem is essentially single objective optimization problem with some inevitable drawbacks. The optimization results are sensitive to the weights which are hard to be determined. The Pareto optimal set cannot be obtained in one trial by using different weights. To overcome the drawbacks mentioned above, some authors have developed parallel multi-objective evolutionary algorithms (MOEA). The classic MOEAs are Deb’s NSGA-II [19] and Zitzler’s SPEA-II [20]. Besides, there are many multi-objective optimization algorithms such as multi-objective PSO [21], [22], MODE [23], [24], multi-objective gravitational search algorithm (MOGSA) [25], [26], multi-objective bat algorithm (MOBA) [27]. Basu [24] introduced the Pareto dominated based selection operator into DE to update new generation and achieved the compromise solutions of hydrothermal dispatch problem with conflicting objectives. Tian et al. [26] integrated the non-dominated sorting and crowding distance with GSA to solve multi-objective SHTS problem. In the proposed NSGSA, the Pareto optimal solutions of problem can be obtained in one trial without weights. However, due to the selection operator based on crowding distance, the Pareto fronts of NSGSA are not well-distributed enough. In spite of these algorithms are capable of optimizing the conflicting objectives simultaneously to get a set of Pareto optimal solutions in one trial, the premature convergence still exists in the evolution process. And due to the mechanisms to select and store Pareto optimal solutions are unadvanced, the distribution and diversity of the Pareto optimal set have the space for improvement.

As one of the clean and renewable energy sources, wind power has low operation cost and zero emission. Under the background of advocating energy conservation and emission reduction, an increasing number of researches express an interest in wind power. Affected by weather conditions, wind speed is uncertain and stochastic so that the distribution of wind speed is generally expressed by probability density function (PDF) in studies. Weibull distribution is a simple probabilistic model with two parameters which is widely used to deal with the wind power [28]. The uncertainty and volatility of wind power would affect the safe and stable operation of power system. To deal with the stochastic character of wind power, many researchers have proposed different methods [29], [30], [31]. Nevertheless, these studies did not consider the operation cost of wind power and the benefit to environmental protection, which would impact the authenticity and objectivity of the results. One method to deal with wind power uncertainty is calculating the extra operation cost of system caused by the wind uncertainty [32]. Hetzer et al. [33] suggested a metric to measure the extra cost of system caused by wind power overestimation and underestimation. If the actual wind power output is less than the predicted, system needs to use spinning reserve to meet load balance; if the actual wind power output is larger than the predicted, system has to compensate for the wasted power output. Liu [34] established a load dispatch problem model considering wind power. This reference converted the uncertainty property of wind power to constraints of problem and applied a Wait-and-See approach in stochastic programming to obtain the optimal load dispatch. In Ref. [35], the researcher solved the wind-thermal coordination scheduling problem and analyzed the impact of wind power outputs on total emission under different system load demand. Mondal and Bhattacharya [36] developed a multi-objective economic emission load dispatch model and used GSA to solving it successfully. But the studies described above all coordinate the wind power and thermal power in scheduling problem, the integration of wind power with hydrothermal scheduling problem is rarely reported in literatures.

NSGA-III is a newly MOEA proposed by Deb and Jain [37]. The basic framework of it is similar to that of NSGA-II [19], both of them use crossover and mutation operator to generate offspring population, and apply fast non-dominated sorting approach to divide individuals into different non-dominated ranks. The significant improvement of NSGA-III is developing the reference-point based selection mechanism to replace the crowding distance, which enhances the diversity and distribution of Pareto solutions. Deb and Jain [37] have proved the efficiency of NSGA-III by multi-objective benchmark functions. In Ref. [38], NSGA-III was applied to obtain the optimal Pareto solutions of multi-objective optimization problem with simple constraints successfully. However, the performance of NSGA-III in solving scheduling problem of power system is rarely verified.

In view of the good performance of NSGA-III for multi-objective problems, this paper has applied it to solve multi-objective hydro-thermal-wind scheduling (MO-HTWS) problem. When calculating the operation cost of system, this paper computes the extra cost of wind power overestimation and underestimation caused by wind uncertainty. To handle the complicated constraints of problem, an adaptive strategy which modifies the decision variables into feasible region according to the violation is proposed. Furthermore, this paper extends NSGA-III by introducing the constraint violation attribute into individuals to be the primary metric to determine dominance relationship. Finally, a daily scheduling problem of the example system consisting of four cascaded reservoirs, six thermal plants and two wind power plants is applied to verify the feasibility and effectiveness of NSGA-III. From the comparison with NSGA-II and MOPSO, it is known that the Pareto optimal fronts of NSGA-III obtain less fuel cost and better emission.

The rest of this paper is organized as follows: Section 2 builds the mathematical model of MO-HTWS problem with considering wind power cost. Section 3 briefly describes the concepts of multi-objective problems. Section 4 proposes an extended NSGA-III for multi-objective problem with constraints. Section 5 provides the specific implement steps of NSGA-III for solving MO-HTWS problem. Section 6 employs an example system to verify the performance of NSGA-III. Section 7 gives the conclusions. Acknowledgements are given in the end.

Section snippets

Problem formulation

The goal of MO-HTWS problem is to find the generation scheduling including the water discharges of hydro plants, planned outputs of thermal and wind power units. This generation scheduling scheme can minimize the fuel cost and emission of electrical power system during the scheduling period subject to the various equation and inequation constraints.

Basic concepts of Multi-Objective Optimization problem (MOOP)

In scientific research and engineering application area, we always meet the problem to seek minimum or maximum value of objective functions, which is called optimization problem. Generally, the objectives of optimization problem are not single one but formed from several associated even conflicting functions. Thus researchers define the concepts of MOOP. The goal of MOOP is to find the best compromise solutions of all objectives, which is called Pareto optimal solutions. In this section, the

Extended NSGA-III algorithm for MOOP with constraints

NSGA-III is a novel reference-point based non-dominated sorting GA proposed by Deb and Jain [37]. The basic framework of NSGA-III is similar to the previous NSGA-II which is popular in solving MOOP. Both of them apply crossover and mutation operator to generate offspring population, and employ fast non-dominated sorting approach to determine the non-dominated rank of individuals. Meanwhile, these two algorithms utilize an elite preservation strategy to select new generation from parent and

Implementation of extended NSGA-III for solving MO-HTWS problem

In this section, the extended NSGA-II is implemented to solve MO-HTWS problem with various constraints. The detailed steps are as follows:

Step 1 Read in the conditions and constraints of MO-HTWS problem, which consist of the number of hydro plants Nh, thermal plants Ns and wind plants Nw; power output limits of each hydro plant (Phjmin, Phjmax) and each thermal plant (Psimin, Psimax); the rated output of wind power wR,m, rated wind speed vR, cut-in wind speed vIN, cut-out wind speed vOUT; the

Description of the example system

In this section, a daily scheduling model of a hydro-thermal-wind system is chosen to verify the performance of NSGA-III for MO-HTWS problem. This example system comprises four cascaded reservoirs, six thermal plants and two wind power plants. The scheduling period is one day consists of 24 intervals. The hydraulic connection of four hydro plants, the detailed coefficients and constraint limits of hydraulic system are taken from reference [40], the data of thermal and wind plants are taken from

Conclusions

This paper has applied a recent NSGA-III for solving the MO-HTWS problem successfully. When dealing with the wind power uncertainty, this paper calculates the penalty and reverse cost of system operation based on the difference between the actual and planned wind power, and reformulates the model of MO-HTWS problem. In order to solve the high dimension MO-HTWS problem with various constraints, this paper has extended the standard NSGA-III by employing the individual repair strategy with

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 51379080) and Hubei Key Laboratory of Cascaded Hydropower Stations Operation & Control in China Three Gorges University (Nos. 2013KJX06, 2013KJX02).

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