Elsevier

Applied Soft Computing

Volume 12, Issue 8, August 2012, Pages 2137-2146
Applied Soft Computing

Mixed integer programming of multiobjective hydro-thermal self scheduling

https://doi.org/10.1016/j.asoc.2012.03.020Get rights and content

Abstract

This paper presents a method for hydro-thermal self scheduling (HTSS) problem in a day-ahead joint energy and reserve market. The HTSS is modeled in the form of multiobjective framework to simultaneously maximize GENCOs profit and minimize emissions of thermal units. In the proposed model the valve loading effects which is a nonlinear problem by itself is linearized. Also a dynamic ramp rate of thermal units is used instead of a fix rate leading to more realistic formulation of HTSS. Furthermore, the multi performance curves of hydro units is developed and prohibited operating zones (POZs) of thermal unit are considered in HTSS problem. Also, in the proposed framework, the mixed integer nonlinear programming (MINLP) of HTSS is converted to mixed integer programming (MIP) problem that can be effectively solved by optimization softwares even for real size power systems. The lexicographic optimization and hybrid augmented-weighted ɛ-constraint technique is implemented to generate Pareto optimal solutions. The best compromised solution is adopted either by using a fuzzy approach or by considering arbitrage opportunities to achieve more profit. Finally, the effectiveness of the proposed method is studied based on the IEEE 118-bus system.

Highlights

► This paper presents a method for hydro-thermal self scheduling (HTSS) problem in a day-ahead joint energy and reserve market. ► The HTSS is modeled in the form of multiobjective framework to simultaneously maximize GENCOs profit and minimize thermal units’ emissions. ► The mixed integer nonlinear programming (MINLP) of HTSS is converted to mixed integer programming (MIP) problem. ► The modified augmented ɛ-constraint method is implemented to generate Pareto optimal solutions.

Introduction

Hydro-thermal scheduling (HTS) consists of determining the optimal manner of hydro and thermal units during a time scheduling [1]. In traditional power system HTS, the objective function is to minimize the operational cost of a hydrothermal system, while in restructured power systems, it is used for minimizing the operational cost or maximizing profit which is called hydro-thermal self scheduling (HTSS) in the case of profit maximization [2]. Indeed, in the HTSS problem the thermal and hydro units/generators will be scheduled for generation and reserve for a horizon in a specific manner that satisfies constraints and optimizes the objectives. The valve loading effects is modeled as an absolute sinus function of power generated by thermal units [3], [4], [5] which is non-linear and cannot be considered in the MIP formulation of HTSS. For this reason it is neglected in the MIP formulation due to its non-linearity [1], [2], [6], [9], [10]. The main contribution of this paper is to linearize the valve loading effects in the MIP formulation of HTSS. In [6], a dynamic ramp rate is proposed to restrict the power output between two successive operating periods. Inspired by [6], in this paper, a formulation is proposed to simultaneously consider dynamic ramp rate and POZs. In [7], an MIP model is presented for self-scheduling of a hydro GENCO in a pool-based electricity market using multi performance curves. This concept is important for the cases that the storage capacity of reservoirs is small and power generated is in effects of hydro unit head. However the mentioned accurate model is rarely used in other MIP based HTS research works mainly for the sake of simplicity [2], [9]. But, in our proposed model the multi performance curves for hydro units is accurately considered, making the formulation more realistic.

After passage of the Clean Air Act Amendments in 1990 [8], conventional pure economic scheduling no longer satisfies the requirements of HTS problem and emission concerns should be inevitably considered. In [9], [10], [11], emissions of thermal units are considered as a constraint of objective function. However, in our proposed HTSS model, it is formulated as a multiobjective function to simultaneously optimize these competing objectives functions of maximizing profit and minimizing emissions. It is worth to mention that, in [9], [10], [11], the valve loading effects and dynamic ramp rate are not taken into account.

In [12], optimal bidding strategy based on the bi-level optimization has been presented and the ɛ-constraint method has been used to maximize the social welfare and minimize the generation emissions of thermal units. In [13], the weighted sum method for multiobjective programming problem has been used to convert two objective functions of profit and emission to one objective problem. In [14], the rolling window procedure has been used for the fuel and emission constrained self-scheduling of a power producer in the day-ahead joint energy, spinning reserve and fuel markets.

Different solution methods for the HTSS problem are comprehensively classified into heuristic and analytical methods in [15].

In this paper the MIP formulation of HTSS that is efficiently solved by analytic based optimization software.

The contributions of this paper can be summarized as follows:

  • (a)

    The HTSS is modeled as a multiobjective problem considering profit and emission as two objective functions of HTSS which is formulated as a MIP optimization. Also, emission arbitrage is considered to obtain more profit.

  • (b)

    Presentation of linear formulation for valve loading effects and using dynamic ramp rate limit instead of fix ramp rate limit.

  • (c)

    Using flexible method for multi POZs of thermal units and multi performance curves for hydro units.

  • (d)

    Converting the MINLP of HTSS problem to MIP problem and solving it by an effective analytical method using lexicographic optimization and hybrid augmented-weighted ɛ-constraint technique.

Section snippets

MIP formulation for HTSS

The proposed multiobjective framework for HTSS contains two objective functions can be written as follows:Multiobjectivefunctions=F1profitmaximizationF2emmisionminimizationwhere F1, and F2 are the objective functions of the HTSS, described in details in the following.

Multiobjective mathematical programming (MMP)

In multiobjective mathematical programming (MMP) there is more than one objective function and there is no single optimal solution that simultaneously optimizes all the objective functions. In these cases, the decision makers (DMs) are looking for the “most preferred” solution. In MMP, the concept of optimality is replaced with that of efficiency or Pareto optimality. The efficient (or Pareto optimal, non-dominated, non-inferior) solution is the solution that cannot be improved in one objective

Case study

The proposed HTSS is studied based on the IEEE 118-bus system (Fig. 6). This system contains 54 thermal units which are10 oil-fired, 11 gas-fired and 33 coal-fired units. To model hydro units, eight hydro units are considered that the required data of hydro units are taken from [7]. The POZ data and valve loading coefficients and also market prices for energy and reserve (spinning and non-spinning) are taken from [24]. Based on [2], the start-up cost for thermal units is linearized in 10

Conclusions

In this paper a realistic MIP model is presented for HTSS in which valve loading effects cost, dynamic ramp rate, POZs, fuel limitation are considered, all in linear model. It also includes multi performance curves for hydro units. The proposed method is efficiently solved by analytic method using lexicographic optimization and hybrid augmented-weighted ɛ-constraint technique. The best compromise solution among the Pareto optimal solution can be selected either by the fuzzy method or arbitrage

References (25)

  • K. Meng et al.

    ‘Quantum-inspired particle swarm optimization for valve-point economic load dispatch’

    IEEE Trans. Power Syst.

    (2010)
  • T. Li et al.

    ‘Dynamic ramping in unit commitment’

    IEEE Trans. Power Syst.

    (2007)
  • Cited by (0)

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