Application of modified pigeon-inspired optimization algorithm and constraint-objective sorting rule on multi-objective optimal power flow problem

Highlights

• Modified pigeon-inspired algorithm (MPIO) and constraint-objective sorting rule (COSR) are proposed for MOOPF problems.

• MPIO-COSR algorithm is effective to reduce the active power loss, fuel cost and emission of power systems.

• MPIO-COSR algorithm is examined on the IEEE 30-node, 57-node and 118-node systems by eight simulation cases.

• The superiority of MPIO-COSR algorithm is validated by comparing with NSGA-II and MPIO-PFM algorithms.

• Proposed MPIO-COSR algorithm obtains evenly-distributed POS & fast-convergence & zero constraint-violation.

Abstract

To solve the non-differentiable optimal power flow (OPF) problems with multiple contradictory objectives, a modified pigeon-inspired optimization algorithm (MPIO) is put forward in this paper. Combining with the common-used penalty function method (PFM), the MPIO-PFM algorithm is proposed and applied to optimize the active power loss, emission and fuel cost (with valve-point loadings) of power system. Eight simulation trials carried out on MATLAB software validate MPIO-PFM algorithm can obtain superior Pareto Frontier (PF) comparing with the typical NSGA-II algorithm. Nevertheless, some Pareto solutions obtained by MPIO-PFM algorithm cannot satisfy all system constraints due to the difficulty in choosing the proper penalty coefficients. Thus, an innovative approach named as constraint-objective sorting rule (COSR) is presented in this paper. The bi-objective and tri-objective trials implemented on IEEE 30-node, 57-node and 118-node systems demonstrate that the Pareto optimal set (POS) obtained by MPIO-COSR algorithm realizes zero-violation of various system constraints. Furthermore, the generational-distance and hyper-volume indexes quantitatively illustrate that in contrast to NSGA-II and MPIO-PFM methods, the MPIO-COSR algorithm can determine the evenly-distributed PFs with satisfactory-diversity. The intelligent MPIO-COSR algorithm provides an effective way to handle the non-convex MOOPF problems.

Introduction

Unlike the single-objective optimization, the multi-objective optimization (MOP) problem takes more than one conflicting goals into account concurrently. It is impossible to make each goal achieve the best state at the same time. The mathematical model of MOP problem, which usually has great computational complexity and obvious non-linear characteristics, is strictly limited by various constraints [1], [2].

The MOP problems are very common in practical fields such as power systems [3], [4]. Electricity with self-evident importance is the most basic energy in modern society. Research shows that the multi-objective optimal power flow (MOOPF) problems have undeniable significance to achieve the safety and economy operation of power system. Essentially, the MOOPF is a non-linear minimization problem with high-dimensional feature [5], [6], [7], [8].

Intelligent algorithms have been successfully applied on handling the nonlinear MOOPF problems with the gradually maturing of computer technology. For instance, the novel quasi-oppositional modified Jaya algorithm [9], the hybrid DA-PSO optimization algorithm [10] and the multi-objective firefly algorithm [11] are capable to deal with the MOOPF problems effectively. However, there is still much room in the bi-objective MOOPF optimizations on large-scale systems and the tri-objective ones.

The basic and improved pigeon-inspired optimization algorithms (PIO) with fast-convergence and strong-robustness have been applied to various fields such as the fuzzy production scheduling problems [12] and large-scale traveling salesman problems [13]. However, the shortcoming of the basic PIO algorithm that is easy to be trapped into the local optimums cannot be ignored.

To effectively solve the complex practical problems, the PIO algorithm needs to be combined with some appropriate improvement strategies. In Ref. [14], the modified PIO algorithm considering the hierarchical learning behavior is proposed to handle the UAV distributed flocking problem among obstacles. In Ref. [15], the cooperative PIO algorithm with distance threshold is put forward to improve the population-diversity in handling the multilevel image thresholding problems. And in Ref. [16], the performance of improved PIO algorithm for the air quality prediction problems has been optimized by integrating the particle swarm optimization (PSO) algorithm. However, it can be seen from these literatures that the PIO algorithm is usually plagued by the local convergence due to the poor diversity. To pertinently improve the population-diversity and avoid the premature-convergence, the adaptive-adjusted map factor (${\mathit{R}}_{mapnew}$) and nonlinearly-adjusted coefficient (${\omega }_{div}$) are integrated into the proposed modified PIO (MPIO) algorithm. The applicability and superiority of MPIO algorithm to deal with the MOOPF problems are verified by eight simulation experiments implemented in this paper.

In this paper, the MPIO algorithm and constraint-objective sorting rule (COSR) as two major contributions are proposed.

First, the MPIO algorithm which adopts the adaptive-adjusted ${\mathit{R}}_{mapnew}$, nonlinearly-adjusted ${\omega }_{div}$ and novel landmark searching model is put forward to solve the MOOPF problems.

Furthermore, the MOOPF problems are severely restricted by equality constraints and inequality ones [11], [17], [18]. The control variables can be limited within a valid range in the initialization phase. But the state variables may not meet all inequality constraints. The penalty function method (PFM), a common technique to handle the constraints of state variables [11], [19], is integrated with the MPIO method to generate the novel MPIO-PFM algorithm.

Comparing with the non-dominated sorting genetic algorithm-II (NSGA-II) which is often used as the comparison benchmark to evaluate the quality of novel algorithms, the proposed MPIO-PFM has greater potential in exploring superior Pareto frontier (PF). However, experimental results indicate the MPIO-PFM algorithm cannot guarantee each solution of obtained Pareto optimal set (POS) achieves the zero system-constraints violation. The presented COSR rule, which takes the constraint-violation value as the screening factor of POS set, can effectively overcome the above shortcoming of PFM method.

By integrating the MPIO algorithm and COSR sorting strategy, an effective MPIO-COSR algorithm is put forward in this paper. Five bi-objective and three tri-objective MOOPF trials which aim to optimize the active power loss, total emission and fuel cost are carried out to verify the applicability and superiority of MPIO-COSR algorithm. In contrast to MPIO-PFM and NSGA-II algorithms, the suggested MPIO-COSR algorithm can not only find the well-distributed POS, but also realize the zero constraint-violation of each power flow solution.

The rest part of this article is set as follows. Section 2 gives the mathematical model of MOOPF problems including objective functions and system restrictions. The constraint processing methods of state variables and the proposed COSR strategy for seeking the evenly-distributed PF are described in Section 3. The basic PIO algorithm and the novel MPIO-COSR algorithm are introduced in Section 4. Section 4 summarizes the application of MPIO-COSR algorithm on MOOPF Problems as well. Section 5 shows the results of eight MOOPF trials on three different standard systems. Section 6 measures the optimization performance of MPIO-COSR algorithm according to the convergence analysis, two quantitative evaluation indexes and computational complexity. Eventually, Section 7 gives the conclusions of this paper.