Multiobjective invasive weed optimization: Application to analysis of Pareto improvement models in electricity markets

Abstract

This paper presents a proposal for multiobjective Invasive Weed Optimization (IWO) based on nondominated sorting of the solutions. IWO is an ecologically inspired stochastic optimization algorithm which has shown successful results for global optimization. In the present work, performance of the proposed nondominated sorting IWO (NSIWO) algorithm is evaluated through a number of well-known benchmarks for multiobjective optimization. The simulation results of the test problems show that this algorithm is comparable with other multiobjective evolutionary algorithms and is also capable of finding better spread of solutions in some cases. Next, the proposed algorithm is employed to study the Pareto improvement model in two complex electricity markets. First, the Pareto improvement solution set is obtained for a three-player oligopolistic electricity market with a nonlinear demand function. Then, the IEEE 30-bus power system with transmission constraints is considered, and the Pareto improvement solutions are found for the model with deterministic cost functions. In addition, NSIWO algorithm is used to analyze this system with stochastic cost data in a risk management problem which maximizes the expected total profit but minimizes the profit risk in the market.

Introduction

Multiobjective optimization is to determine the best Pareto-optimal solution set or a representative subset which is defined as a set of solutions that are nondominated with respect to each other. Note that vector σ = (σ, …, σ_n) dominate vector τ = (τ, …, τ_n) (denoted by σ ≻ τ) if and only if:

$${\sigma }_i\ge {\tau }_{i}i=1,\dots ,n$$

And there is at least one i(1 ≤ i ≤ n) such that,

$${\sigma }_i>{\tau }_i$$

In this regard, evolutionary computing has been one of the main approaches for multiobjective optimization. In fact, due to population-based nature of evolutionary algorithms, there is an intuitive hope to find the Pareto-optimal solutions in a single run. During the last two decades, many techniques have been proposed for multiobjective optimization based on evolutionary algorithms. Vector evaluated genetic algorithm (VEGA) was the first implementation of multiobjective evolutionary algorithms (MOEA) in the mid-1980s [1]. Afterwards, several MOEAs were developed including multiobjective genetic algorithm (MOGA) [2], niched Pareto genetic algorithm (NPGA) [3], weight-based genetic algorithm (WBGA) [4], random weighted genetic algorithm (RWGA) [5], nondominated sorting genetic algorithm (NSGA) [6], [7], [8], strength Pareto evolutionary algorithm (SPEA) [9], improved SPEA (SPEA2) [10], Pareto-archived evolution strategy (PAES) [11], [12], Pareto envelope-based selection algorithm (PESA) [13], region-based selection in evolutionary multiobjective optimization (PESA-II) [14], multiobjective evolutionary algorithm (MOEA) [15], micro-GA [16], rank-density based genetic algorithm (RDGA) [17], dynamic multiobjective evolutionary algorithm (DMOEA) [18], and a real-coding jumping gene genetic algorithm (RJGGA) [19].

Some of the most well-known and efficient MOEAs will be shortly discussed here. PESA is an algorithm which works on cell-based density. In this approach, the objective space is divided into a number of cells or hyper-cubes, and the number of solutions in each cell defines its density. This density metric is used to select more diverse solutions out of the nondominated solutions. PESA is easy to implement and computationally efficient, but it has two disadvantages: the performance depends on the cell sizes and the prior information is needed for the objective space. PAES is another efficient algorithm which is equipped with a random mutation hill climbing strategy. There are two main disadvantages associated with PAES. First, it is not a population-based approach, and also its performance depends on the cell sizes. SPEA is a well-known and popular MOEA, but it has a complex clustering algorithm. SPEA2 improves SPEA by making sure that the extreme points in Pareto front are preserved, but it has a computationally expensive fitness and density calculation. NSGA is a fast algorithm for multiobjective optimization, but it has some problems related to niche size parameter. NSGA-II is an efficient and well-tested algorithm with only one single parameter (without GA parameters) to be tuned, however, the crowding distance described in this algorithm works in objective space only [20].

Invasive Weed Optimization (IWO) is a novel ecologically inspired algorithm that mimics the process of weeds colonization and distribution. Despite its recent development, it has shown successful results in a variety of practical applications like optimization and tuning of a robust controller [21], optimal positioning of piezoelectric actuators [22], developing a recommender system [23], antenna configuration [24], cooperative identification and adaptive control of a surge tank [25], analysis of electricity markets dynamics [26], [27], cooperative multiple task assignment of the UAVs [28], etc. Due to its wide range applicability and relative fast convergence rate, we are motivated to introduce a multiobjective form of IWO based on the fast nondominated sorting approach proposed in NSGA-II.

The first part of this paper is dedicated to Pareto-optimal solution search for benchmark problems in multiobjective optimization using nondominated sorting IWO (NSIWO). In the second part, application of NSIWO algorithm for investigating the Pareto improvement model [30] in electricity markets is studied. The Pareto improvement model is a newly developed model for analysis of pool-based electricity markets. Actually, this model approximates infinitely repeated games under tacit collusion among the firms [30]. In our experimental simulations, we study two types of electricity market: an unconstrained electricity market with nonlinear system demand function and a transmission-constrained electricity market with linear demand curves. In these problems, we are facing some challenging issues like nonconvexity (because of the nonlinearity in the first problem) and discontinuity (because of the constraints in the second problem), which lead us to the games with local optima. As a result, the capability of the proposed evolutionary algorithm for global search could help us escape from local traps. In addition, we study a constrained form of collusion model (as a special case of Pareto improvement model) for the second market with stochastic cost data. In this problem, we use the proposed NSIWO algorithm to find the Pareto-optimal solutions which maximize the expected total profit but minimize the total risk of the market players. Consequently, after finding the Pareto front, the decision makers may choose the solution that satisfies their needs.

The organization of this paper is as follows. Section 2 presents NSIWO algorithm accompanied by a quick review of IWO and NSGA-II algorithms. In Section 3, the simulation results for multiobjective optimization of a well-known test suit are provided, and the performance of NSIWO algorithm is compared with some other MOEAs. Section 4 explains the Pareto improvement model in electricity markets and shows the application of NSIWO to find the Pareto improvement solution set for two different test cases: a three-player unconstrained electricity market with a nonlinear demand function, and the transmission-constrained IEEE 30-bus power system with deterministic and stochastic cost data. Finally, the conclusions are drawn in Section 5.