# Chaotic Evolutionary Programming for an Engineering Optimization Problem
## 1 Introduction  
The electric power system is a complex engineering system. The planning, operation and control of the interconnected electric power system is a challenging task. Economic load dispatch (ELD) of the power system is one such task, which means planning, scheduling, and operating generators in an economical manner. The transmission losses form an inherent part of the economic operation of the power system. Today's ELD problem posses nonlinear behavior, due to imposed equality and inequality constraints [1]. The ELD of a multi-fuel system involves power generators that utilize more than one type of fuel. Depending on the power demand, the generator can switch the fuel type [2]. The ELD problem has been identified as a multimodal problem, which is a challenge to solve. Since the practical problems are multimodal in nature, the gradient approaches are not suitable for them. In this respect, the solution of nonlinear economic dispatch problems has been effectively obtained using random search algorithms irrespective of the shape of the solution hyperspace. Although these heuristic methods provide a faster and reasonable solution, these do not ensure a global optimal solution in a finite time. The complex dispatch and scheduling problems require effective and efficient optimization algorithms for a beneficial solution [3]. In effect, the global optimization algorithms have been extensively used as a solution procedure for ELD problem having a multi-fuel generator. In the previous decade, global optimization techniques such as genetic algorithm or simulated annealing, which is a form of probabilistic heuristic algorithm, have been used to solve the ELD problem. The other widely accepted stochastic search algorithms are evolutionary programming (EP) [4], particle swarm optimization (PSO) [5], genetic algorithm (GA) [6], simulated annealing (SA), etc. The population based algorithms, also called evolutionary algorithms (EAs), have been widely employed to solve the practical constrained ELD optimization problems. The derivative free mechanism, parallel processing nature, fast convergence rate, performance independent of the hyperspace, etc., are the key factors for the superiority of these methods. In the 1990s, EP dominated the field of optimization algorithms. The features of the EP mechanism are: firstly, the utilization of real-valued variables and parameters; secondly, mutation and selection are the sole operators, i.e., EP uses a single evolution operator. The computational resources required by EP are much less compared to other EAs; hence, it may result in a smaller computational time. The maturity phase of EP consists of classical EP [7], self-adaptive EP [8], fast EP [9], scaled Gaussian mutation EP [10], EP with the mean of Gaussian and Cauchy mutations with an empirical learning rate [7], EP with Gaussian mutation with empirical learning [11] and EP with the better of Gaussian and Cauchy mutations with an empirical learning rate [12]. The objective of all these modifications is to elevate the limitations of multimodal problem solving [12]. [PERSON] et al. [13] showed empirically that EP with cooperative convolution cab be used to solve large-scale problems with superior performance. Further, a fast EP algorithm based on a Levy probability distribution based mutation operator has shown a performance advantage [14]. It has been observed from the literature survey that the modifications of EP are random number or mutation operator oriented. Recently, the application of non-linear dynamics has been suggested for the selection of algorithm control parameters [15] and search algorithm tuning [16; 17]. A random sequence inheriting features of long periodicity and uniformity is suitable to enhance the search ability of the stochastic search algorithm. The random numbers generated by the chaotic map exhibit ergodicity, non-repetitiveness, and non-linearity and are dynamic in nature [18]. The random number sequence possesses an element of regularity and exhibits sensitive dependence on initial conditions [19].