1. Introduction
The air-conditioning systems of large-scale commercial buildings in Taiwan account for about 32%–54% of Taiwan’s electrical energy consumption, and chiller plants consume more than 70% of the overall energy consumed by air-conditioning systems [
1]. In systems where multiple chillers are operated in parallel (multi-chiller systems), each chiller can operate independently; adjusting the chiller operation schedule to provide the venue with a stable refrigeration ton (RT) load and a flexible maintenance schedule [
1] is a common practice in large commercial buildings. Because multi-chiller systems are composed of chillers of varying features or even of various types of chillers, the question of how to adjust appropriate numbers of operating chillers and operational control points so that each chiller operates at optimal efficiency is crucial for saving energy in air-conditioning systems.
In recent years, many articles have discussed the optimal chiller loading (OCL) problem. Chang [
2,
3] proposed using a branch-and-bound method and a Lagrangian multiplier method to solve OCL problems. In addition to traditional numerical methods, numerous heuristic optimization methods have been used to solve OCL problems. Optimal chiller loading that consumed less energy than that of Chang [
2,
3] was obtained by genetic algorithms (GAs) [
4]. Chang et al. [
5] applied an evolutionary strategy (ES) to OCL problems and found a lower power consumption with higher precision for chillers than found previously. Particle swarm optimization (PSO), which uses the social behavior of entities for evolutionary computing, was applied to OCL problems [
6]. PSO was more efficient than binary GA and real-valued GA in solving OCL problems. Differential evolution (DE), which uses the characteristics of intergroup differences for evolution, was also used in OCL problems [
7]; according to the literature, its results were superior to that of PSO. Coelho et al. [
8] proposed using a differential cuckoo search approach to improve the original cuckoo search approach, so that the performance of a differential cuckoo search for OCL problems could be superior to those of GA, PSO, and DE.
Exploration (global search) and exploitation (local search) are the two critical factors that influence evolutionary optimization methods; the balance of the two directly affects the results and efficacy of searches for optimal solutions. Tan et al. [
9] dynamically adjusted evolutionary computations to maintain the balance between scope and convergence of multi-objective optimization. Binkley et al. [
10] used PSO for multimodal optimization; when the velocity of the particle swarm was lower than a threshold, the designed quantity was reduced and the particles were restarted to maintain the diversity of the multimodal design space. Epitropakis et al. [
11] combined the mutation mechanisms of different DE algorithms, first using exploration and then a high-convergence mechanism to search for optimal solutions. Bao [
12] proposed a two-phase hybrid optimization algorithm involving an ant colony algorithm (ACO) and a simulated annealing (SA) optimization algorithm to solve complex optimization problems. Beghi et al. [
13] used a multiphase GA for the management of a multi-chiller system to reduce power consumption and operational costs. Cheng and Tran [
14] used a two-phase DE algorithm project to obtain an optimized schedule of time and cost.
This study proposes a two-stage DE algorithm to solve OCL problems. The framework includes DE with two different types of variables: the first stage uses a modified binary differential evolution (MBDE) algorithm [
15,
16] for exploration; the second stage uses a real-valued DE algorithm for exploitation. A binary encoding method enabled greater exploration than a real-valued encoding algorithm [
17] would have allowed; thus, in the process of searching for the optimal solution, the proposed method was able to quickly find the optimal solution. After the proposed MBDE completes stage 1, the optimal solution discovered by MBDE in stage 1 undergoes conversion into real numbers and is then introduced into the real-valued DE of stage 2 for optimal exploitation. Through the integration of the aforementioned two stages, the proposed two-stage algorithm, which integrates the advantages of binary MBDE and real-valued DE, has excellent exploration and exploitation capabilities. Its design enhances its operational efficacy and its evolutionary methods enhance the results of its searches for optimal solutions.
The remainder of this paper is structured as follows.
Section 2 describes the intricacies of the OCL problem, its objective function, and its restrictions.
Section 3 explains the evolutionary mechanism and the features of the two-stage DE algorithm.
Section 4 compares the results of the proposed method with those of different methods for various examples. Conclusions are drawn in
Section 5.
2. Introduction to Multi-chiller System
Multi-chiller systems can provide flexible operation, reserve capacity, and less frequent system shutdowns for maintenance. Those systems composed of two or more chillers are widely used in the air-conditioning systems of large buildings. In a multi-chiller system, each chiller can operate independently and provide different refrigeration capabilities; the chillers operate efficiently according to different or similar performance curves to meet a wide range of RT requirements in HVAC (heating, ventilation and air conditioning) system. The architecture of the multi-chiller system is as shown in
Figure 1 below.
Generally, the maximum capacity of a chiller is designed to meet the maximum peak load demand, but because of actual venue requirements and change of seasons, maximum peak load generally only occurs in summer, and the system operates at low partial load mode during the remaining time. Therefore, if the designed capacity is too large, the system consumes excessive power. The partial load rate (PLR) of the chiller can be expressed as (1):
The power consumption of a chiller and its PLR share a certain relationship; according to the chiller power consumption equation of [
8] as shown in (2) and (3):
Equations (2) and (3) are chiller power consumption for Example 1 and Example 2 introduced from literate [
8]. In (2) and (3), the coefficients
ai,
bi,
ci, and
di define the relationship between power consumption and PLR for the chillers in Examples 1 and 2; this analysis was based on [
8]. The PLR ranged between 0.0 and 1.0.
The overall objective of OCL optimization was to find the optimal partial loading rate of each chiller of the multi-chiller system that satisfied the cooling requirements of the venue but also minimized overall power consumption. Therefore, the objective function of the proposed OCL optimization was defined as shown in (4):
In (4), the parameter i signifies the ith chiller; n is the total number of chillers in the multi-chiller system; Pi is defined as the power consumption (kW) of the ith chiller. The object of Equation (4) is to find the lowest total power consumption in the multi-chiller system.
OCL problems have two types of restrictions [
8] during solving; the first restriction is that the total output of RT must be equal to the RT required by the venue, as shown in (5).
Qi signifies the rated RT capacity of the
ith chiller;
CL is the total RT required by the venue. This is the basic constraint for OCL problems. If the total RT generated by all chillers is smaller or larger than the RT requirement, the people who stay in the venue will feel not very comfortable.
The second restriction is that the partial loading of each chiller cannot be less than 30% [
8], as defined in (6):