1. Introduction
Micro gas turbine (MGT) generation systems have become very promising in the power system because of the performance of quick start, load following ability, voltage regulation, peak-load shaving and black-start [
1]. An important application for the MGT generation system is to form the combined heat and power (CHP) system with the exhaust heat recovery system. The CHP system is an efficient form of energy utilization, saving a substantial amount of primary energy compared to traditional energy generation [
2].
A number of scholars have studied the optimized operation of CHP systems. Reference [
3] extensively studies the demand response capability of CHP and quantifies the potential energy cost savings. Reference [
4,
5] propose multi-objective models for optimal energy management of grid-connected CHP systems respectively, in which Reference [
4] considers real-time prices through prediction models, while Reference [
5] is more concerned with carbon emissions. The economic efficiency of CHP systems can be further improved because of the value of heat storage units for improving highly renewable generation-dominated power systems [
6,
7]. Moreover, in real-world operation conditions, the uncertainty of operational parameters [
8] and advanced gas turbine cycles [
9] may also have an influence on the optimized operation.
The CHP optimization operation is a multi-timescale problem, which depends on the coordination between the dispatch instructions at the system level and the control system at the equipment level. Reference [
3,
4,
5,
6,
7] mainly focuses on the optimal operation of the steady state, but most of them ignore the transient process and the important characteristics of the equipment. It is also significant to study a smaller time-scale optimal problem, which mainly focuses on the control system and the regulation. On the other hand, compared to wind turbines and photovoltaic systems, MGT generation systems are more flexible in the sense that the output power can be much easily controlled [
3]. Thus, following the demand response is a feasible operation mode for the MGT-based CHP system, which has been proven to be more economical than the traditional heat-led approach in Reference [
3]. Therefore, this paper aims to enhance demand response capability by improving the performance of MGT generation system during the dynamic regulation period.
The modeling of the micro gas turbine (MGT) is essential in this study because it largely determines whether the dynamic response of the model matches that of the actual CHP system. Currently, methods for modelling MGT can be roughly divided into two major categories: Mechanism modeling and black box modeling, where the former is more commonly used in simulation research. A number of studies have investigated the optimization of different micro gas turbine technologies with various demand based on static models [
10,
11,
12], which are not suitable for transient study. Dynamic models can be described in the form of multi-differential algebraic equations, such as the differential algebraic equations model established by Schobeiri [
13] and the small gas turbine differential algebraic equations established by Perez [
14]. Different models are used for different research purposes. To fully quantify the potential of cycle humidification, the model in Reference [
15] focus more on the complex chemical kinetics in the combustion chamber. Reference [
16] proposed a model described in the form of transfer functions, which is widely used in the simulation of MGT generation system. Systems in Reference [
17,
18,
19] are all based on the gas turbine model in Reference [
16], but the speed control system cannot follow the order with a zero steady error so that the economic efficiency is unsatisfactory when applied to follow the power demand response.
Since the control parameters have a decisive impact on the performance of the controller, it is another key issue for studying a suitable control parameter optimization method. Currently, meta-heuristic algorithms have gradually replaced the traditional methods based on amplitude-frequency characteristics and engineering debugging in terms of PI parameter optimization. The particle swarm optimization algorithm (PSO) [
20,
21] and its improved versions [
22,
23] are among the most popular approaches, due to simplicity and flexibility. There have been many novel swarm-based optimization algorithms developed in recent years [
24,
25,
26,
27], and the whale optimization algorithm (WOA) is one of them [
27]. From Reference [
27], compared with other four typical optimization algorithms, including PSO, Gravitational Search Algorithm (GSA), Differential Evolution Algorithm (DE) and Fast Evolutionary Programing (FEP), WOA is very competitive in both local search capability and global search capability. WOA has also been applied to solve engineering problems in power systems [
28,
29]. Thus, it is foreseeable that WOA has good application prospect in PI parameter optimization problems.
When applied to demand response, the MGT generation system is expected to respond to dispatch instructions quickly and effectively. From the economic aspect, however, the signals are largely decided by energy consumption and energy prices, which vary according to the change of settlement. On the one hand, the generation system needs to improve its dynamic performance to follow the dispatch instructions as soon as possible. On the other hand, during this period, it also needs to ensure its security and reduce its energy loss to a minimum level. In order to achieve the balance between response time and energy loss with the consideration of security, a better control strategy is desirable for the MGT generation system.
This paper focuses on the performance in the transient process of power regulation from the perspective of the MGT control system. Speed and power control is introduced to the MGT to realize a zero steady error. An optimization method is developed for identifying the control parameters, combining economy, safety and rapidity. The optimization objective for the controller is to minimize the comprehensive electricity error considering both electricity quantity error and the integral of absolute error (IAE) of power. The constraints on the rate of fuel valve opening and the simulation model are included. An improved whale optimization algorithm (IWOA) with adaptive weights is proposed to resolve the optimization problem. The new model is built in MATLAB/Simulink and studied in different operational scenarios to testify its capability and effectiveness. Results show that the proposed control strategy is able to enhance the demand response capability of the MGT generation system.
The rest of this paper is structured as follows:
Section 2 introduces the dynamic model of the MGT generation system.
Section 3 builds the optimization model for control parameters of the MGT control. Details of the improved whale optimization algorithm are illustrated in
Section 4. An extensive case study is investigated in
Section 5, and
Section 6 concludes the paper.
4. An Improved Whale Optimization Algorithm
From the perspective of optimization, the problem of control parameter optimization can be regarded as a nonlinear multimodal function with strong complexity. It is difficult to solve such kind of optimization problems with traditional algorithms based on the calculation of gradients [
32]. Heuristic algorithms represented by the Particle Swarm Optimization (PSO) algorithm have been applied to such problems and achieved satisfactory results [
22,
23].
The Whale Optimization Algorithm (WOA) was proposed by Mirjalili S and Lewis A in 2016 [
27]. It is assumed that whales are the most intelligent animals containing motion and WOA is a swarm-based optimization method motivated by the specific hunting behavior of humpback whales. According to Reference [
27], WOA is more competitive than PSO in both exploration capability and exploitation capability.
Therefore, the WOA algorithm is used in this paper to solve the optimization problem, and its global search capability is further improved to escape from local extremums. This section mainly introduces the original WOA algorithm and the basic idea of the improved WOA.
4.1. The Original WOA
The basic idea of WOA is to perform optimization by simulating the unique predation and social behavior of whale. It describes the spiral bubble-net feeding maneuver through three mathematical models, simulating the process of encircling prey, bubble-net attack and searching for prey [
27].
4.1.1. Encircling Prey
In this stage, one search agent is selected and defined as the optimal candidate solution in the current generation. Other search agents try to stay away from the optimal agent to implement a global search, which can be described as follows:
where
t represents the current number of iterations;
and
are coefficient vectors;
is in the current position vector in this generation;
is the optimal position vector which needs to be updated in every generation.
The coefficient vectors
and
are calculated as follows:
where
is a random vector with a value in
;
is the upper limit of the iteration number. From Equation (14), it is noticed that
, which realizes the goal of updating the search agent away from the best candidate position, thereby avoiding falling into local optimum.
4.1.2. Bubble-Net Attack
In this stage, two methods (shrinking encircling and spiral updating position) are discussed. The mechanism of shrinking encircling is similar to that of the global search stage, except that the range of
is set to
. The other method, spiral updating position, constructs a logarithmic spiral curve based on the current position and the optimal agent so that the search agent can gradually approach the optimal position. This is described in Equations (17)−(19).
where
is a constant that defines the shape of the logarithmic spiral;
is a random number between [−1, 1];
is a probability factor used to determine whether to select shrinking encircling or spiral updating positions.
4.1.3. Search for Prey
In addition to the bubble-net search strategy, the random se arch for prey is also a feasible solution. If
exceeds the range of [−1, 1], the distance
is updated randomly. Search agents deviate from the original target to find the prey, which gives WOA certain global search capability. This is described in Equations (20) and (21).
It is noted that in Equations (20) and (21) is the position of a random agent in the current population instead of a simple random position.
4.2. The Improved WOA with Adaptive Weights
Although WOA is very competitive, it is insufficient in balancing the exploration and exploitation [
33,
34], which is similar to other swarm-based algorithms. In WOA, the exploration search is implemented by Equation (21) and the exploitation search is implemented by Equation (19). Whether to choose Equation (21) or Equation (19) is determined by the coefficient vector
. In fact, the total probability of performing Equation (19) is
under the precondition
. Therefore, Equation (19) dominates Equation (21) and the unbalance may lead to premature convergence to local extremums in some conditions [
33].
It is noted that for various agents with different fitness values,
is generated in the same way. If agents with lower fitness values focus on global search, while agents with higher fitness values pay more attention to local development, the performance of WOA may be improved. Adaptive inertia weights have been applied to PSO algorithms to balance the global exploration ability and the local optimization ability [
23]. Based on this idea, the adaptive weight is added to the WOA in this paper, forming an improved WOA with adaptive weights (IWOA). Unlike PSO, weights are added to the optimal whale instead of the simulation step. Specific operations are as follows: In the iteration process, all search agents are divided into two halves according to their fitness values
, which are determined by their distance away from the optimal agent. Agents with lower fitness values are assigned to Group 1, while those with higher values are assigned to Group 2. Denote the average fitness values of Group 1 and Group 2 as
and
respectively, where
. On this basis, the inertia weights are classified to 3 cases as follows:
Case 1:
The current individual’s fitness is better than the average fitness of the better group, indicating that the individual is dominant in the population. In this case, it is beneficial to explore the local optimal value by making the optimal whale individual shake in a small range, so the value of ω is set as a random number from 0.8 to 1.2.
Case 2:
The current individual’s fitness is worse than the average fitness of the worse group, indicating that the position is not ideal. In order to broaden the overall optimization ability, ω is set as a random number between 0.3 and 0.6 or a random number between 1.3 and 1.6, which is conducive to the individual to find the prey in other locations, enhancing the overall searchability.
Case 3:
In this case, the current individual is in the general position of the group, so that it can approach the optimal position according to the original algorithm with the value of ω set as 1.
In this way, Equation (19) can be improved to Equation (22).
The flowchart of the IWOA is shown in
Figure 7.