11.1.1 GA Mechanism

The GA mechanism is inspired by the mechanism of natural selection, where stronger individuals would likely be the winners in a competing environment. Normally in a GA, the parameters to be optimized are represented in a binary string. A simplified flowchart for GA is shown in Figure 11.1. The cost function, which determines the optimization problem, represents the main link between the problem at hand (system) and GA, and also provides the fundamental source to provide the mechanism for evaluation of algorithm steps. To start the optimization, GA uses randomly produced initial solutions created by a random number generator. This method is preferred when a priori information about the problem is not available. There are basically three genetic operators used to produce a new generation: selection, crossover, and mutation. The GA employs these operators to converge at the global optimum. After randomly generating the initial population (as random solutions), the GA uses the genetic operators to achieve a new set of solutions at each iteration. In the selection operation, each solution of the current population is evaluated by its fitness, normally represented by the value of some objective function, and individuals with higher fitness values are selected.

Different selection methods such as stochastic selection or ranking-based selection can be used. In the selection procedure the individual chromosomes are selected from the population for the later recombination/crossover. The fitness values are normalized by dividing each one by the sum of all fitness values named selection probability. The chromosomes with a higher selection probability have a higher chance to be selected for later breeding.

The crossover operator works on pairs of selected solutions with a certain crossover rate. The crossover rate is defined as the probability of applying a crossover to a pair of selected solutions (chromosomes). There are many ways to define the crossover operator. The most common way is called the one-point crossover. In this method, a point (e.g., for two binary-coded solutions of a certain bit length) is determined randomly in two strings and corresponding bits are swapped to generate two new solutions. A mutation is a random alteration with a small probability of the binary value of a string position, and it will prevent the GA from being trapped in a local minimum. The coefficients assigned to the crossover and mutation specify the number of the children. Information generated by the fitness evaluation unit about the quality of different solutions is used by the selection operation in the GA. The algorithm is repeated until a predefined number of generations have been produced.

Unlike the gradient-based optimization methods, GAs operate simultaneously on an entire population of potential solutions (chromosomes or individuals) instead of producing successive iterations of a single element, and the computation of the gradient of the cost functional is not necessary. Interested readers can find basic concepts and a detailed GA mechanism in Goldberg¹⁶ and Davis.¹⁷

11.1.2 GA in Control Systems

GA is one of rapidly emerging optimization approaches in the field of control engineering and control system design.¹⁸ Optimal/adaptive tracking control, active noise control, multiobjective control, robust tuning of control systems via seeking the optimal performance indices provided by robust control theorems, and use in fuzzy-logic- and neural-network-based control systems are some important applications of GA in control systems. Genetic programming can be used as an automated invention machine to synthesize designs for complex structures. It facilitates the design of robust dynamic systems with respect to environmental noise, variation in design parameters, and structural failures in the system.¹⁹,²⁰

A simple GA-based control system is conceptually shown in Figure 11.2. The GA controller consists of three components: performance evaluator, learning algorithm, and control action producer. The performance evaluator rates a chromosome by assigning it a fitness value. The value indicates how good the chromosome is in controlling the dynamical plant to follow a reference signal. The learning algorithm may use a set of rules in the form of “condition then action” for controlling the plant. The desirable action will be performed by a control action producer when the condition is satisfied. The control structure shown in Figure 11.2 is implemented for several control applications in different forms.²¹

11.3.1 Multiobjective Optimization

Initially, the majority of control design problems are inherently multiobjective problems, in that there are several conflicting design objectives that need to be simultaneously achieved in the presence of determined constraints. If these synthesis objectives are analytically represented as a set of design objective functions subject to the existing constraints, the synthesis problem could be formulated as a multiobjective optimization problem.

As mentioned, in a multiobjective problem, unlike a single optimization problem, the notation of optimality is not so straightforward and obvious. Practically in most cases, the objective functions are in conflict and show different behavior, so the reduction of one objective function leads to an increase in another. Therefore, in a multiobjective optimization problem, there may not exist one solution that is best with respect to all objectives. Usually, the goal is reduced to compromise all objectives and determine a trade-off surface representing a set of nondominated solution points, known as Pareto-optimal solutions. A Pareto-optimal solution has the property that it is not possible to reduce any of the objective functions without increasing at least one of the other objective functions.

Unlike single-objective optimization, the solution to the multiobjective problem is not a single point, but a family of points as the (Pareto-optimal) solutions set, where any member of the set can be considered an acceptable solution. However, the choice of one solution over the other requires problem knowledge and a number of problem-related factors.²³,²⁴

Mathematically, a multiobjective optimization (in the form of minimization) problem can be expressed as

where x = {x₁, x₂, ..., x_N} ∈ X is a vector of decision variables in the decision space X, and y = {y₁, y₂, ..., y_N} ∈ Y is the objective vector in the objective space. The solution is not unique; however, one can choose a solution over the others. In the minimization case, the solution x¹ dominates x², or x¹ is superior to x², if

The x¹ is called a noninferior or Pareto-optimal point if any other in the feasible space of design variables does not dominate x¹. Practically, since there could be a number of Pareto-optimal solutions, and the suitability of one solution may depend on system dynamics, the environment, the designer’s choice, etc., finding the center point of a Pareto-optimal solutions set may be desired.

GA is well suited for solving multioptimization problems. Several ap proaches have been proposed to solve multiobjective optimization problems using GAs.^{15,16,25–29} The keys for finding the Pareto front among these various procedures are the Pareto-based ranking²⁹ and fitness-sharing¹⁶ techniques. In the most common method, the solution is simply achieved by developing a population of Pareto-optimal or near-Pareto-optimal solutions that are non-dominated. The xⁱ is said to be nondominated if there does not exist any x^j in the population that dominates xⁱ. Nondominated individuals are given the greatest fitness, and individuals that are dominated by many other individuals are given a small fitness. Using this mechanism, the population evolves toward a set of nondominated, near-Pareto-optimal individuals.²⁹

In addition to finding a set of near-Pareto-optimal individuals, it is desirable that the sample of the whole Pareto-optimal set given by the set of nondominated individuals be fairly uniform. A common mechanism to ensure this is fitness sharing,²⁹ which works by reducing the fitness of individuals that are genetically close to each other. However, all the bits of a candidate solution bit string are not necessarily active. Thus, two individuals may have the same genotype, but different gene strings, so that it is difficult to measure the difference between two genotypes in order to implement fitness sharing. One may simply remove the multiple copies of genotypes from the population.³⁰

11.3.2 application to agC Design

The multiobjective GA methodology is conducted to optimize the proportional-integral (PI)-based supplementary frequency control parameters in a multiarea power system. The control objectives are summarized to minimize the ACE signals in the interconnected control areas. To achieve this goal and satisfy an optimal performance, the parameters of the PI controller in each control area can be selected through minimization of the following objective function:

where ObjFnc_i is the objective function of control area i, K is equal to the simulation sampling time (s), and |ACE_i,t| is the absolute value of the ACE signal for area i at time t.

Following use of the multiobjective GA optimization technique to tune the PI controllers and find the optimum values of objective functions (Equation 11.4), the fitness function (FitFunc) can be defined as follows:

Each GA individual is a double vector presenting PI parameters. Since a PI controller has two gain parameters, the number of GA variables could be N_var = 2n, where n is the number of control areas. The population should be considered in a matrix form with size m × N_var; where m represents individuals.

As mentioned earlier, the population of a multiobjective GA is composed of dominated and nondominated individuals. The basic line of the algorithm is derived from a GA, where only one replacement occurs per generation. The selection phase should be done first. Initial solutions are randomly generated using a uniform random number of PI controller parameters. The crossover and mutation operators are then applied. The crossover is applied on both selected individuals, generating two children. The mutation is applied uniformly on the best individual. The best resulting individual is integrated into the population, replacing the worst-ranked individual in the population. This process is conceptually shown in Figure 11.6.

The above-described multiobjective GA is applied to the three-control area power system example used in Section 2.4. The closed-loop system response for the following simultaneous load step increase (Equation 11.6) in three areas is examined, and some results for areas 2 and 3 are shown in Figure 11.7.

It has been shown from simulation results that the proposed technique is working properly, as well as the robust H_∞-PI control methodology addressed in Bevrani et al.³¹ Interested readers can find more time-domain simulations for various load disturbance scenarios in Daneshfar.¹³

11.4.1 Mixed H₂/H_∞

In many real-world control problems, it is desirable to follow several objectives simultaneously, such as stability, disturbance attenuation, reference tracking, and considering the practical constraints. Pure H_∞ synthesis cannot adequately capture all design specifications. For instance, H_∞ synthesis mainly enforces closed-loop stability and meets some constraints and limitations, while noise attenuation or regulation against random disturbances is more naturally expressed in H₂ synthesis terms. The mixed H₂/H_∞ control synthesis gives a powerful multiobjective control design addressed by the LMI techniques.

A general synthesis control scheme using the mixed H₂/H_∞ control technique is shown in Figure 11.8. G(s) is a linear time-invariant system with the following state-space realization:

where x is the state variable vector, w is the disturbance and other external input vector, and y is the measured output vector. The output channel z₂ is associated with the H₂ performance aspects, while the output channel z_∞ is associated with the H_∞ performance. Let T_∞(s) and T₂(s) be the transfer functions from w to z_∞ and z₂, respectively.

In general, the mixed H₂/H_∞ control design method provides a dynamic output feedback (DOF) controller, K(s), that minimizes the following tradeoff criterion:

Unfortunately, most robust control methods, such as H₂/H_∞ control design, suggest complex and high-order dynamic controllers, which are impractical for industry practices. For example, real-world AGC systems use simple PI controllers. Since a PI or proportional-integral-derivative (PID) control problem can be easily transferred to an SOF control problem,¹⁵ one way to solve the above challenge is to use a mixed H₂/H_∞ SOF control instead of a H₂/ H_∞ DOF control method. The main merit of this transformation is to use the powerful robust SOF control techniques, such as the robust mixed H₂/H_∞ SOF control, to calculate the fixed gains (PI/PID parameters), and once the SOF gain vector is obtained, the PI/PID gains are ready in hand and no additional computation is needed. In continuation, the mixed H₂/H_∞ SOF control design is briefly explained.

11.4.2 Mixed H₂/H_∞ SOF Design

The mixed H₂/H_∞ SOF control design problem can be expressed to determine an admissible SOF (pure gain vector) law K_i, belonging to a family of internally stabilizing SOF gains K_sof,

such that

The optimization problem given in Equation 11.10 defines a robust performance synthesis problem, where the H₂ norm is chosen as the performance measure. There are some proper lemmas giving the necessary and sufficient condition for the existence of a solution for the above optimization problem to meet the following performance criteria:

where γ₂ and γ_∞ are H₂ and H_∞ robust performance indices, respectively.

It is notable that the H_∞ and H₂/H_∞ SOF reformulation generally leads to bilinear matrix inequalities that are nonconvex. This kind of problem is usually solved by an iterative algorithm that may not converge to an optimal solution. An ILMI algorithm is introduced in Bevrani³² to get a suboptimal solution for the above optimization problem. The proposed algorithm searches the desired suboptimal H₂/H_∞ SOF controller K_i within a family of H₂ stabilizing controllers K_sof such that

where ॉ is a small real positive number, ${\gamma }_2^{*}$ is H₂ performance corresponding to H₂/H_∞ SOF controller K_i, and γ₂ is the optimal H₂ performance index that can be obtained from the application of the standard H₂/H_∞ DOF control.

11.4.3 AGC Synthesis Using GA-Based Robust Performance Tracking

The design of a robust SOF controller based on a H₂/H_∞ control was discussed in the previous section. Now, the application of GA for getting pure gains (SOF) is presented to achieve the same robust performances (Equation 11.11). Here, like in the H₂/H_∞ control scheme shown in Figure 11.8, the optimization objective is to minimize the effects of disturbances (w) on the controlled variables ( z_∞ and z₂ ). This objective can be summarized as

such that the resulting performance indices $({\text{γ}}_{\text{2}}^{\text{*}},{\text{γ}}_{\infty }^{\text{*}})$ satisfy $|{\text{γ}}_{\text{2}}^{\text{*}}-{\text{γ}}_{\text{2}}^{opt}|<\epsilon$ and ${\text{γ}}_{\infty }^{\text{*}}<1$. Here, ε is a small real positive number, ${\text{γ}}_{\text{2}}^{\text{*}}$ and ${\text{γ}}_{\infty }^{\text{*}}$ are H₂ performance and H_∞ performance corresponding to the obtained controller K_i from the GA optimization algorithm, and ${\text{γ}}_{\text{2}}^{opt}$ is the optimal H₂ performance index that can be achieved from the application of standard H₂/H_∞ dynamic output feedback control. In order to calculate ${\text{γ}}_{\text{2}}^{opt}$, one may simply use the hinfmix function in MATLAB based on the LMI control toolbox.³³

The proposed control technique is applied to the three-control area power system given in Section 2.4. In the proposed approach, the GA is employed as an optimization engine to produce the PI controllers in the supplementary frequency control loops with performance indices near the optimal ones. The obtained control parameters and performance indices are shown in Table 11.1. The indices are comparable to the results given by the proposed ILMI algorithm. For the problem at hand, the guaranteed optimal H₂ performance indices $({\text{γ}}_{\text{2}}^{opt})$ for areas 1, 2, and 3 are calculated as 1.070, 1.03, and 1.031, respectively.

Figure 11.9 shows the closed-loop response (frequency deviation, area control error, and control action signals) for areas 1 and 3, in the presence of simultaneous 0.1 pu step load disturbances, and a 20% decrease in the inertia constant and damping coefficient as uncertainties in all areas. The performance of the closed-loop system using GA-based H₂/H_∞ PI controllers is also compared with that of the ILMI-based H₂/H_∞ PI control design. Simulation results demonstrate that the proposed GA-based PI controllers track the load fluctuations and meet robustness for a wide range of load disturbances, as well as ILMI-based PI controllers.

11.5.1 GA for Finding Training Data in a BN-Based AGC Design

As mentioned in Chapter 8, the basic structure of a graphical model is built based on prior knowledge of the system. Here, for simplicity, assume that frequency deviation and tie-line power change are the most important AGC variables, regarding the least dependency to the model parameters and the maximum effectiveness on the system frequency. In this case, the graphical model for the BN-based AGC scheme can be considered as shown in Figure 11.10, and the posterior probability that can be found is p(ΔP_c|ΔP_tie,Δf).

After determining the most worthwhile subset of observations (ΔP_tie,Δf), in the next phase of BN construction, a directed acyclic graph that encodes assertion of conditional independence is built. It includes the problem random variables, conditional probability distribution nodes, and dependency nodes.

As mentioned in Chapter 8, in the next step of BN construction, i.e., parameter learning, the local conditional probability distributions p(x_i|pa_i) are computed from the training data. According to the graphical model (Figure 11.10), probability and conditional probability distributions for this problem are p(Δf), p(ΔP_tie), and p(ΔP_C|Δf,ΔP_tie). To calculate the above probabilities, suitable training data are needed. In the learning phase, to find the conditional probabilities related to the graphical model variables, the training data can be used in a proper software environment.³⁴

Here, GA is applied to obtain a set of training data (ΔP_tie, Δf, ΔP_C) as follows: In an offline procedure, a simulation is run with an initial ΔP_C vector provided by GA for a specific operating condition. Then, the appropriate ΔP_C is evaluated based on the calculated ACE signal. Each GA’s individual ΔP_C is a double vector (population type) with n_v variables in the range [0 1] (the number of variables can be considered the same as the number of simulation seconds). In the simulation stage, the vector’s elements should be scaled to a valid ΔP_C change for that area: [ΔP_Cmin ΔP_Cmax]. The ΔP_Cmax is the possible maximum control action change to use for an AGC cycle, and similarly, ΔP_Cmin is possible minimum change that can be applied to the governor set point.

11.5.2 Application Example

Consider the three-control area power system (used in the previous section) again. Here, the start population size is fixed at thirty individuals and run for one hundred generations. Figure 11.11 shows the results of running the proposed GA for area 1. To examine the individual’s eligibility (fitness), ΔP_C values should be scaled according to the specified range for the control area. After scaling and finding the corresponding ΔP_C, the simulation is run for a given load disturbance (a signal with one hundred instances) or the scaled ΔP_C (with 100 s). The individual fitness is proportional to the average distances of the resulting ACE signal instances from zero, after 100 s of simulation. Finally, the individuals with higher fitness are the best ones, and the resulting set of (ΔP_tie , Δf, ΔP_C) provides a row of the training data matrix.

As explained in Chapter 8, this large training data matrix is partly complete, and it can be used for parameter learning issues in the power system for other disturbance scenarios and operating conditions. Here, the Bayesian networks toolbox (BNT)³⁴ is used as suitable software for simulation purposes. After providing the training set, the training data for each area are separately supplied to the BNT. The BNT uses the data and the parameter learning phase for each control area parameter, and determines the associated prior and conditional probability distributions p(Δf), p(ΔP_tie).

After completing the learning phase, we are ready to run the AGC system online, and the proposed model uses the inference phase to find an appropriate control action signal (ΔP_C) for each control area as follows: at each simulation time step, corresponding control agents get the input parameters (ΔP_tie,Δf) of the model and digitize them for the BNT (the BNT does not work with continuous values). The BNT finds the posterior probability distribution p(ΔP_C|ΔP_tie,Δf) for each area, and then the control agent finds the maximum posterior probability distribution from the return set and provides the most probable evidence ΔP_C.

The response of the closed-loop system (for areas 2 and 3) in the presence of the disturbance scenario given in Equation 11.6 is shown in Figure 11.12. The performance of the proposed GA-based multiagent BN is compared with that of the robust PI control design presented in Bevrani and coworker.³¹