Optimization of PID Controller Parameters for Automatic Generation Control in Two-Area Heating System Using Firefly Algorithm

The complexity of electric power systems increases, as the demand for electricity increases; therefore providing electrical energy is required with high reliability and stability. For a successful performance of the power system under abnormal conditions, the mismatches must be amended by the complementary control. The severe stress caused by the imbalance between generation and consumption drastically reduces the power system performance, which cannot be described in former studies related to transient and voltage stability. These slow phenomena must be considered in relation to the load-frequency control issue. Today, load-frequency control has turned into an integral part of the electric power system design and operation based on the increasing size, changing structure and interconnected power system complexity. Growing economic pressures on the efficiency and reliability of the power system led to a need to keep the frequency and the exchange of power system at planned values as much as possible. Thus, load-frequency control plays an essential role as a supplementary service in the support for the power exchange and providing better conditions for the electricity trade in a modern power system. This kind of slow phenomena should be considered in relation to the load-frequency control issue. Today, due to the increasing size, structure changes and complexity of interconnected power systems, load-frequency control has become an important issue in the design and operation of the electric power system. Increasing economic pressures for the efficiency and reliability of the power system led to a need to maintain the frequency and exchange power of system at planned values as much as possible. Thus, in a modern power system, load-frequency control plays an essential role as an auxiliary service in supporting power exchanges and providing better conditions for the electricity trade.

Today, load-frequency control has grown in importance in the power system design and operations according to the extension, restructuring, and emergence of sustainable energy sources and the complexity of interconnected power systems. Various control methods, such as optimal control, adaptive control, variable structure control and intelligent control have been proposed for load-frequency control of the power system. While these methods work well, they require system state information or proper on-line identification. Therefore, it may be difficult to apply and implement these methods in practice. In the meantime, PID or PI controllers have been considered much because of their simplicity in implementation.

Automatic generation control is a device for automatic control of generating power of units to perform economical load distribution, system frequency control and distribution of load of communication lines at desirable level. Sometimes automatic generation control called load-frequency control or load control, is performed by energy management systems in energy control centers or energy coordination centers. Modern energy management systems have telecommunication equipment to send control signals to generating units. Supplemental automatic generation control is done in the form of local control in the power plants. At thermal power plants, local control systems make the turbine speed adjustment to change the system's frequency. It also adjusts the input fluid flow to the turbine to adjust the generation when the system frequency is low or when the system frequency is high.

Automatic generation control is a crucial issue in the operation and control of the power system to provide adequate and reliable power with appropriate quality. In large-scale power systems, which usually include interconnected control area, load frequency control is important to keep the system frequency and transmit power between districts close to planned values as much as possible. The mechanical input power provided to the generators is used for controlling the output electrical power frequency and maintaining the transient power among the areas at the planned value. A well-designed power system must be capable of acting well against load changes and perturbations of the system and provide high degrees of expected power quality while keeping the frequency within an acceptable range. The problem of automatic adjustment of the frequency and crossing power of tie line in power systems dates long back to the history. Various control strategies, often associated with the use of intelligent algorithms, have recently been used in the design of load frequency controllers. Among different types of load frequency controllers, the proportional integral derivative controller has been considered much. Recently, the use of intelligent algorithms has gained a special position in optimization issues. One of these convenient and relatively new methods is the firefly algorithm, which its efficiency has been shown in various papers. In this paper, this algorithm is used to design a PID controller for automatic generation control in a double-district heating system.

In the second section, a review of the studies on the subject of paper will be discussed. The third section will briefly explain the firefly algorithm. The fourth section describes the dynamic model of AGC and the fifth section describes the control model and the objective function. Sixth section provides simulation and examination of the results of the firefly algorithm for designing two degrees of freedom controller on a two-area heating system. Finally, this paper ends presenting the conclusion in seventh section.

Firefly against its name is from insect family and a subset of beetles and sub-branch of butterfly. Fireflies produce cool light that lacks infrared and ultraviolet spectra. This light, which its wavelength is varied from 510 to 670 nanometers, can be seen in yellow, green, or light red. Scientists previously believed that these insects with their phosphorus-illuminating light attract the opposite gender for mating or hunting. But research recently conducted by a group of researchers by supervising Belgian scientist, Rafael de Cock, shows that fireflies use lighting as a defense system to combat hunters. So far, about 2000 species of these insects have been identified in temperate and tropical areas. Many live in swampy and forest areas where their larvae have access to food. Among most species, both males and females have the ability to fly. But among some species, the female is unable to fly.

The firefly optimization algorithm, or simply firefly algorithm, is caused by the behavior of natural fireflies that live together in large sets and it is one of the most efficient algorithms in solving hybrid optimization problems. The firefly algorithm is caused by the computerized behaviors of firefly. The main purpose for fireflies flash is acting as a signal system to attract other fireflies. Xin Yang presented this firefly algorithm assuming the following formula:

1. All fireflies have sexuality, so one firefly attracts all other fireflies.

2. Attraction is proportional to its brightness, and for both fireflies, one will be less bright. Attraction (and thus moving), one brighter, however, brightness can increase or decrease as their distance.

3. If there is a brighter firefly than a given firefly, it will move randomly.

Brightening must be related to the objective function. Firefly algorithm is an optimization algorithm inspired by nature. Fireflies live in nature collectively, and less brighter firefly are always moving toward brighter firefly. We consider three main assumptions in this algorithm:

-All fireflies are from the same gender.

-The attractiveness of each firefly is proportional to its brightness.

-The intensity of brightness of each firefly determines an exponential of the objective function of the problem.

For the optimization maximization problem, brightness I of a firefly in a particular area x can be as I (x) α f (x). However, the attractiveness of β is relative, as it should be seen in other eyes by other fireflies. So with the distance rij, the firefly i and the firefly j will change. In addition, the reduction in brightness is proportional to the distance from its source. Light is also absorbed in the media, so attractiveness varies with amount of attraction.

$\mathrm{I}(\mathrm{r})=\frac{I_{s}}{r^{2}}$     (1)

where, Is is the light intensity of the source. For an environment with a constant light absorption coefficient $\gamma$, the intensity of light I vary with distance r:

$\mathrm{I}=I_{0} e^{-\gamma r}$      (2)

where, $\mathrm{I}_{0}$ is the intensity of the main light. In order to avoid the singularity of $\mathrm{r}=0$ in $\frac{I_{s}}{r^{2}}$, the combination of the effect of both the inverse square and the absorption law can be approximated by the following Gosine form:

$\mathrm{I}(\mathrm{r})=I_{0} e^{-\gamma r^{2}}$     (3)

Since the attractiveness of a firefly corresponds to the intensity of light seen by adjacent fireflies, we can define the attractiveness of $\beta$ as follows:

$\beta=\beta_{0} e^{-\gamma r^{2}}$     (4)

where, $\beta_{0}$ is the attractiveness at $\mathrm{r}=0$.

In actual implementation, the attractiveness function $\beta(\mathrm{r})$ can be any uniform subtraction function such as the following general form:

$\mathrm{B}(\mathrm{r})=\beta_{0} e^{-\gamma r^{m}},(\mathrm{~m} \geq 1)$     (5)

The distance property defines $\Gamma=1 / \sqrt{\gamma}$ that the attractiveness varies significantly from $\beta_{0}$ to $\beta_{0} e^{-1}$. For constant $\gamma$, the property is extended:

$\Gamma=\gamma^{-1 / m} \rightarrow 1, \mathrm{~m} \rightarrow \infty$     (6)

Conversely, for the length scale $\Gamma$ in an optimization problem; the parameter $\gamma$ can be used as a common initial value:

$\gamma=\frac{1}{\Gamma^{m}}$     (7)

The distance between both fireflies i and j at xi and xj, respectively, is the Cartesian distance:

$r_{i j}=\left\|x_{i}-x_{j}\right\|=\sqrt{\sum_{k=1}^{d}\left(x_{i, k}-x_{j, k}\right)^{2}}$     (8)

where, K X_ik is the component of the coordinate distance xi of the firefly I, we will have in two dimensions:

$r_{i j}=\sqrt{\left(x_{i}-x_{j}\right)^{2}+\left(y_{i}-y_{j}\right)^{2}}$     (9)

Moving a firefly i to a more attractive (brighter) firefly j is defined as follows:

$x_{i}=x_{i}+\beta_{0} e^{-\gamma r_{i j}^{2}}\left(x_{j}-x_{i}\right)+\alpha \epsilon_{i}$     (10)

In which the second term is related to attractiveness, the third term is the random parameter $\alpha$ and $\epsilon_{i}$ is the random vector of the numbers represented by a Gaussian or uniform distribution [29]. For example, the simplest form is $\epsilon_{i}$ and can be replaced by rand- $1 / 2$, where rand is a uniform distribution of generating random numbers in the interval $[0,1]$. For most implementations, we can consider $\beta 0=1$ and $\alpha \in[0,1]$. In this formula, the parameter $\gamma$ indicates the types of attractions and its value is in determining the convergence rate and how the firefly algorithm behaves. In the theory of $\gamma \in[0, \infty)$ but in practice $\gamma=\mathrm{O}$ (1) obtained by the feature of length $\gamma .$ So in many applications, it usually varies from $0.1$ to 10.

The distance of r defined above is not limited to Euclidean distances, but we can define other distances depending on the type of problem in multidimensional spaces. For example, for the scheduling problem, r can be defined as delay time or time interval. For complex networks such as the Internet and social networks, the r distance can be considered as a combination of degrees of local clustering and average proximity of points. In fact, any feature that can effectively express interest rates can be considered as the r interval [30].

The ordinary scale $\Gamma$ should be accompanied with the interested scale in optimization problems. If $\Gamma$ is a ordinary scale for the optimization problem, for many fireflies $\mathrm{n}>>\mathrm{m}$ which $\mathrm{m}$ is the number of local optimizations, the initial locations of this $\mathrm{n}$, firefly should be relatively uniform distributed throughout the search space. The repetition of fireflies should converge within all local optimization points. By comparing the best solutions across all these optimization points, the general optimization point will be easily obtained. Recent methods suggest that the firefly algorithm determines the general optimization point when $\mathrm{n} \rightarrow \infty$ and $\mathrm{t} \gg 1$. Further improvement on the algorithm coverage is to change $\alpha$ parameter, which decreases gradually in the proximity points. For example we can use the following formula:

$\alpha=\alpha_{\infty}+\left(\alpha_{0}-\alpha_{\infty}\right) e^{-t}$     (11)

where, $t \in\left[0, t_{\max }\right]$ is the approximate time for simulation and $\mathrm{t}_{\max }$ is the maximum number of generations, $\alpha 0$ is the initial random parameter while $\alpha_{\infty}$ is the final value. We can also use a similar function for the feedback scheduling problem:

$\alpha=\alpha_{0} \theta^{t}$     (12)

where, $\theta \in(0,1]$ is a random reduction constant.

3.1 Implementation of the algorithm

By placing a population of n member of fireflies in the search space, the optimization problem randomly begins. Initially all fireflies have the same amount of luciferin as I. Each algorithm replicate consisted of a phase of updating luciferin and a phase of updating the location of fireflies.

Updating Luciferin: The amount of Luciferin of each firefly in each replicate is determined according to the location fitting of the firefly, i.e., some is added to the previous firefly Luciferin in each replicate, according on the amount of fit and proportion to it, to model the gradual decrease with time, an amount of current luciferin with a coefficient of less than 1 is decreased from it. So we'll have:

$l_{i}(t+1)=(1-\rho) l_{i}(t)+\gamma J\left(x_{i}(t+1)\right)$     (13)

where, $\mathrm{J}$ (xi $(\mathrm{t}+1)$ ) is the value of the objective function (location fit) of the firefly i in the replicate $\mathrm{t}$ from the algorithm, $\rho$ and $\gamma$ are also constant values.

$p_{i j}(t)=\frac{l_{j}(t)-l_{i}(t)}{\sum_{k \in N_{i}(t)} \quad l_{k}(t)-l_{i}(t)}$    (14)

For each firefly I, the probability of moving to a more radiant neighbor is defined as follows:

where, Ni (t) is the set of neighboring fireflies of the firefly i at time t.

Updating location: According to these equations, the discrete time motion of a firefly can be expressed as follows:

$x_{i}(t+1)=x_{i}(t)+s\left(\frac{x_{j}(t)-x_{i}(t)}{\left\|x_{j}(t)-x_{i}(t)\right\|}\right)$     (15)

where, xi (t) is the vector m of the location of the firefly i at time t, ||. || The Euclidean soft operator and s shows the step size of motion.

There is no replicate in the formula for the attractiveness of fireflies, and only by increasing the distance in subsequent replicates, it can stimulate the subsequent attraction. The initial attractiveness is the amount of user interest at any moment. In the Luciferin optimization formula, the previous luciferin amount is equal to the user's interest at any moment, and the location fit of the firefly i at the time t, which is the objective function of the problem, was obtained from the location updating of firefly or displacement relation of firefly. And so the amount of Luciferine updated at any moment can be calculated.

The degree of freedom of the control system is defined as the number of closed-loop conversion functions that can be independently set. Since various efficiency factors must be met in designing a control system and must be desirable in different ways, a two-degree of freedom control system has advantages over a single-degree of freedom control system [14]. The 2-DOF controller produces an output signal based on the difference between a reference signal and the measured output of the system. A weighted difference signal is calculated for each proportional multiplication, integral, and derivative operations according to predefined weight settings. The output of the controller is the sum of the output of the proportional multiplication, integral, and derivation operations on the related difference signals, where the reference signal of each operation is weighted according to the selected gain parameters [15].

The PI controller is often used in industry today. The non-derivative controller (D) is used when there is no need for rapid response, and they should be existed during the operation of noise processing and large perturbations and in the system with large transmission delays. The PID controller is used when stability and high response speed are required. Derivative mode improves system stability and enables us to increase proportional gain and reduce integral gain, which in turn increases the controller response speed.

According to the benefits stated, the use of PID controller [16, 19] and integral double-derivative controller (IDD) [6] for AGC has been reported in the technical literature. However, when the input signal has sharp edges, the derivative part generates unacceptable control signals as the plant unit input. Also, any noise in the input control signal leads to a large plant unit control input. These reasons cause to face with difficulty and complexity in practical applications. The practical solution to these problems is to put the filter-derivative in input and adjust its poles in way that no abnormality of response due to noise occurs due to its attenuating effect on high-frequency noise. To improve the efficiency when there is noise or random error in measuring process variables, a derivative filter is used. The derivative filter limits the large output shifts caused by the measured noise. The derivative filter can help to reduce the constant volatilities of the controller output, which can lead to the weariness of the control elements. Given the above, a derivative filter is used in this thesis.

2.png

Figure 2. 2-DOF PID controller structure

The structure of the parallel 2-DOF PID controller is shown in Figure 2 [15]. Where R (s) represents the reference signal, Y (s) represents the feedback from measured output of the system, and U (s) represents the output signal. KP, KI and KD are proportional, integral and derivative coefficients, respectively. PW and DW are the reference signal weights for the proportional and derivative controller and the N is derivative filter coefficient. A 2-DOF PID control system is shown in Figure 3. In this figure, C (s) is a controller with a degree of freedom, D (s) perturbations of laod, and F (s) act as a pre-filter on the reference signal.

3.png

Figure 3. 2-DOF PID controller

For a PID controller, two degrees of freedom C (s) and F (s) are given with the following equation:

$F(S)=\frac{\left(P W K_{P}+D W K_{D}\right) S^{2}+\left(P W K_{P} N+K_{1}\right) S+K_{1} N}{\left(K_{P}+K_{D} N\right) S^{2}+\left(K_{P} N+K_{1}\right) S+K_{1} N}$     (22)

$C(S)=\frac{\left(K_{P}+K_{D} N\right) S^{2}+\left(K_{P} N+K_{1}\right) S+K_{1} N}{S(S+N)}$     (23)

To design a 2-DOF PID controller based on a modern innovative optimization technique, the objective function is first defined according to the criteria and constraints. Selection of the objective function to optimize the controller is generally based on efficiency criteria that are dependent on the system response. In time domain systems, the desirable criteria are peak overshooting, rise time, setting time and steady state error. In the technical literature, the efficiency of different objective functions has been compared and reports indicate that ITAE is a better objective function in AGC studies [17, 18]. Therefore it is used as the objective function in this paper. Eq. (24) shows the expansion of the ITAE objective function.

$J=I T A E=\int_{0}^{t_{s i m}}\left(\left|\Delta F_{1}\right|+\left|\Delta F_{2}\right|+\left|\Delta P_{T i e}\right|\right)$     (24)

In Eq. (24), ΔF1 and ΔF2 are system frequency deviations; ΔP_Tie is the exponential changes of the power crossing tie lines, t_sim is simulation time range. The constraints of problem are the boundaries of controller parameter. So the design problem can be formulated as the following optimization problem.

Minimize J

Subject to

$K_{P \min } \leq K_{P} \leq K_{P \max }, K_{I \min } \leq K_{I} \leq K_{I \max }$ and $K_{D \min } \leq K_{D} \leq K_{D \max }$

$P W_{\min } \leq P W \leq P W_{\max }, D W_{\min } \leq D W \leq D W_{\max }$ and $N_{\min } \leq N \leq N_{\max }$     (25)

where, J is the objective function and K_{P min}, K_P max, _{KI min}, K_{I max}, K_{D min} and K_D max are the minimum and maximum value of parameters, PW_min, PW_max, DW_min and DW_max are the minimum and maximum value of the reference signal weights of proportional and derivative controller. N_min and N_max are the minimum and maximum values of the derivative filter coefficient.

The PID controller parameters are selected in the range of -2.0 to 2.0. The time constant of the N-derived filter is generally greater than one. PW, DW and N should be in the range of 0 to 5 and 10 to 300.