2017 2nd IEEE International Conference on Computational Intelligence and Applications
Using Jaya Algorithm to Optimal Tuning of LQR Based Power System Stabilizers Adel Akbarimajd, Sajjad Asefi, Hossein Shayeghi Department of Electrical Engineering, Faculty of Technical Engineering University of Mohaghegh Ardabili, Ardabil, Iran e-mail:
[email protected],
[email protected],
[email protected]
Abstract—In this study a powerful optimization algorithm is used for optimal tuning of linear quadratic regulator (LQR) based PSS in order to stabilize a synchronous machine which is connected to an infinite bus. Jaya algorithm is a newly proposed AI optimization algorithm inspired from TLBO. The robustness of optimized controller by the Jaya algorithm has been compared regarding to commonly used types of PSS, i.e. lead-lag and PID. By analyzing the results based on the solutions given by Jaya algorithm, the excellency of this algorithm in solving such problems is obvious. ITAE criterion is applied as objective function in this investigation. Keywords-Ai optimization; jaya algorithm; LQR; PID controller; power system stabilizer
I.
INTRODUCTION
The main idea a power system follows is providing constant ranges of frequency and voltage magnitude for each consumer irrespective of the variation in load. Reliability and security are two important aspects which customers expect from a supply of electric energy while they don’t take into account the facts that power systems include extensive network, long length cables, and different types of transformers, and even power stations are so far from power consumers. Hence, implementing a controller for power system means that even after being subjected to a disturbance or occurring of a fault, the power system must remain a reliable power source. Traditionally, to enhance the damping of low frequency oscillations in a power system, the most commonly used tool is a power system stabilizer (PSS) with the excitation system. There are so many work available in the literature to design PSS for power systems, most of which are based on deMello and Concordia’s pioneering work [1, 2]. Negative effects on transferred power by local oscillations (even at low frequency) are obviously clear at transmission lines. Disturbances such as sudden load changes or faults lead to an imbalance between electrical power delivered by the generator and the mechanical power being produced by the turbine. The imbalance results in a shaft torque with an accelerating or decelerating effect on the shaft line. Hence, considering the problem of transient stability which concerns maintaining synchronism between generators in the case of a severe disturbance is necessary [3]. In order to reduce effect of instability and improve the system stability performance it is useful to introduce supplementary stabilizing signals at low frequency oscillations, to increase the damping torque of the
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synchronous machine [4]. Different approaches have been proposed in the literature to provide the damping torque required for improving the stability, the first proposed method is lead-lag power system stabilizer, second method proposed for power system stabilizer is based on PID controller and the third proposed power system stabilizer method is based on linear quadratic regulator (LQR). Heffron-Phillip’s model is used for the PSS design in this paper. The proposed PSS judges system disturbances such as changes in system configuration or variation in loads etc, based on the deviations in power flow, voltage and voltage angle at the secondary bus of the step-up transformer. The PSS tries to control the rotor angle measured with respect to the local bus rather than the angle δ measured with respect to the remote bus to damp the oscillations. Knowledge of external parameters, such as equivalent infinite bus voltage and external impedance value is required for designing of the proposed power system stabilizer [5]. We have used Jaya algorithm to optimize the parameters of the applied PSS in the system. Jaya is a simple and powerful optimization algorithm which have the ability to solve whether constrained or unconstrained optimization problems. The fundamental concept of algorithm is that the solution obtained for a problem should move towards the best solution and should avoid the worst solution. This does not need any algorithm-specific control parameters except common control parameters [6]. Furthermore we have used Lead-Lag, PID and LQR power system stabilizers for our system and after optimization the results have been compared regarding Integral of the Time-Weighted Absolute Error (ITAE) criterion. Although the lead-lag and PID approaches have been widely used and they produce satisfactory results regarding the damping of local modes of oscillation, their outcome may not be considered as the best possible. Because of the mentioned consideration, researches were directed towards optimal control theory, and in particular towards optimal linear quadratic regulator (LQR) control theory. The shift towards optimal control was motivated by the following reasons: (i) Providing a systematic way in order to design a state feedback LQR; (ii) Guarantee to stabilize the system using the resulting stabilizer and to provide a sufficient level of performance and for that reason the closed loop poles placement does not arise;
an algorithm may help us in finding a sufficient answer and reach to a good level of accuracy. As mentioned before, in this paper, a proposed robust approach based on LQR control theory is presented to overcome the above-mentioned problems. We have used a new proposed algorithm, named Jaya, to optimally choose diagonal elements of Q matrix. By applying ITAE criterion as a cost function for the algorithm, we have obtained best answers. Comparing to commonly used power system stabilizers such as PID or lead-lag, we can clearly illustrate the effectiveness of the applied method. It is shown that a straightforward application of the design method can produce a PSS that significantly improves the small disturbance stability of the power system. Finally, the power system stabilizers (PSS) are added to excitation systems to enhance the damping during low frequency oscillations. The main aim of research dealing with power system stabilizer (PSS) design for synchronous generator excitation systems is to assure damping of the power system transient processes under various operating conditions [7-10].
(iii) Robustness while being subjected to small parameter changes is due to providing sufficient stability margins via LQR; and (iv) Well developed design algorithms exist. The LQR control design method can be considered to be a foundation of the modern optimal control theory and is based on the minimization of a cost function that penalizes states’ deviations and actuators’ actions during transient periods. The main advantage of LQR control is its flexibility and usability when specifying the underlying trade-off between state regulation and control action. LQR has been proved to be a significant method which could achieve optimum performance. As a result, reports are available in the literatures which concern the designing of power system stabilizers (PSSs) by implementing of optimal LQR theory. In order to simply show the new concepts behind different problems related to power systems, papers considered single machine infinite bus (SMIB) systems. Later, if applicable, these studies will be extended to the multi-machine cases. In the papers referenced here, the way by which the state and control weighting matrices (Q and R respectively) were selected and may be summarized by the following iterative procedure: (i) The chosen structure for Q and R matrices is to be diagonal; (ii) The diagonal elements in Q and R matrices are numerical values which assigned arbitrarily; (iii) Determining an optimal state or output feedback LQR; and (iv) Investigation of the closed loop system performance. The criterion for finishing design is that the performance to be satisfactory. Otherwise, the above mentioned process from (i) to (iv) is repeated until obtaining satisfactory performance. In the above mentioned procedure there are a number of deficiencies exist. These are: (a) Not offering a principled way for choosing Q and R matrices and being a method based on trial and error. (b) The matrix Q, if found by using the iterative procedure above, is diagonal. Some of the former researchers indicated, in the discussion following their paper, that by choosing nonzero off diagonal elements in Q, there would be no advantages gained, but they do not provide any theoretical evidence to support their claim. As demonstrated in [7], this statement is simply not true; hence, in this paper by using a new proposed algorithm, named Jaya, we will optimally choose diagonal elements of Q matrix. (c) There is no analytical treatment of any kind has been given to the choice of R. (d) If we do not use trial and error, choosing the state and control weighting matrices will highly depend on the designer's experience. In summary, it can be said that in all of the PSS optimal LQR design approaches mentioned above, the major problem is the lack of a principled procedure for the design of a quadratic cost function that would allow reflecting the physical characteristics of the system under investigation. The practice has been to use one's knowledge of the system, and/or follow an iterative trial and error procedure that may produce even best possible result. It is quite easy that using
II.
MODELING OF POWER SYSTEM
In the next two subsections presented below, we have provided the model of the power system and its complete block diagram which can be used for simulations. A. Power system model For the stability analysis, a single machine connected to an infinite bus through transmission line (SMIB) has been adopted. The system includes synchronous generator which is connected to a power grid (infinite bus) via a transmission line. Exciter automatic voltage regulator (AVR) is one of the equipment used in generator [11]. Fig. 1 shows a schematic diagram for the test system. System data are given in [12].
Figure 1. Single line diagram of SMIB
B. Block diagram of Simulink We have considered that the synchronous generator is represented by model 1.1, i.e. containing field circuit and one equivalent damper winding on the q axis (4th – order model). Due to linearizing the nonlinear model for an operating condition, a linear dynamic model can be expressed. Block diagram of the linearized dynamic model of the SMIB power system with PSS is shown in Fig. 2. K1, K2,… K6, are Heffron-Phillips constants that can be easily obtained by data presented in [12]. The linearized equations can be rewritten in the state space form as follows:
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control system for a given set of performance objectives. A control system which minimizes the cost associated with generating control inputs is called an optimal control system. The control energy can be expressed as UT*R*U, where R is a positive definite square, symmetric matrix called the control cost matrix. Such an expression for control energy is called a quadratic form, because the scalar function, UT*R*U, contains quadratic functions of the elements of U. Similarly, the transient energy can also be expressed in a quadratic form as XT*Q*X, where Q is a positive semi-definite square, symmetric matrix called the state weighting matrix [16]. The term LQR actually is summary of the introduction of linear optimal adjusting with quadratic operational indicator. Quadratic-linear method (LQ) perhaps is the most important outcome of modern control in design of state feedback controllers. Many engineering problems have been solved by this method and different numerical solutions methods have been devised for problem solving. Despite the pole placement method in which we do not optimize a specific criteria and characteristic for placement of the poles, in this method a criterion or specification is defined as:
Figure 2. Block diagram of the linearized dynamic model of the SMIB
where, A and B are the system and input matrices, respectively. The overall linearized state-space model of the power system (SMIB including PSS) has been developed using the state-space equations and is provided in [12]. Note that, as D is a zero matrix, it has not been mentioned in the above equation. III.
PROPOSED STABILIZERS
Below three types of power system stabilizers which have been used in this investigation, are discussed. A. Lead-Lag power system stabilizer The transfer function of commonly used structure of the PSS that implemented in this study is given by the relationship below. It comprises a block of KPSS gain followed by a high-pass filter (so called washout filter) of time constant Tw and lead-lag structured phase compensation blocks with time constants T1 and T2. It is to be noted that the reduction of the power system oscillations after a wide perturbation in order to enhance the stability of power system is one of the reasons for suggesting stabilizers designation. The output of stabilizer is a voltage signal that adds supplementary control loops to the generator AVR, i.e. input voltage signal of the exciter system. The input signal of such a structure is usually the deviation of the synchronous speed Δω [13-15].
(5) The optimal control problem consists of solving for the feedback gain matrix, K, such that the scalar objective function, J, is minimized if all state variables can be measured. R and Q matrices are positive semi-definite matrices, and in fact quadratic form can be seen in the above integral elements (6) If the positive semi-definite matrix Q is Q ≥ 0, in this case for all x will have xTQx ≥ 0. If Q is positive definite (Q > 0) then we have for all x 0 that xTQx > 0. Meanwhile,
It is worth noting that in this study, the time constant Tw is considered as 10.0 s.
(7)
Equation (7) indicates a norm or positive number, which determines absolute error of non-zero state variables with appropriate weighting factors assigned to each element. If B. PID power system stabilizer value of this criterion be small, shows that in general, the The main application of a PID type power system range of error is very small and quickly dissipates in an stabilizer is to create a proper torque on the rotor of the exponential manner. On the other hand, due to the quadratic generator in order to compensate the phase lag between the feature of this criterion, in fact it is proportional to the energy machine electrical torque and the exciter input. The transfer error. With regard to the reduction of values of state function of the PID-PSS is given by variables, reduction of the required control efforts u is (4) overlooked, therefore, minimizing J1 will result in a large control effort. By considering C. LQR (8) The LQR design problem has been extensively investigated during past decades. Consequently, the theory of Total energy of the control signal is entered in the linear optimal control has been very well developed. Optimal performance index and this will prevent unlimited control allows us to directly formulate the performance enlargement of u. So, the performance index will be Eq. (5). objectives of control system and produces the best possible
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Energy loss over time enlarges the size of control effort energy in this optimization and thus compromises between error energy and control signal energy according to the amounts of R and Q matrices. It is to be noted that system closed-loop performance criteria such as - Internal stability - Regulation (small error) - Disturbance rejection - Noise - Insensitivity to model are often more complex than it would be gathered in a function like J. But this performance characteristic by having the appropriate general features is also very attractive mathematically, because modern optimization and problem solving methods exist for this criterion. Summary of method • To formulate the regulation or tracking problem in the form of LQR equations, we will display Δx in incremental form and we will determine performance indicator based on the incremental state variables Δx and input Δu. • We know that Riccati equation of this problem has only one answer P ≥ 0, so the following equation can be solved to achieve this response.
Figure 3. Flowchart of the proposed algorithm
(9) • Use the following equation to calculate k control gain matrix. (10) • Knowing the control gain k, the state feedback control algorithm, in addition to feedforward uer in accordance with the previously provided procedures, is constructed to apply the ultimate form of control. LQR problem solving leads to determine the response of Riccati equation. Fortunately, there are diverse and very convenient manual and numerical methods for solving Riccati equation. This equation can be solved in MATLAB using ARE, LQR commands. D. Objective function There are four kinds of performance criteria, generally considered in the control design. These are the integral of absolute error (IAE), integral of squared error (ISE), integral of time weighted squared error (ITSE) and integral of time multiplied absolute error (ITAE). For all types of investigated controllers, i.e. lead-lag, PID and LQR we just have used ITAE due to its quick settling feature. Below the equation related to this error criterion is provided.
X'j,k,I =Xj,k,i + r1,j,i (Xj,best,i -|Xj,k,i│ - r2,j,i (Xj,worst,i -|Xj,k,i)
(12)
where, Xj,best,i and Xj,worst,i shows the value of the variable j for the best and worst candidate respectively. X'j,k,i is the updated value of Xj,k,i while r1,j,i and r2,j,i are two random numbers for the jth variable during the ith iteration in the range [0, 1]. The term “r1,j,i ( (Xj,best,i - │Xj,k,i│)” indicates the willing of the solution to move closer to the best solution and the term “-r2,j,i (Xj,worst,i- │Xj,k,i│)” indicates the willing of the solution to avoid the worst solution. X'j,k,i is accepted if it gives better function value. All the accepted function values at the end of iteration are maintained and these values become the input to the next iteration. The flowchart of the proposed algorithm can be seen in Fig. 3. The main idea behind the name of Jaya algorithm is the fact that, the algorithm has the willing get closer to success (i.e. obtaining the best solution) and tries to move away from failure (i.e. avoiding the worst solution). Reaching the best solution somehow is considered as achieving victory and hence it is named as Jaya (a Sanskrit word meaning victory).
(11) where ts is total simulation time and e(t) have been selected to be rotor speed deviation (Δω) in this study. IV.
learner phase. But Jaya algorithm has only one phase and it is very easy to use, whether the problem is constrained or unconstrained. Hence, the working of the proposed algorithm is much different from that of the TLBO algorithm. Assume f(x) as the objective function to be minimized, which here we have used the error criteria ITAE. At any iteration i, ‘m’ indicates number of design variables (i.e. j=1,2,…,m) and ‘n’ indicates number of candidate solutions (i.e. population size, k=1,2,…,n). Amongst the entire candidate solutions, best candidate best obtains the best value of f(x) (i.e. f(x)best) and the worst candidate worst obtains the worst value of f(x) (i.e. f(x)worst). If Xj,k,i is the value of the jth variable for the kth candidate during the ith iteration, then the modification of its value will be done by Eq. (12).
PROPOSED ALGORITHM
Regarding to success of the TLBO algorithm, another algorithm-specific parameter-less algorithm [14] is employed in this paper. As we remember from TLBO algorithm, there are two phases, one is teacher phase and the other is learner
V. RESULTS AND DISCUSSIONS Below we are going to show the obtained results by applying optimized LQR to our system.
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Fig. 4 shows compensated and uncompensated system response due to 1% change in the mechanical torque (ΔTm). It is obvious that the LQR can successfully damp the oscillation. The way we procced to find Q matrix was using Jaya algorithm to find optimal diagonal elements of this matrix and finally examine the gained results by ITAE criterion. (13) The following values are the optimal value of each element: q1 = 10; q2 = 10; q3 = 1.3237; q4 = 0; In order to find out the robustness of the presented controller, we have compared it with commonly used controllers, i.e. PID and lead-lag, which one can see the result in Fig. 5. It is worth noting that PID and lead-lag controllers are also optimized with Jaya algorithm, too. As it is obvious from Fig. 5, the applied LQR controller acts better than the two other controllers in all aspects. It has lower overshoot, undershoot and settling time and is faster than the two other controllers.
VI. CONCLUSIONS In this paper, a newly presented optimization algorithm named Jaya was applied to determine and compare robustness of commonly used and newly presented power system stabilizers using ITAE criterion. The diagonal elements of Q matrix which obtained using algorithm are presented. To investigate the results, the deviation of rotor speed regarding one percent change in Tm with the given system data are shown in this paper. It is obvious that the algorithm works well in solving our stability problem by providing reasonable values for transient characteristics. Finally, the effectiveness of optimized LQR controller can be easily understood due to obtained results. [1]
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