Robust Power System Stabilizer Design Based on H∞/μ

In this article, H∞ /μ controller is relied on to control the power system stabilizer(PSS) using state space approach for a single machine infinite bus (SMIB) system. Design a robust feedback controller for the system using the H∞/μ technology supported by the Matlab / Simulink. The H∞/μ design method leads to a robust controller with a fixed structure and fixed parameters. The uncertainties of the model are taken into account when specifying the weights. The controller demeanor obtained was analyzed through the input represented by the step response and the output response of the power system (PS) in the case of normal operation and then the system with changed parameters. The suggested controller proved its effectiveness by maintaining the stability of the system with acceptable limits of disturbances.


INTRODUCTION
Stabilizers for power systems have been used for many years to dampen electromechanical vibrations. It works through the generator excitation system, which generates an electrical torque corresponding to the generated speed difference [1]. However, it is easy to implement the stabilizers of the (PS), since their function mainly depends on the vibration modes, if it is in local mode or between zones [2]. Vibrations from the low frequency cause the system to become unstable and reduce its performance, so (PSS) is utilized to generate a control signal to mitigate the impact of these vibrations. It is also the (PSS) responsible for maintaining the stability of the system in the event of changing machine parameters resulting from the change in load in different operating conditions.The conventional power system stabilizer(CPSS) for electrical systems is used in the existing electrical system and has contributed to improve the dynamic stability of electrical systems [3]. Since power systems are non-linear systems and parameters of (CPSS) operate on a linear (PS) around the nominal operating point of that(CPSS) cannot provide a guaranteed performance of the (PS) in the practical work environment [4]. The vastly used (CPSS) is prepared utilize the phase compensation theory in the frequency domain and is put in as a lead-lag compensator. [5]. Many researchers have been contributed their efforts on the tuning techniques of PSS parameters. These techniques include robust control [6], optimization methods [7] and artificial intelligence approaches such as fuzzy logic [8], fuzzy sliding mode control techniques [9 ],neuro-fuzzy [10]. Genetic Algorithm (GA) [11] and Particle Swarm Optimization (PSO) [12]. In this paper, a robust controller was developed that uses H∞/μ control of the (PSS) to dampen vibrations in the (SMIB) system. The

SYSTEM DESCRIPTION
The linearized model of the studied (PS) consisted of (SMIB). This is shown in a functional diagram, as shown in Figure 1, it can be expressed state space formulation as follows [13].

DESCRIPTION AND REPRESENTATION OF THE SYSTEM UNCERTAINTY IN THE INTERCONNECTION MATRIX
Two physical parameters (TA, T'do) can be taken into account. these parameters are mostly unknown. Is being added  to the parameters (TA, T'do) to exemplify the uncertainty parameters. where  is the weight of the parameter's uncertainty and  is 1 or -1. Figure 2 shows the procedure with which the uncertainty parameter (T'do) is entered. The perturbations entered are TA, T'do,. the (1+) can be added to the (TA, T'do) parameters to display the fluctuations in the uncertainty as shown in Figure 3.

THEORY OF SYNTHESIS  /
The / control synthesis method contains three stages: first is  optimal control synthesis Al-Rafidain Engineering Journal (AREJ) Vol.25, No.2, December 2020, pp. 30-36 is calculated based on the synthesis configuration, which consists of a state-space model, second: analysis, third: a D-scale, which are nested in an iterative scheme [14].

Singular value structure and synthesis
Using linear fractional transformations (LFTs) the general structure is built for µ-analysis and synthesis as shown in a diagram in Figure 4 [15].it is shown all interconnection of inputs, outputs and a controller with disturbance and reorganized according to this diagram. For analysis, the controller K is acquired in the structure plant P to form the interconnected matrix structure shown in " Figure 4 Thus, is a measure of the smallest structure that gives rise to the instability of the feedback-loop constant matrix shown in " Fig.4-b".Given a required uncertainty level, the objective of this design is to look for a control law, which can minimize the µ level closed-loop system and secure the stability of the system for all prospective uncertainty attributive. The performance and stability conditions of a system in the presence of a structured uncertainty in relation to the µ are given by: Ideally, the value of the controller K is calculated as follows ‖ ( , )‖ µ ℎ 1 However, since there is no efficient mechanism to get this K directly, the D-scale matrix is calculated indirectly. = { ( 1 , 2 , … , 0 ) ↑ ∈ } During the minimization process, the fixation of D or K is specifically referred to as iteration D-K. It has no practical meaning and can be widespread [16].

Fig. 4 -analysis and synthesis structure
The goal of designing a controller in a model of an interconnected (PS) is to dampen the angular velocity deviation. Therefore, the angular velocity (speed) deviation (Δω) of power system is treated as controller inputs.The state-space model will be separated from the uncertainty as follows: Where the matrix ∆ is given by : The figuration of the controller design based on μ is shown in Figure 5. In this diagram, Po is the interconnection between the nominal installation and all parametric uncertainties. In order to take into account the modeling error, a parameter of uncertainty (Wc) was added as an Al-Rafidain Engineering Journal (AREJ) Vol.25, No.2, December 2020, pp. 30-36 input to the system. Whereas, (Wp) represents the system performance specifications [17]. Fig. 5 The figuration of the controller design based on μ.

Uncertain system robust performance
At this stage, some parameters such as Wc and Wp are added to the system as improved parameters, as shown in Figure 6. where, Wc and Wp are represent input and output uncertainties weight.

Design procedure / Algorithm
The H∞/µ controller design using -synthesis can be summarized in the following steps: 1. Forming the interconnection matrix. This step includes linearization of the non-linear model. The 2. -synthesis. When the interconnection matrix has been defined, an  controller is designed. This involves the solution of two Riccati equations iterated over a scalar parameter in a one dimensional search. The result is a controller K. When the plant matrix P is closed-loop with the controller K results in a closed-loop system matrix M. 3. -analysis. In this step -analysis of the closed loop system matrix M is carried out. the structured singular values of M is calculated [18]. 4. Rational approximation of Ds-scaling. In this step, the Ds-scaling calculated in the -analysis (step3) is approximated by rational transfer functions. 5. Ds-K iteration. The interconnection matrix P is improved with the coherent transport utilities. synthesis, -analysis and Ds-scaling approximations are repeated until no longer changes occur in it. 6. Changing weights. If Ds and K have converged but the requirements are not fulfilled, then the weights must be changed. The design objectives must be stopped and a new Ds-K iteration must be done.-analysis can be used to know which design objectives are driving the problem.
The algorithm steps are given in the flowchart shown in Figure (7).

SIMULATION AND RESULTS
Using the MATLAB / SIMULINK program, the proposed / controller performance has been tested by performing multiple simulation tests and comparing them to a CPSS system to check if the proposed controller is better and more robust than the CPSS or not. For comparison, simulation tests of the reaction to the speed deviation depending on the nominal condition and the variation of the uncertainty parameters of the (PS) were carried out. Figure 8 shows the control of the speed deviation responses of the CPSS power system and the / controller. Based on the simulation results, the / controller offers the best performance compared to the CPSS.
In order to be a high performance / controller, it must be strong and robust during the change in the parameters (TA, T'do) examined by simulations by changing one parameter at a time, while the other parameters remain unchanged. Figures 9 and 10 show the responses of (PS) based on / controller when TA and T'do are incremented by 30% and 50% of the main value. Figures 11 and 12 show the responses of (PS) based on / controller when TA and T'do are changed from the main value by -30% and -50%.   (Figure 8). It is obvious that changing the uncertainty of the TA parameter has little effect on the behavior of the model. Figure 9 and Figure 10. While the parameter T'do ( Figure 12) changes the behavior of the model compared to other parameters. Table 1:The effect of changing one of the parameters on the performance of the system, with the other parameters remaining constant for each time.

CONCLUSION
In this work, the designed H∞/μ controller deals with a (PS) model with the presence of uncertain parameters using Matlab Simulink. This controller is applied to control the speed deviation. The performance of both the designed H∞/μ controller and CPSS was tested through simulation and it was found that the designed H∞/μ controller maintains the stability of the power system in a manner Durable when changing uncertain parameters. It was concluded that the dominant had a high dynamic behavior of the power system with a rapid (ts) and very small overshoot compared to CPSS. The designed H∞/μ controller can realize robustness and good performance.