Design of Robust UPFC Controller Using Η Control Theory in Electric Power System

Abstract: An industrial plant, such a power system, always contains parametric uncertainties. In the design of a controller the uncertainties have to be considered. Otherwise, if the real plant differs from the assumed plant model, a controller, designed based on classical controller design approaches, may not ensure the stability of the overall system. In this paper design of robust control for the UPFC controllers including power flow and DC voltage regulator, using a Η loop-shaping design via a normalized coprime factorization approach, where loop-shape refers to magnitude of the loop transfer function L = GK as function of frequency is presented. As an example, we have designed a case for the system to compare the proposed method with a conventional method (classical P-I controller). AS the results of the linear and nonlinear simulations, the validity of the proposed method has been confirmed.


INTRODUCTION
The Flexible AC Transmission Systems (FACTS) based on power electronics offer an opportunity to enhance controllability, stability, and power transfer capability of AC transmission systems [1] . The Unified Power Flow Controller (UPFC), which is the most versatile FACTS device, has the capabilities of controlling power flow in the transmission line, improving the transient stability, mitigating system oscillation and providing voltage support [2][3][4] . PID is the most commonly used control algorithm in the process industry. Also, this technique is used to control the FACTS devices [5] . However, the nonlinear nature of well as the uncertainties that exist in the system make it difficult to design an effective controller for the FACTS that guarantees fast and stable regulation under all operating conditions. A major source of difficulty is that open-loop plant may change. In particular, inaccuracy in plant may cause problems because the plant is part of the feedback loop. To deal with such a problem, instead of using a single model plant, an uncertain model should be considered. This problem has led to the study of applying adaptive controllers for instance [6,7] , nonlinear controllers for instance [8] in the power system stability control. Also, during past decade, the Η optimal robust control design has received increasing attention in power systems. Most of above methods have been applied in power systems and some of these efforts have contributed to the design of supplementary control for SVC using mixed sensitivity [9] , applying µ-synthesis for SVC in order to voltage control design [10] and supplementary control design for SVC and STATCOM [11] . For many control problems, a design procedure is required that offers more flexibility than mixed sensitivity, but should not be as complicated as µ-synthesis and should not be limited in its application like LTR procedures. The Η loop-shaping design is such a controller procedure, used to design a robust controller for FACTS control to improve the system damping [12] .
In this paper as an example, a Single Machine Infinite Bus (SMIB) power system installed with a UPFC is considered for case study and Η loop-shaping method is used to design a robust controller for UPFC controller including power-flow and DC-voltage regulator in this system. To show influence of proposed method, the proposed method is compared to conventional method (the parameters of conventional P-I controller are optimized using genetic algorithm). As the validity of the proposed method has been confirmed by linear and nonlinear time domain simulation results.  Fig. 1: SMIB power system equipped with UPFC (VSCs) and a DC link capacitors. The four input control signals to the UPFC are m E , m B , E , and B , where, m E is the excitation amplitude modulation ratio, m B is the boosting amplitude modulation ratio, E is the excitation phase angle and B is the boosting phase angle.

MATRIALS AND METHODS
Non-Linear Dynamic Model: By applying Park's transformation and neglecting the resistance and transients of the ET and BT transformers, the UPFC can be modeled as [13][14][15] : Where, K 1 , K 2 …K 9 , K pu ,K qu and K vu are linearization constants. The state-space model of power system is given by: (14) Where, the state vector x, control vector u, A and B are: The block diagram of the linearised dynamic model of the SMIB power system with UPFC is shown in Fig. 2. In the second step, the resulting shaped plant is robustly stabilized with respect to coprime factor uncertainty using Η optimization. An important advantage is that no problem-dependent uncertainty modeling, or weight selection, is required in this second step [16] .
The stabilization of a plant G is considered, that has a normalized left coprime factorization as follows [16] : That has a normalized left coprime factorization as follows: A perturbed plant model G d can then be written as: Where ∆ M and ∆ N represent the uncertainty in the nominal plant model G. The objective of robust stabilization is to stabilize a family of perturbed plants defined by: (17) Where ε>0 is then the stability margin. For the perturbed feedback system of Fig. 3, the stability The lowest achievable value of y and corresponding maximum stability margin ε are given by: Where . H denotes the Hankel norm, p denotes radius (maximum eigenvalue), and for a minimal stat space realization (A, B, C, D) of G, Z and X are the unique positive definite solution to the algebraic Riccati equations: (20) (21) Where R = I + DD T and S = I + D T D. A controller, which guarantees that For a specified y > y is given by: and . It is important to emphasizes that, since y min is computed from (19) and an explicit solution has been derived by solving just two Riccati equations and the y iteration needed to solve them, the general Η problem has been avoided [16][17] . The controller design procedure can be summarized as follows.
Loop Shaping: Using pre-and post compensators W1 and W2, the singular values of the plant are shaped to give a desired open loop shape as shown in Fig. 4. Some trial and error is involved Here W2 is usually chosen as a constant. W1 contains dynamic shaping. Integral action for low frequency performance, phaseadvance for reducing the roll-off rates at crossover, and phase-lag to increase the roll-off rates at high frequencies, should all be placed in W1 if desired. The weights should be chosen so that no unstable hidden modes are created in G s .

Robust Stabilization:
Robustly stabilize the shaped plant G s . First, calculate the maximum stability margin max <1/y min . If the margin is too small, max <0.25, return to step 1 and adjust the weight. Otherwise, select y>y min by about 10% and synthesis a suboptimal controller using (22). When max >0.25 (respectively, y min <4) the design is usually successful. A small value of max indicates that the chosen singular value loop shapes are incompatible with robust stability requirements. The loop shape does not change much following robust stabilization if y is small [16][17] .
If all the specification is not met: Return to step 1 and make further modification to the weights.
Reduce the order of controller: Check the frequency response plot of K s-red against that of K s .

Final feedback controller K:
This is achieved by combining K s-red with the shaping function W1 and W2 such that K = W1K s-red W2.

UPFC CONTROLLERS
The UPFC control system comprises following controllers: • Power flow controller • DC voltage regulator controller • Power system oscillation-damping controller ( )

Power flow and DC voltage regulator controllers:
The UPFC is installed in one of the two lines of the SMIB system. Figure 5 shows the transfer function of the power flow controller. The power flow controller regulates the power flow on this line. k pp and k pi are the proportional and integral gain of the power flow controller. The real power output of the shunt converter must be equal to the real power input of the series converter or vice versa. In order to maintain the power balance between the two converters, a DC-voltage regulator is incorporated. DC-voltage is regulated by modulating the phase angle of the shunt converter voltage. A P-I type DC-voltage regulator is considered Fig. 6. k dp and k di are the proportional and integral gain of the DC-voltage regulator.

Power system oscillation-damping controller:
A damping controller is provided to improve the damping of power system oscillations. This controller may be considered as a lead-lag compensator [18][19] or a fuzzy controller block. However an electrical torque in phase with the speed deviation is to be produced in order to improve damping of the system oscillation. The transfer function block diagram of the damping controller is shown in Fig. 7. It comprises gain block, signal-washout block and lead-lag compensator. The variable ( min ) is the inverse of the magnitude of coprime uncertainty, which can be tolerated before getting instability. min >1 should be as small as possible, and usually requires that min is less than a value of 4 [16][17] . By applying this, min = 1.3104 for power flow controller and min = 1.3236 for the DC voltage regulator are obtained. In order to show influence of Η loop-shaping method, the proposed method is compared to conventional method. In conventional method, the parameters of the power-flow controller (k pp and k pi ) are optimized using genetic algorithm [20] . Optimum values of the proportional and integral gain settings of the power-flow controller are obtained as k pp = 2 and k pi = 0.35.
The parameters of DC voltage regulator are now optimized using genetic algorithm. When the parameters of power-flow controller are set at their optimum values. The optimum gain setting of P-I type DC voltage regulator are k dp = 0.25 and k di = 0.35.
Using a commercially available software package [21] , two controllers satisfying design objectives are obtained. For easy implementation, the order has been reduced by model reduction technique. The transfer functions of the controllers are:

Design of PSS:
The damping controllers are designed to produce an electrical torque in phase with the speed deviation according to phase compensation method. The four control parameters of the UPFC (m B , m E , δ B and δ E ) can be modulated in order to produce the damping torque. In this paper m B is modulated in order to damping controller design. The speed deviation is considered as the input to the damping controllers. The structure of UPFC based damping controller is shown in Fig. 7. It consists of gain, signal washout and phase compensator blocks. The parameters of the damping controller are obtained using the phase compensation technique. The detailed Step-by-step procedure for computing the parameters of the damping controllers using phase compensation technique is given [16,17] . Damping controller m B was designed and obtained as follows (wash-out block is considered). Power flow controller damping with damping ratio of 0.5,

RESULTS AND DISCUSSION
In order to examine the robustness of the UPFC power-flow and DC voltage regulator controller in the presence of wide variation in loading condition (three cases), the system load is varied over a wide range. Dynamic responses are obtained for the following three typical loading conditions for and The performance of the designed Η -UPFC and classic-UPFC controllers with damping controller m B after sudden change in reference power on transmission line 2, reference mechanical power and reference voltage are shown in Figs. 10 to 15. It can be observed from these figures, which the proposed controller designed (Η -UPFC) significantly damp power system oscillations compared to conventional (classical P-I) UPFC controllers (C-UPFC).
In order to investigate the performance of the proposed controller and the system behavior under large disturbances and various operating conditions, a transitory 3-phase fault of 5-10 ms duration at the generator terminals is considered. Dynamic performance is obtained using the non-linear model under the system of the nominal and heavy loading condition with Η -based and optimal settings of the UPFC controllers (Power-flow controller, DC-voltage regulator and damping controller). Figure 16 shows the power system responses under the above operation condition. It can be observed from this figure, which the proposed controller designed significantly damp power system oscillations compared to conventional (classical P-I) UPFC controllers (C-UPFC).

CONCLUSION
In this paper, the design of robust controller based on Η theory with application to an UPFC has been carried out for power system. The performance of the controller has been evaluated in comparison with conventional UPFC by linear and nonlinear time domain simulations. The following issues have been addressed: • representation of non-linear characteristics of the system by uncertainty model principle. • The non-linear characteristics of the system can easily by incorporated into the controller design by suitable selection of weighting functions. • The results of these studies show that the proposed controller design using Η method compared to conventional method, has an excellent capability in damping of power system oscillations. • Performance of damping controllers under large perturbations show the superiority of proposed Η -based controller over its conventional counterpart. Also, effectiveness of the proposed control strategy in damping the local low frequency oscillations with UPFC is confirmed.

APPENDIX
The nominal parameters and operating condition of the system are given below: