Modeling and control of power and energy produced by a synchronous generator Using Polynomial Fuzzy Systems and Sum-of-Squares Approach

Parisa Ansari Bonab1, Seyyed Mohammad Hosseini Rostami2, Ahmad Jafari3, Babak Sheikhi4, Jin Wang5, Xiaofeng Yu6 1 Electrical and Robotic Engineering Department, Shahrood University of Technology, Shahrood, Iran, Email: parisaansari1@gmail.com 2 Electrical Engineering Department, Shahid Beheshti University, Tehran, Iran, Email: s_hosseinirostami@sbu.ac.ir 3 Electrical Engineering Department, Mazandaran Electric Power Distribution Company Sari, Mazandaran Email:Ahmadjafari.eng@gmail.com 4 Electrical Engineering Department, Science and Research Branch, Islamic Azad University, Tehran, Iran, Email: babak.sheikhy1367@yahoo.com 5 School of Computer and Communication Engineering, Changsha 410004, China, Email: jinwang@csust.edu.cn 6 School of Business, Nanjing University, Nanjing 210093, China, Email: xiaofengyu@nju.edu.cn

sets [1]. The fuzzy set theory has been developed by Lotfizadeh to control plants. The fuzzy logic controller (FLC) is known as the most efficacious solution for a number of control issues. FLCs efficiency relates to the fact that they are less sensitive to parametric variations; therefore, they are more robust than the conventional classical controllers (such as PI, PD, and PID) in controlling system output [2]. FLC has been used as an effective control process because of several remarkable reasons such as quick decision-making capability, usability in nonlinear systems, and intuitive definition of controller behavior [2][3][4][5][6].
In the last two decades, the Takagi-Sugeno (T-S) fuzzy model based control methodology has received much attention as a powerful tool to deal with complex nonlinear control systems. The main contribution of T-S fuzzy model in representing nonlinear systems is that nonlinear systems are shown as a combination of local linear subsystems weighted by membership functions [7]. In addition, this fuzzy modeling method offers another excellent approach for describing higher order nonlinear systems, and then reduces the number of rules in their modeling [8].
The T-S fuzzy model can represent any smooth nonlinear systems by fuzzily blending linear sub-systems; moreover, their stabilization conditions based on Lyapunov stability theory can be represented in terms of linear matrix inequalities (LMIs) [7,8,9]. By the same token, designs have been carried out using LMI optimization techniques. In the T-S fuzzy model based control, for designing a fuzzy controller for the system the parallel distributed compensation (PDC) concept based on a common quadratic Lyapunov function has the main contribution [7], [9].
It is worth mentioning that, nowadays, numerous researches [10][11][12][13][14] try to utilize the T-S fuzzy model as the stabilization conditions of nonlinear systems because of the above mentioned reasons.
Recently, in [15] a more general version of T-S fuzzy model has been introduced, it is named the polynomial fuzzy model. The main distinction between the T-S fuzzy model and the polynomial fuzzy model is that the former method only deals with constants in the system matrices; however, the latter one provides a perfect opportunity to deal with the polynomials in the system matrices.
This great advantage of polynomial fuzzy model results in remarkable applications in nonlinear systems. Therefore, representation of the nonlinear systems with a number of polynomial terms can be controlled more efficiently [15,16]. There is a problem in handling the polynomial fuzzy model.
To put it in other words, it is clear that T-S fuzzy model utilizes LMI optimization techniques, a numerical solution is obtained by convex optimization methods such as the interior point method.
However, it cannot be used to solve stability analysis and control design problems directly in the polynomial fuzzy model [15,16]. Despite the great success and popularity of LMI-based approaches, still there exists a large number of design problems that either cannot be represented in terms of LMIs, or the results obtained through LMIs are too conservative and the polynomial fuzzy model is one of that problems. Hence, the paper [15] introduced a sum-of-squares (SOS) optimization technique to perform stability analysis and control design for the polynomial fuzzy model. The problems represented in terms of SOS can be numerically solved by free third-party MATLAB toolboxes such as SOSTOOLS [17] and SOSOPT [18].
In this paper SOS approach for modeling and control of the synchronous generator using polynomial fuzzy systems is presented. The proposed SOS-based approach was selected for modeling and control of this vital system since this method has proved its high efficiency and obvious superiority over T-S fuzzy model. One of the advantages as discussed above is that the polynomial fuzzy model framework is a general version of T-S fuzzy model, hence is more effective in representing nonlinear control systems. The second one is that, one polynomial Lyapunov function that contains quadratic Lyapunov function was employed to stabilize the fuzzy polynomial system and its stability conditions. Hence, the obtained stability conditions from proposed SOS-based approach are more general than those based on the existing LMI-based approaches to T-S fuzzy model and control. The derived stability conditions were represented in terms of SOS can be numerically solved via the recently developed SOSTOOLS [19]. These SOS conditions cannot be generally solved via convex optimization methods. SOSTOOLS [19] is a free, third-party MATLAB toolbox that solves SOS problems. The techniques behind it are based on the SOS decomposition for multivariate polynomials, which can be efficiently computed using semidefinite programming. SOSTOOLS is developed as a consequence of the recent interest [15].
Synchronous generators are the most important parts and electrical energy suppliers of all power systems. They usually operate together (or in parallel), forming a large power system supplying electrical energy to the loads or consumers. Synchronous generators are built in large units, their rating ranging from tens to hundreds of megawatts. They convert mechanical power to ac electric power. The source of mechanical power, the prime mover, may be a diesel engine, a steam turbine, a water turbine, or any similar device.
One stable model of synchronous generator improves the performance and the stability of nonlinear power systems, and provides several benefits such as saving time, energy, and money. Therefore, a helpful and powerful control strategy such as SOS-based polynomial fuzzy control strategy for modeling and control of synchronous generator will be a cost effective, time and energy saving strategy to improve the performance and the stability of nonlinear power systems, as well as enhances the dynamic response of the operating system. The rest of the paper is organized as follows: In section 2 dynamic model of the synchronous generator is presented. Next, a general form of the fuzzy logic controller is introduced. In the section 4 the polynomial fuzzy model and the polynomial Lyapunov function are described, precisely.
Then, the stability analysis via SOS are explained. In section 6, designing the polynomial fuzzy controller is shown. Finally, in section7, the synchronous generator behavior in presence of the introduced polynomial fuzzy controller and without it is analyzed carefully. In the last section, conclusion explains the whole paper briefly.

Dynamic Model of the Synchronous Generator
The detailed nonlinear model of a synchronous generator is a sixth-order model. However, the third-order model is of crucial interest for studying control systems of the generator as well as their synthesis [20]. Therefore, the detailed nonlinear model is usually reduced to a generalized one-axis nonlinear third-order model. Generator structure diagram and the simplified model of the synchronous generator are shown in figure1. and figure2. respectively. In Figure 1 Variables and parameters in the equations of the third-order model of the generator are introduced in the below  The state variables of the generator are defined as follow: Hence, state variables vector for the generator will be : The control input () ut also considered as follows: The nonlinear equations of the system, define the following constants for the generator: The equations (1) and (2) can be rewritten by using (8) as follows: In these equations the rotor angle is the first state variable, the second one is the rotor speed deviation, and the last state variable describes the voltage. Considering the above described equations, it is clear that the systems is complex and difficult to control because of nonlinear terms. Therefore, employing a powerful approach to control it is a significant issue.

Fuzzy Logic Control
In modeling and control of systems with no accurate mathematical model, fuzzy Logic Controls (FLCs) can be of great help. The fuzzy-model-based control methodology provides a natural, simple, and effective design approach to complement other nonlinear control techniques that require special and rather involved knowledge [21,22]. The structure of a fuzzy controller is shown in Figure3. It consists of fuzzification inference engine and defuzzification blocks:

Figure 3. Basic configuration of fuzzy systems with fuzzifier and defuzzifier
Example 1: Consider the following nonlinear system: ; 31 If the nonlinear items 2 21 xx and 2 21 ( 3) xx + in equation (11) are replaced by 1 z and 2 z respectively, the following equation is obtained: , 0 Therefore: The membership functions are given as follows: With defuzzification process: This model exactly shows nonlinear system in area

Polynomial fuzzy model
In To solve the above-mentioned problems and prepare further beneficial results, a simpler and more general method is introduced as the polynomial fuzzy model [15]. For proving the stability of this model a polynomial Lyapunov function should be defined. The stability conditions for polynomial fuzzy systems based on polynomial Lyapunov functions could be reduced to SOS problems, which avoids the difficulty of the LMIs and could be solved readily. Therefore, instead of the LMI toolbox, these problems can be solved via SOSTOOLS. In the following, the polynomial fuzzy model of a general nonlinear system is described. Suppose a nonlinear system as follows: ( ) ( ( ), ( )) x t f x t u t = (22) As discussed above, a so called polynomial fuzzy model is introduced to represent the nonlinear system (22).
. Then (24) reduces to (25): Where, (25) shows the T-S fuzzy model of above nonlinear system. Therefore, (24) or polynomial fuzzy model is a more general representation compared to T-S fuzzy model (25). Analyzing the stability of mentioned polynomial system could be simple in using a polynomial Lyapunov function, in this case the stability results and conditions could be relaxed. The proposed polynomial Lyapunov function is defined as below:

Polynomial Lyapunov function
and ( ( )) P x t is a constant matrix, then (26) reduces to the quadratic Lyapunov function ) () ( T xx t Pt.Therefore, it is clear that (26) is a more general representation.

5.1.Sum of squares
One of the most important objectives of this paper is utilizing the SOS method as the computational method to provides significantly more relaxed stability results than the existing LMI approaches to T-S fuzzy models and avoid the difficulty of solving the LMI. A multivariate

5.2.Stability conditions
In this section, the stability of system (24) is analyzed. The zero equilibrium of the system (24) with 0 u = is stable if there exists a symmetric polynomial matrix Where () Txis a polynomial matrix whose ( , ) i j th entry is given by If () Pxis a constant matrix, then the stability holds globally.
Proof. See [15]. Lyapunov function. Thus, the proposed SOS approach to polynomial fuzzy models contains the existing LMI approaches to T-S fuzzy models as a special case. Therefore, the SOS-based polynomial fuzzy models provide significantly more relaxed stability results than the existing LMI approaches to T-S fuzzy models.

Designing Polynomial Fuzzy Controller
A fuzzy controller with polynomial rule consequence can be constructed from the given polynomial fuzzy model (23).
The ith rule of polynomial fuzzy controller is as follows: Control rule i : Where mn i FR   is the polynomial feedback gain in rule j . Thus, the following polynomial fuzzy controller is applied to the nonlinear plant represented by the polynomial fuzzy model: Therefore, from (24) and (31) the closed-loop system can be represented as:  In the next step the stability of the closed-loop control system (32) should be considered.

6.1.SOS design conditions
Where  is independent of x . () Tx is a polynomial matrix and , ij are entries that are given by: Proof. See [15].

Nonlinear synchronous generator system
The model of simple transmission system containing power plant is shown in figure 6. The  cos sin The nominal values of system parameters are shown in Table 2:   As shown in Figures 7 and 8, the angle and the speed deviation of the rotor goes to infinity with respect to time, which exhibit unstable behavior. Also in Figure 9, the variable 3 x goes to zero with an uncontrolled initial value. Hence, we should improve and control it.
Since phase plot is a useful graphical tool to understand the stable or unstable behavior of equilibrium points of nonlinear systems, figures 10, 11, 12 are prepared to show the behavior of a nonlinear system with 0 u = . As shown, 1 x and 2 x exhibit unstable behavior, therefore, nonlinear system is unstable . To meet the objective of control and stabilize the system, modeling and designing a polynomial fuzzy controller is necessary.
In this step one more variable is defined as According to derivative of the chain rule: By replacing sin 2 (2sin )(cos ) x x x = the above equation could be written as follow: It is clear that: Membership functions are given as follow: Therefore, the fuzzy model of the system is obtained as follows: ( ) ( ) The fuzzy controller is obtained from the equation (31): Finally, the closed-loop system could be obtained from (32) :   x and 2 x plane approaches to the origin. In this case, the equilibrium point is called the stable node. Consequently, the paths are said to be stable (as t increases, the paths lead to the origin).

Design example
In this step, to show the results more accurately, it is assumed that the system's state equations are as follows: State equations of the system include nonlinear terms. By plotting the system time response and, and as shown in figures [19][20][21][22][23][24][25]1 x and 2 x show an unstable behavior. and 25 show the values of variables and go to infinity. Therefore, the nonlinear system is unstable. In this system using SOSTOOLS, the following are obtained:  Consequently, the paths are said to be stable (as t increases, the paths lead to the origin).

8.Conclusion
This paper discussed the synchronous generators as a highly complicated system and its importance in power systems and their stability. Then, it presented SOS approach to modelling and control of the synchronous generator in terms of polynomial fuzzy systems as an efficacious method.
First of all, the state equations of the synchronous generator were described. Secondly, a polynomial fuzzy modeling and control framework that is more general and effective than the T-S fuzzy model