A Novel Coordinated Sliding Mode Controller Based on Lyapunov Method for SVC, Excitation and Steam Valving ()

Jiwei Du, Hu Peng, Ben Wang

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, China.

School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China.

Suining Breach of Sichuan Power Electric Corporation, Suining, China.

**DOI: **10.4236/epe.2012.46065
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School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, China.

School of Electrical Engineering, Southwest Jiaotong University, Chengdu, China.

Suining Breach of Sichuan Power Electric Corporation, Suining, China.

A novel coordinated controller is proposed in the paper for SVC, excitation and steam valving for a single machine infinite system. Firstly, the nonlinear mathematic model of the system including the itation and steam valving is exactly linearized via state feedback. Then, the quasi-linearized system after the exact lineariztion is controlled by the sliding model controller based on Lyapunov direct method. At last, the novel coordinated controller is compared with a traditional linear controller and a nonlinear optimal controller respectively by simulations. The simulation results show that the proposed controller gives better dynamic response and stronger robustness.

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J. Du, H. Peng and B. Wang, "A Novel Coordinated Sliding Mode Controller Based on Lyapunov Method for SVC, Excitation and Steam Valving," *Energy and Power Engineering*, Vol. 4 No. 6, 2012, pp. 523-528. doi: 10.4236/epe.2012.46065.

1. Introduction

[1,2] presented a novel nonlinear optimal excitation controller and a nonlinear variable structure excitation controller. [3] applied nonlinear robustness excitation controller with voltage regulation. [1,4-6] adopted SVC controller to restrain the system sub-synchronous oscillation, and damp the voltage fluctuation. [7-9] designed a cooperated controller for a SVC and excitation controller to improve system stability. The excitation controller and steam valving controller are proposed to increase the performance of the system in [10].

In this paper, a novel coordinated sliding mode control based on Lyapunov method is used to generate the control schemes for the SVC, Excitation and Steam Valving. In order to achieve superior performance for the power system, the coordination between the SVC, Excitation and Steam Valving is studied to avoid poor interaction.

2. Mathematical Model

A single generator—infinite bus power system with SVC is shown in Figure 1.

For a generator set connected to a single generator infinite bus power system with SVC, traditional transient behavior equations can be described as follows [1,6]:

(1)

where, T_{d}_{0} is the direct axis transient short circuit time constant, P_{m}_{0} the initial mechanical power, P_{H} the mechanical power generated by the HP turbine in per unit, H the inertia coefficient, D the damping constant, the transient EMF in the quadrature axis of the generator. C_{H} and C_{ML} are the power distribution coefficients of the HP and the MP/LP steam turbine subsystems. T_{HΣ} and T_{MΣ} are the equivalent time constants of the HP and MP/LP steam value control system respectively. δ is the rotor angle in radian/s and ω the radial speed of the machine. T_{C} is the time constant of SVC regulator. is the direct axis transient. x_{d} is the direct axis reactance of the generator, B_{L} is the susceptance of the inductor in SVC, B_{C} is the susceptance of the capacitor in SVC, u_{1} is the control input of the excitation system. u_{1} is the control input of SVC. u_{1} is the control input of the HP steam turbine subsystems.

P_{m} is the mechanical power. P_{e} is the active electrical power delivered by the generator, and given by:

The equations above describe a five-order three-input three-output affine nonlinear system and can be rewritten in a normal form as follows:

(2)

where

,

,

,

3. Global Linearization

The output functions have a significant relationship with state variables which will be controlled. The output functions may be state variable or the system actual output variable, but system actual output variable must be the function of state variables. In this paper, we choose output functions considering the following aspects:

1) Consider the Stability of δ, we can choose

2) Consider the impact of stream valve on active electrical power, we can choose,

3) Consider the Stability of SVC Connection point voltage, y_{3} is adopted as follows:

According to circuit principle, is given by:

where

,

is the reference voltage.

First, calculate the relative degree of the system as follows:

The matrix:

is nonsingular in whole state space, so, the relative

Figure 1. The single machine infinite system with SVC.

degree set, and.

as a new state vector, can be given by:

(3)

Thus, a Brunowsky canonical form is written as:

(4)

the model for the power system described in Equation (2) can be rewritten as:

(5)

A and B are constant matrix, and V is a vector of virtual inputs.

where

Getting easily:

(6)

where,

the output for the power system described in Equation (2) is represented as follows:：

(7)

4. Sliding Mode Controller Based on Lyapunov Method

Consider the equation of the system described in Equation (5), A is decomposed into three Jordan sub-matrix as

The system is decomposed into three sub-system for controllers design independently. At First, for the corresponding subsystem A_{11}, Sliding mode control based on Lyapunov method is used to design the anticipated dynamic characteristics.

Here, is Controllable, L^{T}Z is introduced to obtain

(8)

where,. Here, Eigenvalues are –2 and –1 ± 2j. According to pole allocation method for the controllable canonical form system, we can get

For the system described in Equation (8), we can choose

(9)

Afterward,

(10)

where, P is the solution of the Lyapunov equation as

(11)

Q in (11) is a positive definite matrix.

Switching hypersurface equation is chosen as

(12)

where

.

On the sliding mode hypersurface, we can get easily according to Equations (10) and (11), we can know as

(13)

In Equation (13), V is positive and is negative, so sliding mode movement is asymptotically stable. The control variable is out of service on the switching hypersurface.

To get better performance, the index asymptotic law control is adopted as follows:

(14)

Thus,

(15)

According to Equations (14) and (15), we can obtain the virtual input

(16)

For the subsystem A_{22} and A_{33}, we can design independently two controller v2 and v3 according to pole allocation of linear system. Switching hypersurface equations are taken as follows:

where, k_{4} and k_{5} are constants, respectively.

The index asymptotic law controls for the subsystem A_{22} and A_{33} are adopted as

where, k_{2}, k_{3}, ε_{2} and ε_{3 }are positive constants.

Then,

(17)

To eliminate the chattering phenomenon on the sliding mode surface, we substitute the sign function with a saturation function as

where, Δ is the boundary of the saturation function, and is the switching control term.

Therefore, according to Equations (16) and (17), we can formulate the final control laws of the whole system as

(18)

5. Simulation Results

A simulation case is given as follows: H = 8, D = 5, Vs = 1, x_{d} = 0.779, x`_{d} = 0.14, T_{d0} = 6.9, x_{L1} = 0.38, x_{L2} = 0.54, x_{T} = 0.01, T_{c} = 0.2, B_{L0} = 0.3, B_{c} = 1.22, V_{ref} = 1, C_{H} = 0.4, T_{H}_{∑} = 0.45, ε_{1} =ε_{2} = ε_{3} = 0.1, k_{1} = k_{2} = k_{3}_{ }= 5, k_{4} k_{5} = 1, Q = I, Δ = 0.2, 0 ≤ E΄_{q} ≤ 3.3, 0 ≤ B_{L} ≤ 1.5. The parameters of equilibrium point are chosen as P_{m0} = 0.79，δ_{0} = 60^{˚}，ω_{0} = 314.16.

Case 1: by comparing traditional linear controller (TLC) with the controller (NSVC) designed in this paper, we study the controller’s dynamic performances. It is assumed that a short circuit fault occurs at the high voltage side of the transformer at 1s and is cleared at 1.2 s. The swing curves of the dynamic system are given in Figure 2.

It can be seen from Figure 2 that the novel sliding mode controller based on Lyapunov method for coordinating SVC, excitation, and steam valving improves power angle stability, damps the system frequency oscillation and prevents active electrical power and SVC voltage from fluctuation.

Case 2: the proposed controller is compared with NOPC in references [1]. It is assumed that the short circuit fault occurs on x_{L2} at 1 s, and the faulting line is

Figure 2. Responses of system under the disturbance 1.

opened at 1.2 s to supply the power with a single line. The second line returns successfully at 1.9 s. The swing curves of the dynamic system are given in Figure 3.

Figure 3 shows that the novel coordinated controller gives better dynamic response and stronger robustness.

Conflicts of Interest

The authors declare no conflicts of interest.

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