Application of a New Multivariable Sliding Mode Controller for the

2011 2nd Power Electronics, Drive Systems and Technologies Conference

Application of a New Multivariable Sliding Mode Controller for the Single Machine Infinite Bus Systems H. Khomami Pamsari

M. Alizadeh Bidgoli, M. Rajabzadeh, S.M.T. Bathaee

S. Ozgoli

Islamic Azad University of Lahijan Branch Islamic Azad University Young Research Clup Lahijan, Iran [email protected]

K. N. Toosi University of Technology Tehran, Iran [email protected], [email protected], [email protected]

Tarbiat Modares University of Technology Tehran, Iran [email protected]

In the event that the post-fault system experiences “small” oscillations, simplified linear models can be used with advantage in analysis and linear controller design. Linear power system stabilizers (PSS) [1] are often used to provide supplementary damping through excitation control to improve the dynamic stability limit. Such linear control generally provides asymptotic stability in a small region of the equilibrium and is only appropriate for the effect of small disturbances. In [2], an adaptive control method was used on transient stabilization of multimachine power systems. The authors used an approximate linearization model as a design model and the controller was a linear controller, which was designed for a specific operating point. Therefore, the usual stabilizer design based on approximate linearization models is not adequate for large disturbances. When a large fault occurs, the behavior of the system changes significantly and in many cases a linear controller cannot maintain adequate stability. In order to improve the transient stability of a power system with large disturbances, and voltage regulation the application of nonlinear controller will be necessary.

Abstract: This paper presents two multivariable nonlinear stabilizers, designed for a single machine infinite bus (SMIB) modeled by a standard ninth-order model. Multivariable feedback linearization (MFBL) and multivariable sliding mode control (MSMC) are proposed to regulate the output voltage and track the reference rotor angle at post-fault conditions. It is the first time that The MSMC method is designed and simulated to ensure both the stability of power system and voltage regulation despite of model uncertainties. An appropriate sliding surface has been found to achieve the desired aims. The proposed sliding mode controller has been simulated on the SMIB in the presence of a large disturbance, namely, a symmetrical three-phase short circuit fault at the terminal of the machine, and compared to the performance of the MFBL controller. Simulation results show that the MSMC technique has better performance to improve transient stability and voltage regulation in comparison with the MFBL controller. Keywords: Multivariable Feedback Linearization Control (MFBLC), Multivariable Sliding Mode Control (MSMC), Single Machine Infinite Bus (SMIB).

I.INTRODUCTION

The application of nonlinear control techniques to solve the transient stabilization problem has been given much attention, Most of these controllers are based on feedback linearization technique [6]. It was shown in the literatures that the dynamics of the power system could be exactly linearized by employing nonlinear state feedback. The essence of this technique is to first transform a nonlinear system into a linear on by a nonlinear feedback, and then uses the well known linear design techniques to complete the controller design.

The electric power system is a complex interconnected system whose structure may be affected by load changes and disturbances. It comprises generators connected to load centers via high voltage transmission circuits. Following a disturbance these generators start oscillating. Supplementary signals in the exciter voltage control loop can effectively damp these oscillations and improve the power system stability [7]. The system is nonlinear and its operating conditions vary with loading changes and disturbances. Hence, to accommodate for a wide range of operating conditions, a nonlinear control design approach is more suitable, and because of the size and widespread of the system over large geographic areas, decentralized control schemes are more appropriate.

In [3, 4] the feedback linearization and backstepping strategy were used for third order model of power system. In the literatures usually application of adaptive nonlinear controller are used in power system with uncertainty. Xiangyang and et all [5], focused on the power systems with unknown disturbance and unknown parameters. Their aim is to design a nonlinear excitation controller ensuring asymptotical stability and the disturbance attenuation performance. So the paper integrated the backstepping design approach and sliding mode control method to form new control law. In [6] the model of the synchronous machine used is a 7-order model and the feedback system is globally asymptotically stable in the sense

To maintain a high degree of reliability, the effect of major disturbances, such as a 3-phase fault, must be considered in power system design. Designing controllers for power systems to prevent an electric power system from losing synchronism after a large sudden disturbance and post-fault voltage regulation are very important [2].

978-1-61284-421-3/11/$26.00 ©2011 IEEE

211

of the Lyapunov stability theory. The combination of nonlinear controllers has been designed through a backstepping technique with the tuning and adaptation of their gains using reinforcement learning to power system stability enhancement in [8]. In these literatures, equations of power system of SMIB with the single input (excitation input) have been used. Whereas the 3-order model of system does not comprise the internal dynamics of power system, therefore [9, 10] has used 9-order model of power system and in order to design a controller, the turbine and excitation as inputs are applied.

symbols

In this paper, a detailed 9-order model of a system which consists of a hydraulic turbine and a single machine infinite bus system is considered. The model of the synchronous machine is a 7th order model (5 for the electrical dynamics and 2 for the mechanical dynamics) which takes into account the stator dynamics. Therefore two multivariable nonlinear controllers which use the excitation and the turbine's servomotor input to control (simultaneously) the power angle and the terminal voltage at the generating plant is proposed. At first

, , , ,

L 0 L L 0

0 L 0 0 L

ı ı ı ı ı

ωL 0 0 0 R

i i i i i

V V V 0 0

Fig.1. single machine infinite bus power system modeling

The following equations can be obtained from the figure 1. V V (2) V The d q terminal voltages and are constrained by the load equation. In the Park transformed coordinates, we can write V V V

Now consider a single machine infinite bus power system as shown in Fig. 1. In this model a hydraulic turbine is connected to a synchronous generator and the generator is connected to the infinite bus through a transmission line with impedance . The model of the synchronous generator on which the controller is tested uses the currents as state variables and is based on a classical representation of a machine with three stator windings, one field winding and two damper windings. This model takes into account both field effects and damper- winding effects introduced by the different rotor circuits. As a result, we obtain seven nonlinear differential equations to which the classical Park’s transformation is applied [11]. L 0 L L 0

0 ωL 0 R 0

Direct and quadrature axis current in per unit Direct and quadrature axis voltage in per unit Excitation field current Excitation field voltage Direct-axis and quadrature-axis damper Winding currents Stator resistance Field resistance damper windings resistances Direct and quadrature self-inductances Rotor self-inductance Direct and quadrature damper windings self inductances Direct and quadrature magnetizing inductances

,

II.PROBLEM FORMULATION

0 L 0 0 L

0 ωL R 0 0

Description

, ,

Multivariable feedback linearization (MFBL) is proposed to regulate the output voltage and track the reference rotor angle at post-fault conditions[9,10] then multivariable sliding mode control (MSMC) is designed and simulated to ensure both the stability of power system and voltage regulation despite of model uncertainties as a new approach. An appropriate sliding surface has been found to achieve the desired aims. The proposed sliding mode controller has been simulated on the SMIB in the presence of a large disturbance, namely, a symmetrical three-phase short circuit fault at the terminal of the machine, and compared to the performance of the MFBL controller. Simulation results show that the MSMC technique has better performance to improve transient stability and voltage regulation in comparison with the MFBL controller.

L 0 L L 0

ωL R 0 0 0

R ωL 0 0 0

R

i i i

L

d i i dt i

d i i i L ωL i i dt i sin( δ a) V cos( δ a) Where voltage of infinite is bus and

V V

R

(3)

is angle of it.

Substituting above equations in Eq.1, the state equation will be obtained for these five generator electrical state variables i , i , i , i , i . The mechanical dynamics of the rotor are also given by the swing equations:

(1)

ω

1 (T 2H

δ

ω

T

(4) (5)

1

The park transform of

212

Dω)

is obtained as follows

T L

L i i

L i i

symbols

L

i i

L

i i

(6)

T g g 0 g 0 0 0 0 0 0 0 0 0 0 0 0 0 g Where coefficients A , i 1,9, j 1,6 of state equations study follow from [13].

F(x)

Description Rotor angle in radians Rotor angular speed in per unit Electromagnetic torque Mechanical torque Inertia coefficient in seconds Damping constant in per unit

Since there are two controls, we are able to influence independently two outputs and satisfy therefore the two objectives before mentioned, i.e. rotor angle stability enhancement and voltage regulation improvement. In order to reach these goals, the first output to be chosen is the terminal voltage V as

There are two mechanical state variables ω , δ , therefore the generator totally has seven state variables. Now the turbine model should be considered.

y

The model of hydraulic turbine considered in this study follows from [12]. Suppose that the input gate of turbine is controlled by a servomotor. The state variables of hydraulic turbine are as follows .

.

.

(7)

. symbols

(8)

V (x)

V (x)

V (x)

The expressions of V , V as a function of the state variables can be obtained by combining Eq. (1) and (3). V (x) c x c x x c

c x c x x cos( x a)

V (x) c x x c x x c x c sin( x a)

Description flow in the conduit gate opening

(10)

c x

c x

c x x

(11)

(12)

The second output is rotor angle stability (δ) which is the 7th state variable.

Finally these two state variables in additional with the state variables of generator form state variables of nine-order power system as follows

y

δ

x

III.MULTIVARIABLE SLIDING MODE CONTROLLER x ,x ,x ,x ,x ,x ,x ,x ,x

x

i , i , i , i , i , ω, δ, q, G

SMC is commonly favored as a powerful robust control method for its independence from parametric uncertainties and external disturbances under matching conditions. In general, SMC comprises a discontinuous control input that drives the control system toward a specified sliding surface.

T

T

We can formulate the complete model of the turbine– generator system in the nonlinear state space form dx dt

F(x)

In this section, we use the same equations of the power system to design a multivariable sliding mode controller to as a power system stabilizer. Whereas, there are two-input twooutput, thus each output control signal should be applied to one of the inputs. In this strategy the aim is to find a proper surface such a way that, Lyapunov theory satisfied. The sliding mode controller block diagram is shown in fig.3. It is obvious that we need two sliding surface for each output, which is designed as follows.

G(x)u

Where u u , u T is the vector of control inputs and F(X) and G(X) are given by

G(x)

A x A A x A A x x A A x A A x x A A x x

A x x x A A x x x A A x x x A A x x x A A x x x A A x x

A x x cos( x A x x cos( x A x sin( x A x cos( x A x sin( x A x x A x A A x wR (x 1) A x x A x A x

A x a) A x a) A x x a) A x x a) A x x a) A x x x x

(9) Fig. 2. Nonlinear control block diagram based on sliding mode strategy

We consider a single input system of the form x

f(x)

g(x)u

(32)

The two steps necessary to synthesize a sliding mode controller:

213

( ) ( )

- define commutation surface s(x), that order is smaller than the system order and which represent the desired dynamics. s

(D

(40)

λ) y (33) y Where y, y and n are respectively output, desire output and relative degree of system for this output.

The first input control law satisfies the precedent condition can be considered in the following form:

- calculate the control u(x, t) with the objective that every state which is outside the commutation surface must join in a finite time. The system takes the dynamics of this surface and the system evolution joins the equilibrium point [16].

(41) ( ) Where u is the control vector, u is the equivalent control.

( )

L s(x) c grad s(x), f (34) L s(x) grad s(x), g c k sign (s) Where n is relative degree of output and L is lie derivation [15, 18].

u

c

A. sliding surface

and input control

y y

The surface is chosen as function of the tracking error, so the sliding surface s (t) can be defined as: s

V

V

s

V

V

y

V (x)

s

(35) V V (x)

(36) V (x)

( )

1 1

y

λ y

y y δ 2δ

2λy (42)

(δ

δ

)

δ

2δ

δ 2δ w . F (x)

(δ

)

δ

δ δ 2. w . (x

G

G

G 1)

(x

( ).

(43)

δ

(44)

)

(45)

( )

( ).

( )

.

2.

.

( ) (46)

Where

V (x)

( )

y

( ). .( 1)

(37)

∂V x ∂x ∂V 1 ∂V (F(x) G(x)u ) V V ∂x ∂x V dV c c c x c c x c x dx c x c sin( x a) dV c x c x c c x c c x dx c x c x c cos( x a)

λ) y y y y y y s

( ).

Therefore from (11), we obtain surface derivative equation as follows: y

0.

Therefore, the sliding surface s (t) can be defined as:

s

0

(D

s y

To make the state variable to the sliding surface, the following condition should be satisfied [15]: 0,

( )

B. sliding surface and input control Design Since the second output of system is third order, so the sliding surface s (t) can be defined as:

Design

This paper has two objectives, rotor angle stability and voltage regulation, so In order to reach these objectives by sliding mode control. Firstly, we must choose two proper sliding surfaces; afterwards we find inputs as ensure the system trajectory will take finite time to reach the surface.

is switching gain and suppose

Where

( ) ( ) (38)

( ) (39) ( )

214

(

) (47)

1 0.975 11.5 3

( )

. ( )

78.72 0.01 0.1

10

1.5

Vt-FBL Vt-SMC

0 Vt(pu)

1

0.5

To make the state variable to the sliding surface, the following condition should be satisfied [15]: 0,

0 0

0

The second input control law satisfies the precedent condition can be considered in the following form ( ) Where u derive from s

( ).

( )

2.

( ) ( )

.(

1)

2.5

R.A-FBL R.A-SMC

16

. ) (48)

( ) Where ( )

2

Figure 4: Terminal Voltage

0 as follow

.

1 1.5 Time(s)

18

Rotor Angle(deg)

(

0.5

14 12 10 8 6 0

switching gain is positive constant and suppose

0.5

1

1.5 Time(s)

2

2.5

Figure 5: Rotor Angle δ

IV.SIMULATION RESULTS 1.003

The nonlinear strategies are applied to a generator and simulated on the complete 9th order model of turbine generator system in a single machine infinite bus configuration as shown in Fig. 4 using MATLAB (power system blockset) [17]. The stability of the system is studied by simulating a 3φ shortcircuit at the secondary of the generator transformer at t=0.4s and this fault cleared after t=0.6s.

Speed-SMC Speed-FBL

Speed(pu)

1.002 1.001 1 0.999 0.998 0.997 0

0.5

1

1.5 Time(s)

Figure 6: Rotor Speed Figure 3: Single Machine Infinite Bus Configuration

The system parameters are in below and all of values are given in P.U. parameters

value 4.64 10 1.116 0.956 9.17 10 6.35 10 3.19 0.06

parameters

value 6.84 10 0.416 0.232 2.17 10 1.083 0 0.261

215

2

2.5

0.5

[4]

[5]

S1

0

[6]

-0.5

-1 0

[7]

0.5

1 1.5 Time(s)

2

2.5

Figure 7: Sliding Surface

[8]

200

[9]

S2

100 0

[10]

-100 [11]

-200 0

0.5

1 1.5 Time(s)

2

[12]

2.5

Figure 8: Sliding Surface

[13]

V.CONCLUSION

[14]

In this paper the MSM method is used to design a controller to improve power system transient stability and regulate voltage after during a large fault. Nonlinear excitation control is designed explicitly for excitation and turbine-governor model. The considered machine in this paper comprises a hydraulic turbine and a synchronous generator. The model used for the control is a nonlinear ninth-order model. The proposed nonlinear controller is based on a robust nonlinear method. The new controller has been tested on a single machine infinite bus power system and compared to the performance of a multivariable feedback linearization controller after 200 ms 3φ fault occurred close to the generator terminals. The simulation results show that the MSM controller is able to improve both the power system damping and the post-fault regulation of the generator terminal voltage better than MFBL controller. Since this paper is not considered uncertainties of power system parameters in designing of controller, it can improve performance of MSM controller when uncertainties of parameters are considered.

[15] [16]

[17] [18] [19]

VI.REFERENCES G. Eason, B. Noble, and I. N. Sneddon, “On [1] [2]

[3]

E. V. Larsen and D. A. Swann, “Applying power system stabilizers,” IEEE Trans. Power Appr. syst., vol. 100, pp. 3017–3046, 1981. J. Y. Fan, T. H. Ortmeyer, and R. Mukundan, “Power system stability improvementwith multivariable self-tuning control,” IEEE Transactions on Power Systems, vol. 5, pp. 227–234, 1990. G. Damm, R. Marino and F.L. Lagarrigue, “Adaptive Nonlinear Output Feedback for Transient Stabilization and Voltage Regulation of Power Generations with Unknown Parameters” Int. J. Robust Nonlinear Control, John Wily, Vol. 14, pp. 833-855, 2004.

216

R. Yan, Z.Y. Dong, T.K. Saha and R. Majumder “Power System Transient Stability Enhancement with an Adaptive Control Scheme Using Backstepping Design”, IEEE Conf 2007. Y.U. Xiangyang, D.U. Xiaoning, H. Yongxuam and N. Haipeng, “A Novel Adaptive Sliding Mode Excitation Controller via Backstepping Approach”, IEEE Conf on Mechatronics and Automation, pp. 540-548, 2008. M. Ouassaid, A. Nejmi, M. Cherkaoui and M. Maaroufi, “A New Nonlinear Excitation Controller for Transient Stability Enhancement in Power Systems”, World Academy of science, Engineering and Technology 2005. S.Y. Li, S.S. Lee, Y.T. yoon and J.K. Park, “Nonlinear Adaptive Decentralized Stabilization Control Stabilization Control for Multimachine Power Systems” Int. J. Conrol, Automation and Systems, pp. 389-397, 2009. A. Karimi, S.Eftekharnejad and A. Feliachi, “Reinforcement Learning Based Backstepping Control of Power System Oscillations” Int. J. Elsevier on Electric Power Systems Research, pp. 1511-1520, 2009. O. Akhrif, F.A. Okou, A. Dessaint and R. Champagne “Application of a Multivariable Feedback Linearization Scheme for Rotor Angle Stability and Voltage Regulation of Power systems”, IEEE Trans on Power System, Vol. 14, No. 2, pp. 620-628, May 1999. A. Kazemi, M.R. J.Motlagh and A.H. Naghshbandy” Application of a New Multi-variable Feedback Linearization Method for Improvement of Power System Transient Stability” Elsevier Journal on Electrical power and Energy Systems, Vol. 29, pp. 322-328, 2007. M. Anderson and A. Fouad, “Power System Control and Stability”, IEEE Press, 1993. IEEE Committee Report, “Hydraulic Turbine and Turbin Control Models for system Dynamic Studies”, IEEE Trans. On Power System, Vol. 7, No. 1, February 1992. Okou. Akhrif, “Conception d'un régulateur non-linéaire detension et d'angle de charge pour un générateur synchrone”, M.S. Thesis, ETS, Montreal, 1996. K.M. Hangos, J. Bokor and G. Szederkényi, “Analysis and Control Process Systems”, Springer Press 2004. J.J.E. Slotine and W. Li, “Applied Nonlinear Control”, Prentice Hall, New Jersey 1991. S.P. Banks, M.U. Salamci and K. Özgören, “On the Global Stabilization of Nonlinear Systems via Switching Manifolds” Turk J Elec Engin Vol. 7, No. 1-3, 1999. The Mathworks, “Power System Blockset-User’s guide”, January 1998. H. Khalil, “Nonlinear Systems”, Prentice-Hall: New Jersey, 1996. Y.Wang, D.J. Hill, R.H. Middleton and L.Gao, “Transient Stability Enhancement and Voltage Regulation of Power Systems”, IEEE Trans on Power Systems, Vol. 8, No. 2, May 1993.