Generalized Lyapunov Function for Stability
Analysis of Interconnected Power Systems
M. A. Mahmud, Student Member, IEEE, M. J. Hossain, Member, IEEE, H. R. Pota and M. S. Ali
Abstract—The formulation of Lyapunov function is necessary to analyze the stability of a system. This paper presents an idea for formulating generalized Lyapunov function for the stability analysis of interconnected power systems. Lyapunov function is formulated based on the total energy of power system where the system is considered as a single machine infinite bus (SMIB) system. The negative definiteness of the derivative of proposed Lyapunov function is proved through the application of zero dynamic control law. This paper also presents the formulation of control Lyapunov function to design a excitation controller for the generator. The performance of control Lyapunov function based excitation is tested on the SMIB system.
I. INTRODUCTION
Power systems are interconnected systems which has in-terconnection between generators, transformers, transmission lines, loads, and many other protective equipments. Most of the nonlinearities in power systems appear due to these inter-connections. The stability of power systems is very important for secure system operation because an unsecured system can undergo non-periodic major cascading disturbances, or blackouts, which have serious consequences.
The basic phenomenon of power system stability is pro-posed in [1] which has explored a variety of machine loading, machine inertias, and system external impedances with a determination of the oscillation and damping characteristics of voltage or speed following a small disturbance in mechan-ical torque. Based on this phenomenon, a vast amount of experience has been accumulated in the last four decades. But still now, the changing behaviors of power systems continues to provide new challenges to the system designers and operators [2]. Therefore, high-performance controllers are designed and implemented on power systems to obtain the stable operation.
Power system stabilizer (PSS) is widely used by power industries all over the world to maintain the stability of power system under disturbances which provides some additional damping to the system through the excitation of the syn-chronous generator [3], [4]. Some advanced methodologies are proposed in [5], [6], [7] to provide the stability of the power systems under large disturbances. These PSSs [3-7] are designed based on the linearized model of power system which can provide stable operation over a fixed set of operating points. The limitations of linear controllers are clear investigate M. A. Mahmud, H. R. Pota and M. S. Ali are with the School of Engineering and Information Technology (SEIT), The University of New South Wales at Australian Defence Force Academy (UNSW@ADFA), Canberra, ACT 2600, Australia. E-mail: [email protected] and [email protected]. M. J. Hossain is with the School of Information Technology and Electrical Engineering (ITEE), University of Queensland, St Lucia, Brisbane, QLD 4072, Australia. E-mail: [email protected].
in our previous work [8]. Moreover, the linearized model of power systems does not represent exact behaviors of the systems.
Nonlinear controllers for power systems maintain the sta-bility of the system over wide range of operating points. The performances of nonlinear controllers are identified in [9], [10]. Also, in our previous work [11], the design of nonlinear controllers are proposed and the stability of power system is analyzed. But the computation burden is more in these design techniques [9-11].
Lyapunov function of a system determines the stability of the system. Stability analysis of power systems using Lyapunov’s direct method is proposed in [12]. An extended lyapunov function is proposed in [13]. In these papers [12], [13], power system models are considered as second order model but for the control design purpose, power systems are modeled as order model. An excitation controller for synchronous generator is presented in [14] by using the third order model of the generator to enhance the transient stability and dynamic performance of the system where the control law is derived mainly based on the adaption law. Control Lyapunov function provides a stabilizing control law with meaningful optimal property. A nonlinear optimal control formula through control Lyapunov function is discussed in [15] using different methods. Control Lyapunov function for controllable series devices and power systems are proposed in [16] by consid-ering the second order model of power systems. In all these literatures [12-16], the negative derivativeness of Lyapunov function is not investigated clearly.
The aim of this paper is to formulate a generalized Lya-punov function to analyze the stability of interconnected power system. To proof the negative definiteness of the formulated Lyapunov function a zero dynamic control for the considered SMIB system is derived and the negative definiteness of the derivative of Lyapunov function is proved. Concepts of control Lyapunov function is also proposed in this paper. the performance of the control Lyapunov function is evaluated through some simulation on SMIB system.
The organization of the paper is as follows. The mathe-matical modeling of SMIB system is shown in Section II. Section III presents the formulation of Lyapunov function. The negative definiteness of the derivative of formulated Lyapunov function is prove in Section IV. Section V gives and idea about the control Lyapunov function and formulation of control Lyapunov function for SMIB system. With the formulated control Lyapunov function, the SMIB is simulated and the simulation results are given in Section VI. Finally, the paper is concluded with future trends and further recommendation in Section VII.
Fig. 1. Power System Model
II. POWER SYSTEM MODEL
Fig. 1 shows a single machine infinite bus system which is the focus of this paper. Since SMIB system qualitatively ex-hibits the important aspects of the behavior of a multimachine system and is relatively simple to study, it is extremely useful in studying general concepts of power system stability [9].
For a control scheme to be effective, some details of the system must be available. The details of any system can best be described by the mathematical model. Power system can be modeled at several different levels of complexities, depending on the intended application of the model. With some typical assumptions, the classical third-order dynamical model of a SMIB power system as shown in Fig. 1 can be modeled by the following set of differential equations [3], [17]:
Generator mechanical dynamics:
˙δ = ω − ω0 (1) ˙ ω = − D 2H(ω− ω0) + 1 2H(Pm− Pe) (2) where δ is the power angle of the generator, ω is the rotor speed with respect to synchronous reference, H is the inertia constant of the generator, Pm is the mechanical input power
to the generator which is assumed to be constant, D is the damping constant of the generator, and Pe is the active
electrical power delivered by the generator. Generator electrical dynamics:
˙
Eq′ = 1
Tdo
(Ef− Eq) (3)
where Eq′ is the quadrature-axis transient voltage of the
generator, Eq is the quadrature-axis voltage of the generator,
Tdois the direct-axis open-circuit transient time constant of the
generator, and Ef is the equivalent voltage in the excitation
coil.
The algebraic electrical equations of synchronous generator are given below:
Eq = xdΣ x′dΣE ′ q− (xd− x′d)Id Id = Eq′ x′dΣ − Vs x′dΣcos δ Iq = Vs x′dΣsin δ Pe = VsEq′ x′dΣ sin δ Qe = VsEq′ x′dΣ cos δ− Vs2 xdΣ Vt = √ (Eq′ − Xd′Id)2+ (Xd′Iq)2 where xdΣ = xd+ xT + xL, xdΣ′ = x′d + xT + xL, xd is
the direct-axis synchronous reactance, x′d is the direct axis transient reactance, xT is the reactance of the transformer, xL
is the reactance of the transmission line, Id and Iq are direct
and quadrature axis currents of the generator respectively, Vs
is the infinite bus voltage, Qe is the generator reactive power
delivered to the infinite bus, and Vtis the terminal voltage of
the generator.
Substituting the electrical equations into the mechanical and electrical dynamics equation (1)-(3) of the system, the complete mathematical model of SMIB system can be written as follows: ˙δ = ω − ω0 (4) ˙ ω = − D 2H(ω− ω0) + 1 2HPm− 1 2H VsEq′ x′dΣ sin δ (5) ˙ Eq′ = −1 Td′E ′ q+ 1 Tdo xd− x′d x′dΣ Vscos δ + 1 Tdo Ef (6) where Td′ = x′dΣ
xdΣTdo is the time constant of the field winding. The numerical values of the system parameters are given in Appendix A.
III. FORMULATION OFLYAPUNOV FUNCTION FORSMIB SYSTEM
Lyapunov functions are the functions which are applicable to the stability analysis of the equilibrium states for dynamical systems. The Lyapunov stability theory includes two methods, Lyapunov’s first method and Lyapunov’s direct method. Lya-punov’s first method is a technique which simply uses the idea of system linearization (lowest order approximation) around a given point. Lyapunov’s direct method is the most important tool for design and analysis of nonlinear system. Normally, Lyapunov function means the direct method. Lyapunov func-tion can be defined as follows [18]:
Definition 1: (Lyapunov Function)
If a function V (x) is positive definite and if its time derivative ˙
V (x) along any state trajectory of the system ˙x = f (x) is
negative definite, i.e., ˙V (x) ≤ 0 then V (x) is said to be a
Lyapunov function.
As mentioned earlier, control is essential for secure oper-ation. In the planning stage, i.e., before implementing any control it is essential to assess the results. Therefore, the operator would like to analyze the dynamic stability through simulations which take longer time. But this can be easily done by using Lyapunov function which is also known as energy function.
From the definition of Lyapunov function or energy function it is seen that the derivative of this function is negative which means that for each state x, the control law u will reduce the energy V of the system. If in each state it is possible to find out a control law to reduce the energy of the system, then it is possible to bring the energy of the system to zero, i.e., to bring the system to a stop which means the stable condition of the system.
An energy-based Lyapunov function for any system corre-sponds to the total system energy made up from the sum of the
system kinetic and potential energy. The Lyapunov function for any third-order power system model can be formulated as follows:
V = Vk+ Vp+ VEq′ (7)
Here, Vk is the kinetic energy due to the speed of the
synchronous generator, Vpis the potential energy due the rotor
oscillations, and VE′
q is the energy due to the quadrature-axis transient voltage of the synchronous generator. All these energy can be written as follows:
Vk = H(ω2− ω20) = H(∆ω) 2 (8) Vp = − ∫ δ 0 (Pm− Pe)dδ (9) VEq′ = xdΣ 2(xd− x′d) 1 x′dΣ[E ′ q] 2 (10)
From equation (8) we get, ˙
Vk = 2H∆ω
dω
dt =−D∆ω + (Pm− Pe)∆ω (11)
In a similar way, from equation (9) we get, ˙ Vp = ∂Vp ∂δ dδ dt + ∂Vp ∂Eq′ dEq′ dt
Now, we can obtain the following equation,
∂Vp ∂δ =−(Pm− Pe) and ∂Vp ∂Eq′ = ∫ δ 0 ∂Pe ∂Eq′dδ =− Vs x′dΣcos δ Therefore, ˙ Vp = −(Pm− Pe)∆ω− Vs x′dΣcos δ dEq′ dt (12)
By taking the derivative of equation (10), the following relationship can be obtained,
˙ VEq′ = ∂VE′q ∂Eq′ dE′q dt = xdΣ (xd− x′d) 1 x′dΣE ′ q dEq′ dt (13) Now, by multiplying Tdo xd−x′d dE′q
dt on both sides of the
equa-tion (6) Tdo xd−x′d (dEq′ dt ) 2=− xdΣ (xd−x′d) 1 x′dΣEq′ dE′q dt +x1′ dΣ Vscos δ dEq′ dt + 1 xd−x′d Ef dEq′ dt (14)
The first term on the right side of the above equation is equal to ˙VEq′, thus the above equation can be written as
˙ VEq′ =−xTdo d−x′d (dE ′ q dt ) 2 +x1′ dΣ Vscos δ dEq′ dt + 1 xd−x′d Ef dEq′ dt (15)
Thus the derivative of Lyapunov function of the system can be written as ˙ V = −D∆ω − Tdo xd− x′d (dE ′ q dt ) 2+ 1 xd− x′d Ef dEq′ dt
The above equation can be written as ˙ V = −D∆ω − 1 xd− x′d 1 Tdo (Tdo dE′q dt )[Tdo dEq′ dt − (Ef− Ef 0)]
Again, we know that
Tdo
dEq′
dt = (Ef− Eq)
Using the above relation into derivative ˙V of the Lyapunov
function, we get ˙ V = −D∆ω − 1 xd− x′d 1 Tdo Eq2 + 1 xd− x′d 1 Tdo EqEf (16)
Equation (16) is the final Lyapunov function. Now, it is essential to make this derivative as negative. This will happen if the third term on the right side of equation (16) is negative and this can be done by applying suitable control law through the excitation voltage Ef of the generator. The derivation of
appropriate control law and the negative definiteness of (16) is shown in the following section.
IV. NEGATIVEDEFINITENESS OF THEDERIVATIVE OF LYAPUNOV FUNCTION
This section will prove the negative definiteness of the proposed Lyapunov function for which a suitable control law is essential. The derivation of control law through zero dynamic design approach and the negative definiteness of the Lyapunov function is shown in the following subsections:
A. Derivation of Zero Dynamic Control Law
The detail of zero dynamic design approach can be found in [19], [20]. Zero dynamic control law for SMIB system can be obtained through the following steps:
Step 1: System modeling
The system model needs to be written in the following form: ˙
x(t) = f (x) + g(x)u (17) where x∈ Rn is the state vector; u∈ R is the control vector;
f (x) and g(x) are the n-dimensional vector fields in the state
space. For SMIB system,
x =[δ ω Eq′]T f (x) = ω− ω0 −D 2H(ω− ω0) + 1 2HPm− 1 2H VsEq′ x′dΣ sin δ −1 Td′Eq′ + 1 Tdo xd−x′d x′dΣ Vscos δ g(x) =[0 0 T1 do ]T
and
u = Ef.
Step 2: Selection of output function
For SMIB system, the output function is considered as y = ω for which the relative degree of the system is r = 2 which is less than the order n = 3 of the system. Therefore, zero dynamic control law con be obtained if the zero dynamic of the system is stable.
Step 3: Zero dynamics and its stability
At this stage, the zero dynamic of SMIB system needs to be calculated. This is optional for the considered case as the stability of the derived control law will be checked through the formulated Lyapunov function. Moreover, power systems are inherently stable.
Step 5: Derivation of control law
After doing some manipulation according to the theory of zero dynamic as proposed in [19], [20], the control law for SMIB system can be written as
u = Eq− Td′ Eq Pe (Qe+ Vs2 x′dΣ)∆ω− D Eq Pe ∆ ˙ω (18) This control law will be used to proof the negative def-initeness of the derivative of the Lyapunov function, i.e. equation (16).
B. Negative Definiteness of Equation (16)
To prove the negative definiteness of (16), it essential to substitute (18) into (16). By substituting (18) into (16), the following relationship can be obtained,
˙ V = −D∆ω − 1 xd− x′d 1 Tdo Td′ Eq Pe (Qe+ V2 s x′dΣ)∆ω − 1 xd− x′d 1 Tdo DEq Pe ∆ ˙ω
In a similar way, it can be shown that the derivative of the proposed Lyapunov function for LQR control technique as in LQR u =−Kx. Therefore, with proper excitation control, the derivative of (7) is negative which specifies the stable operation of the system.
V. CONTROLLYAPUNOVFUNCTION
The aim of this section is to find out an optimal excitation control law for the synchronous generator using the concept of control Lyapunov function. Let, the derivative of the derived Lyapunov function can be written as
˙
V = LfV + LgV ˜u (19)
where LfV is the Lie derivative of V along f (x), LgV is the
Lie derivative of V along g(x), and ˜u is the optimal control
law which needs to be determined.
If it is possible to make (19) negative at every point by choosing an appropriate ˜u, then the stability of the system with
the proposed ˜u will be achieved, i.e., ˜u needs to be selected
in such a way that it satisfies the following condition: min
u [LfV + LgV ˜u] < 0 (20)
If the above condition is satisfied, then V can be said as Lyapunov control function and the optimal control law for this Lyapunov control function can be written as [15], [16]
˜
u = LfV (21)
Equation (16) can be written very similar to (19). Therefore, for SMIB system LfV and LgV can be written as
LfV =−D∆ω − 1 xd− x′d 1 Tdo E2q LgV = 1 xd− x′d 1 Tdo Eq
Therefore, using the concept of control Lyapunov function, the control law that will stabilize the system can be written as follows: ˜ u = Ef =−D∆ω − 1 xd− x′d 1 Tdo Eq2 (22) If this value will be substituted in (16), the it will be negative. Thus, the proposed excitation control law obtained by control Lyapunov function ensures the stability of the system. In equation (22), Eq = xadIf where xad is the mutual
reactance between excitation coil and stator coil and If is is
the per unit excitation current with the no-load rated excitation current as the base value. Therefore, the control law (22) is in terms of all measured variables. The performance of this controller is tested through some simulations in the following section.
VI. SIMULATIONRESULTS
The performance of the control Lyapunov function based excitation controller is analyzed on SMIB system as shown in in Fig. 1. The parameters used in the simulation are given in Appendix A. The system is simulated under the following two cases:-(i) sudden outage of one of the transmission lines, and (ii) a bus fault at the generator bus. In the simulation, all the faults are applied at t = 1 s and cleared at t = 2 s.
Case 1: Sudden outage of one of the transmission lines
Fig. 2 shows the generator terminal voltage when one of the transmission line is tripped. During the faulted period 1 s to 2 s, the generator terminal voltage goes down and again the system becomes stable.
Fig. 3 and Fig. 4 present the rotor angle and speed deviation of the synchronous generator, respectively when one of the transmission line is out of service. Both Fig. 3 and Fig. 4 show the stable operation at post-fault condition.
Case 2: Bus fault at the generator terminal
In this case, a bus fault is applied to the terminal of the generator. Fig. 5 presents the generator terminal voltage under this situation. During the faulted period 1 s to 2 s, the generator terminal voltage of the generator goes down to zero but the system becomes stable with some oscillations in voltage.
Fig. 6 and Fig. 7 present the rotor angle and speed deviation of the synchronous generator, respectively when there is fault at generator terminal. From Fig. 6 and Fig. 7, it is seen that there are more oscillations within the system but the system is stable at some other operating points.
Fig. 2. Generator terminal when one of the transmission line is tripped
Fig. 3. Rotor angle when one of the transmission line is tripped
Fig. 4. Speed deviation when one of the transmission line is tripped
Fig. 5. Generator terminal when fault is applied at generator bus
Fig. 6. Rotor angle when fault is applied at generator bus
VII. CONCLUSION
Formulation of generalized Lyapunov function for intercon-nected power system is shown in this paper. To formulate the Lyapunov function, the total energy of the system is taken into account. A zero dynamic excitation control law is obtained to prove the negative definiteness of the derivative of the proposed Lyapunov function. The proposed Lyapunov function can the stability of interconnected power systems will all proper control techniques. An excitation control law is determined by using the concept of control Lyapunov function. The performance of control Lyapunov function based excita-tion controller is verified through simulaexcita-tions under different faulted conditions of the system. Simulation results presents the stable system operation. But more enhanced performance can be obtained by using other robust excitation controller. Future work will deal with the extension of the proposed techniques to handle large power systems and power systems with loads as well as robust controller design for power system.
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APPENDIX
The parameters used for the SMIB system are given below: Synchronous generator parameters:
xd = 1.863 pu, xad = 1.712 pu, x′d = 0.257 pu, H = 8 s,
Tdo= 6.9 s, D = 5, ω0= 314.159.
Transformer Parameter: xT = 0.127 pu.
Transmission Line Parameters: xL= 0.4853 pu.
Infinite bus voltage, Vs= 1.00 pu.
Mechanical Power Input, Pm= 0.9 pu.
M. A. Mahmudwas born in Rajshahi, Bangladesh in 1987. He has received the B.Sc. in Electrical and Electronic Engineering with honours from Ra-jshahi University of Engineering and Technology (RUET), Bangladesh, in 2008. He is currently a PhD candidate at the University of New South Wales, Australian Defence Force Academy.
His research interests are dynamic stability of power systems, solar integration and stabilization, voltage stability, distributed generation, nonlinear control, electrical machine, and HVDC transmission system.
M. J. Hossainwas born in Rajshahi, Bangladesh, on October 30, 1976. He received the B.Sc. and M.Sc. Eng. degrees from Rajshahi University of Engineering and Technology (RUET), Bangladesh, in 2001 and 2005, respectively, all in electrical and electronic engineering. He has finished his Ph.D. degree from the University of New South Wales, Australian Defence Force Academy in 2010. Cur-rently, he is working as a research fellowship in the University of Queensland.
His research interests are power systems, wind generator integration and stabilization, voltage stability, micro grids, robust control, electrical machine, FACTS devices, and energy storage systems.
H. R. Potareceived the B.E. degree from SVRCET, Surat, India, in 1979, the M.E. degree from the IISc, Bangalore, India, in 1981, and the Ph.D. degree from the University of Newcastle, NSW, Australia, in 1985, all in electrical engineering.
He is currently an Associate Professor at the University of New South Wales, Australian Defence Force Academy, Canberra, Australia. He has held visiting appointments at the University of Delaware; Iowa State University; Kansas State University; Old Dominion University; the University of California, San Diego; and the Centre for AI and Robotics, Bangalore. He has a continuing interest in the area of power system dynamics and control, flexible structures, and UAVs.
M. S. Ali was born in Bangladesh in 1983. He obtained his B.Sc degree in electrical and electronic engineering from the Ahsanullah University of Sci-ence and Technology, Bangladesh in 2008. Cur-rently, he is a Ph.D student in electrical engineering at UNSW@ADFA, Australia. His research interest is in the areas of distribution network with distributed generation, green energy, distribution network with dynamic load and control, implementation of FACTS devices on distribution network, PHEV and micro-grid.
