Proceedings of the 33rd Chinese Control Conference July 28-30, 2014, Nanjing, China
Time-Delay Power Systems Control and Stability with Discretized Lyapunov Functional Method Xin Wang1 and Keqin Gu2 1. Department of Electrical and Computer Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026, USA E-mail:
[email protected] 2. Department of Mechanical Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026, USA E-mail:
[email protected] Abstract: Control design and stability analysis are developed for time-delay power systems. The coupled differential-difference equations are used to model the practical power system dynamics. A robust state feedback controller design stabilizing the unstable power plant is proposed first. Then, the stability analysis with Discretized Lyapunov Functional method is conducted. The overall system stability is determined from the solution of a linear matrix inequality feasibility problem. Numerical simulation results of the proposed schemes are presented and discussed to illustrate the effectiveness and robustness in power systems control and stability applications. Key Words: Time Delay, Small Signal Stability, Power System, Discretized Lyapunov Functional Method
1
Notations
The notations used in the description of power generation system model are listed below: δ ωs ν Tdo P SB ωB H
Rotor angle Synchronous speed Normalized frequency ν = ωωs Equivalent transient rotor time constant Total number of poles Rated three phase voltage ampere Rated speed in electrical radians per second In this work, ωB = ωs The shaft inertia constant is scaled by defining
Kd Id Iq Xd Xq Xd Eq Eq Ef d Ef d Tm Rs VT Vref Re Xe z1 , z2 , z3 Upss V∞
H = 2 SBB P The damping factor Direct axis current Quadrature axis current Direct axis reactance Quadrature axis reactance Direct axis transient reactance Quadrature axis voltage Quadrature axis transient voltage Excitation voltage Transient excitation voltage Mechanical torque Stator resistance Terminal bus voltage Reference voltage Transmission line resistance Transmission linear reactance State variables of power system stabilizer Power system stabilizer control signal The infinite bus voltage
1
2
J(ω
2
)2
Introduction
Time delays emerge in a wide variety of power systems. These delays in time can degrade the power quality and even cause severe damage to the overall power system stability, therefore, they should be taken into consideration in practical
controller design and stability analysis. Time delays loom in the actuators, sensors and control loops of the power systems. Most modern power system control facilities, such as Automatic Voltage Regulator (AVR), Excitation control system and Power System Stablizer (PSS), are all implemented using digital hardware, such as Programmable Logic Controllers (PLC) and Programmable Automation Controllers (PAC). In common practices, there exist delays introduced by the actuators, transducers, digital amplifiers, analog to digital converters, microprocessor computations, memory and cache, etc. Moreover, considering that impact of modern communication network, the delay can be significant. Most data is sent through remote buses, the measurement delays from these digital equipment can drastically increased more than 100 ms. Therefore, the delays in time are not negligible. Time delays have become important issues in the smart grid systems. In recent years, with the development of the Wide-Area Measurement System (WAMS) technology, synchronized real time measurements can be obtained from the Phasor Measurement Units (PMU). However, the communication channel delays of the measurement signals from PMUs also render significant impact on the overal power system stability. Therefore, it is important to establish a systematic approach to tackle the issue of time delay power system control. During the last decades, controller design and small signal stability anslysis of time delayed power systems have attacted a lot of attention, and many results on this topic have been reported in literatures [1, 13, 14]. Existing stability analysis and controller design on small signal stablity of delayed power systems can be categorized as: Frequency Domain Approach [13] and Time Domain Approach [1]. Methods of robust stability analysis and controller design of time delay systems can be found in [2–11] using the linear matrix inequality (LMI) framework. This paper contributes to the modeling of the infinite bus power generation system dynamics, and the state feedback control scheme considering delays for small signal stability of the power system. Including the time delays in power sys-
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tem model, the differential-difference equations can be obtained. The coupled differential-difference equations based control scheme renders computational advantage over the non-coupled differential-difference equations based control approach. By using the Lyapunov Krasovskii Functional approach, the Linear Matrix Inequality solution is derivated to obtain the control feedback gain. It is proven that the proposed time delay power system control scheme guarantee the asymptotic stability. The remainder of paper is organized as follows: Section II defines the dynamics model of the power system. The retarded measurement of normalized speed propagates in the power system stabilizer scheme. The coupled differentialdifference model is used to characterize the retarded power system dynamics. The control scheme is proposed in Section III. Detailed computer simulation results of the controlled power system are presented and discussed in Section IV. Finally, some concluding remarks are given in Section V.
3
The following linearized dynamics for Automatic Voltage Regulator (AVR) and Excitation control system is adopted: KA ΔEf d ΔE˙ f d = (ΔVref − ΔVT + ΔUpss ) − TA TA By applying (6) to (8), we have ΔE˙ f d
1 K4 1 ΔEq − Δδ + ΔEf d K3 Tdo Tdo Tdo
−
Δδ˙
=
Δν˙
=
(2) ωs Δν K2 Kd ωs 1 K1 − ΔEq − Δδ − Δν + ΔTm 2H 2H 2H 2H (3)
Δz˙1
=
Δz˙2
=
Δz˙3
=
ΔUpss
=
(1)
=
K4
=
K2
=
K1
=
x = [Δδ, Δν, ΔEq , ΔEf d , Δz1 , Δz2 , Δz3 ]T u = [ΔTm , ΔVref ]T
x(t) ˙ where
A0
=
(4) where
ΔVT = K5 Δδ + K6 ΔEq
(6)
where
=
⎛
0 ⎜A21 ⎜ ⎜A31 ⎜ ⎜A41 ⎜ ⎜ 0 ⎜ ⎝ 0 0
ωs A22 0 0 0 0 0
0 A23 A33 A43 0 0 0
0 0 A34 A44 0 0 0
0 0 0 A45 A55 A65 A75
0 0 0 A46 0 A66 A76
with
K6
A0 x(t) + A1 x(t − τ ) + B0 u(t)
(5)
The linearized terminal bus voltage
=
=
(13)
K5
(12)
Therefore, the overall linearized model of power system becomes
{(Xd + Xe )cosδ + Re sinδ}]
(11)
and inputs:
(Xd − Xd )(Xq + Xe ) 1+ Δ V∞ (Xd − Xd ) [(Xq + Xe )sinδ − Re cosδ] Δ 1 o [Iq Δ − Iqo (Xd − Xq )(Xq + Xe ) Δ −Re (Xd − Xq )Ido + Re Eqo ] 1 − [Iqo V∞ (Xd − Xq ){(Xq + Xe )sinδ Δ −Re cosδ} + V∞ {(Xd − Xq )Ido − Eqo }
Δ = Re2 + (Xe + Xq )(Xe + Xd )
−(Kw Δν(t − τ ) + Δz1 )/Tw T1 [(1 − )(Kw Δν(t − τ ) + Δz1 ) − Δz2 ]/T2 T2 T3 T1 {(1 − )[Δz2 + ( (Kw Δν(t − τ ) + T4 T2 Δz1 )] − Δz3 }/T4 T3 T1 Δz3 + [Δz2 + (Kw Δν(t − τ ) + Δz1 )] T4 T2 (10)
Define the states variables:
where 1 K3
ΔEf d KA K5 − Δδ − TA TA KA K6 KA KA ΔEq + ΔUpss + ΔVref TA TA TA
−
A typical Power System Stabilizer (PSS) control scheme include a washout filter and two lead-lag blocks. The retarded measure of ν propagates in the PSS equations. The linearized form of power system stabilizer is:
The dynamic model of a synchronous generator infinitebus power system is derived in this section. The linearized dynamic model of synchronous generator is given as: =
=
(9)
Dynamical model of power plant
ΔE˙ q
(8)
1 Vdo { Xq [Re V∞ sinδ + V∞ cosδ(Xd + Xe )] Δ VT Vqo + [Xd (Re V∞ cosδ − V∞ (Xq + Xe )sinδ)]} VT Vqo Vqo 1 Vdo { Xq Re − Xd (Xq + Xe )} + Δ VT VT VT (7)
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K1 A21 = − 2H K2 A23 = − 2H
A33 = − K 1T A41 A44 A46 A55 A66 A76
3 do KA K5
= − TA = − T1A A T3 =K TA T4 = − T1w = − T12 = (1 − TT34 ) T14
d ωs A22 = − K2H K4 A31 = − T do
A34 = − T1 A43 A45 A47 A65 A75 A77
do
= − KTAAK6 A T3 T1 =K TA T4 T2 A =K TA = (1 − TT12 ) T12 = (1 − TT34 ) TT12 T14 = − T14
⎞ 0 0 ⎟ ⎟ 0 ⎟ ⎟ A47 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠ A77 (14)
⎛
A1
=
0 0 ⎜0 0 ⎜ ⎜0 0 ⎜ T3 T1 Kw ⎜0 ⎜ T4 T2 ⎜0 w −K ⎜ Tw ⎜ T1 (1 − T2 )Kw T12 ⎝0 0 (1 − TT34 ) TT12 Kw T14 ⎛
B0
4
=
0
0 0 0
0 0 0 0 0 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ ⎟ 0 0 0 0 0⎠ 0 0 0 0 0
V (x(t))
+ +
0
−r 0 −r
0
φT (ξ)S(ξ)φ(ξ)dξ
ψ = x(t),
(16)
φ(0) = y(t),
(26)
φ(ξ) = y(t + ξ)
(27)
Differentiating LFK along the system trajectory, we have V˙ (ψ, φ, r) = 2ψ T P ψ˙ + 2ψ˙ T
A0 x(t) + A1 x(t − τ ) + B0 u(t)
Assume all the state variables can be directly obtained from measurements or state estimations. The linear constant gain state feedback controller is proposed:
u(t) = K0 x(t) + K1 x(t − τ )
0
Q(ξ)φ(ξ)dξ −r T
+2ψ T Q(0)φ(0) − 2ψ Q(−r)φ(−r) 0 ˙ −2ψ T Q(ξ)φ(ξ)dξ +2φT (0) −2φT (−r) −
(18)
0 −r
0 −r
φT (ξ)[
−r
0
R(0, η)φ(η)dη −r 0
R(−r, η)φ(η)dη −r
∂R(ξ, η) ∂R(ξ, η) + ]φ(η)dξdη ∂ξ ∂η
+φT (0)S(0)φ(0) − φT (−r)S(−r)φ(−r) 0 ˙ − φT (η)S(η)φ(η)dη
Applying control to the system dynamical equation, we have =
Q(ξ)φ(ξ)dξ −r
φT (ξ)R(ξ, η)φ(η)dηdξ
−r
(17)
x(t) ˙
0
where
Consider the following general model for time-delay power system: =
ψ T P ψ + 2ψ T
State feedback control design and stability analysis
x(t) ˙
=
(15)
⎞
⎟ ⎟ ⎟ ⎟ KA ⎟ TA ⎟ 0 ⎟ ⎟ 0 ⎠ 0
⎜ 1 ⎜ 2H ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0
The control and stability of linear retarded system with discrete delays can be studied via Lyapunov-Krasovskii Functionals (LKFs).
⎞
(A0 + B0 K0 )x(t) + (A1 + B0 K1 )x(t − τ )
(28)
−r
(19)
Since
Denote the following notations:
0
A1
=
B1 C 1
(20)
B 0 K1
=
B2 C 2
(21)
=
2[Ax(t) + By(t − r) − x] ˙T [P1 x(t) ˙ − P3 x(t) − BP4 y(t − r)] +2[Cx(t) − y(t)]T [P2 y(t) − B T P1 x(t)] (29)
with B2
=
B0
(22)
C2
=
K1
(23)
In the control design, we constrain P3 = P1 , P4 = P2
Then the overall power system can be modeled by the following coupled differential difference equations: x(t) ˙ y(t)
= =
Ax(t) + By(t − r) Cx(t)
A B y(t) C
=
2[Ax(t) + By(t − r) − x] ˙T [P1 x(t) ˙ − P1 x(t) − BP2 y(t − r)] +2[Cx(t) − y(t)]T [P2 y(t) − B T P1 x(t)] (31)
(24)
Adding to V˙ the right part of the expression (38), and apply the following variable transform:
=
A 0 + B 0 K0 = B1 B 2
y1 (t) = y2 (t)
C1 = K1
Hence, we have 0
where
(30)
ψˆ = P1 ψ Pˆ1 = P −1 1
(25)
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Pˆ = Pˆ1T P Pˆ1 ˆ = Pˆ2T RPˆ2 R
φˆ = P2 φ Pˆ2 = P −1 2
ˆ = Pˆ T QPˆ2 Q 1 Sˆ = Pˆ T S Pˆ2 2
(32)
We have ˙ V˙ (ψ, φ, r) = ζ T Ξζ + 2ψˆT −2ψˆT
+2φˆT (0)
ˆT
−2φ (−r) −
0 −r
0
−r 0
ˆ p , Sˆp , R ˆ pq Thus, the LKF is completely determined by Pˆ , Q for p, q = 0, 1, 2, ..., N .
ˆ ˆ φ(ξ)dξ Q(ξ)
Recall that the system is asymptotically stable if and only if there exist a Lyapunov -Krasovskii function, such that the Lyapunov-Krasovskii functional condition
ˆ ˆ˙ φ(ξ)dξ Q(ξ)
−r 0
V (x(t)) ≥ x(t)2
ˆ ˆ η)φ(η)dη R(0,
−r 0
and its derivative satisfy the Lyapunov-Krasovskii derivative condition (47) V˙ (x(t)) ≤ − x(t)2
ˆ ˆ R(−r, η)φ(η)dη
−r
ˆ η) ∂ R(ξ, ˆ η) ∂ R(ξ, ˆ + ]φ(η)dξdη φˆT (ξ)[ ∂ξ ∂η −r 0 ˆ ˆ˙ φ(η)dη − φˆT (η)S(η) 0
for some > 0. (33)
−r
where ζT
= ⎛
ψˆT
Ξ11 ⎜ Ξ=⎜ ⎝
˙ ψˆT Ξ12 Ξ22
φˆT (0) Ξ13 0 Ξ33
φˆT (−r)
(34)
The LKF condition V (x(t)) ≥ x(t) is satisfied for some > 0 if Sp > 0 for p = 1, 2, ..., N and
˜ P Q (48) ˜ + S˜ ∗ R where
⎞ Ξ14 Ξ24 ⎟ ⎟ 0 ⎠ Ξ44
˜ Q
=
˜ R
=
S˜
=
ˆ0, Q ˆ 1 , ..., Q ˆN ] [Q ⎛ˆ ˆ 0N ⎞ ˆ R00 R01 . . . R ⎟ ⎜ .. ˆ 0N ⎟ ˆ ˆ ⎜R . R ⎟ ⎜ 10 R11 ⎜ . .. .. ⎟ .. ⎝ .. . . . ⎠ ˆN 0 R ˆN N ˆN 1 . . . R R 1 1 1 diag{ Sˆ0 , Sˆ1 , ..., SˆN } h h h
(35) with Ξ11
=
Ξ12 Ξ13
=
Ξ14
=
Ξ22
=
Ξ24
=
Ξ33
=
Ξ44
=
=
−2Pˆ1T AT − 2Pˆ1T C T B T Pˆ1T AT + Pˆ + Pˆ1 ˆ Q(0) + Pˆ1T C T + B Pˆ2 T ˆ −Pˆ1 AT B − Q(−r) − B Pˆ2 T −2Pˆ1 B Pˆ2 + Pˆ1T B ˆ S(0) − 2Pˆ2T ˆ −S(−r) − 2Pˆ2T B T B
(49) (36) (37) (38) (39)
ˆ p (α) =Q = Sˆp (α)
Theorem 1 The system described by the previous equations are asymptotically stable if there exist matrices Pˆ = ˆ i , Sˆi = SˆT , i = 0, 1, ..., N , and R ˆ pq = R ˆT , p = Pˆ T , Q pq i 0, 1, 2, ..., N, q = 0, 1, 2, ..., N , such that
(40) (41) (42)
We apply the discretization to divide the domain of defˆ R, ˆ Sˆ into small regions, and inition of matrix functions Q, choose these matrix functions to be continuous piecewise linear. Divide the delay interval [−r, 0] into N segments [θp , θp−1 ], p = 1, 2, 3, ..., N of equal length h = θp − θp−1 = r/N . Then θp = −ph = − pr N , for p = 0, 1, 2, ..., N . This also divides the square matrix Sˆ = [−r, 0]×[−r, 0] into N ×N small squares Sˆpq = [θp , θp−1 ]× [θq , θq−1 ]. Each square is further divided into two triangles. ˆ ˆ are chosen to The continuous matrix functions Q(ξ), S(ξ) be linear within each segment and continuous matrix function R(ξ, η) is chosen to be linear within each triangle: ˆ p + αh) Q(θ ˆ p + αh) S(θ
(46)
˜ Q ˜ + S˜ R
P ∗
>0
(50)
and ⎛
ˆs D ˆ d − Sˆd −R 0
Ω ⎝D ˆ sT ˆ aT D
ˆa ⎞ D 0 ⎠ 0 ∗ R
− 2ψˆT h
ζ T Ξζ −
=
ˆ p−1,q−1 − R ˆ p,q ) h(R
ˆs D ˆa D
=
ˆ ps D
=
ˆ pa D
=
=
ˆ s, D ˆ s , ..., D ˆs ) (D 1 2 N a a ˆ ,D ˆ , ..., D ˆa ) (D 1 2 N ⎛ ˆp) ⎞ ˆ p−1 − Q −(Q ˆ ˆ ⎜ h (Q ⎟ ⎜ 2 p−1 + Qp ) ⎟ ⎝ h (R ˆ 0,p−1 + R ˆ 0,p ) ⎠ 2 ˆ N,p−1 + R ˆ N,p ) − h2 (R ⎞ ⎛ 0 ˆ ˆ ⎟ ⎜ − h (Q ⎜ h 2 p−1 − Qp ) ⎟ ⎝− (R ˆ 0,p−1 − R ˆ 0,p )⎠ 2 h ˆ ˆ ( 2 RN,p−1 − RN,p )
(61) (62)
(63)
(64)
(65) (66)
(67)
(68)
Based on similar procedures in [10], we can derive the LMI control scheme in Eqn. (50)-(55), which concludes the proof.
ˆ p − 1) + (1 − α)R(0, ˆ p)]φˆp (η)dη− [αR(0,
5
Simulation studies
Assuming that Re = 0, Xe = 0.5pu, VT θ = 1 15o pu, o o ˆ ˆ 2φˆT (−r) [αR(−r, p − 1) + (1 − α)R(−r, p)]φˆp (η)dη and V∞ 0 = 1.05 0 pu. The generator, automatic volt age regulator and excitor parameters are H = 3.2sec, Tdo = p=1 0 9.6sec, KA = 400, TA = 0.2sec, Rs = 0pu, Xq = 2.1pu, 1 N N 1 ˆ ˆ 2 p T Rp−1,q−1 − Rp,q qT ˆ ˆ −h [ ] φ (ξ)dξ] [ φ (η)dη Xd = 2.5pu, Xd = 0.39pu, Kd = 0, and ωs = 377. The h 0 power system stabilizer parameters are Kw = 0.5, T1 = p=1 q=1 0 0.5, T2 = 0.01, T3 = 1, T4 = 0.1, Tw = 10. We can N 1 Sˆp−1 − Sˆp ˆp pT calculate the values: K1 = 0.9224, K2 = 1.0739.K3 = ˆ −h ]φ (η)dη φ (η)[ h 0 0.296667, K4 = 2.26555, K5 = 0.005, K6 = 0.3572 p=1 The system is simulated assuming that the system has no (58) uncertainties. The simulation results are presented in Fig. 1-4. After controller is applied to the system, the system p ˆ p + αh) = φ(−pr/N ˆ + αh) for where φˆ (α) = φ(θ becomes stable. Both Fig. 3 and 4 have shown the effectivep = 1, 2, ..., N . ness of our proposed control approach. Therefore, it can be concluded that the control scheme works well when applied Denote to the power generation system with state delays. ⎞ ⎛ ˆ(1) φ (α) 6 Conclusions ⎜ φˆ(2) (α) ⎟ ˜ ⎟ (59) φ(α) =⎜ ⎝ ... ⎠ Time delays severely influences the stability of the overall (N ) ˆ power systems, which may causes detrimental or disastrous φ (α) N
1
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impact to the entire power grid. The paper contributes to the modeling of the infinite bus power system dynamics, and state feedback control scheme development considering delays for small signal stability of the power system. Using the Lyapunov Krasovskii Function, it is proven that the control scheme guarantee the stabilization of the power system. The control schemes are proposed to achieve stability of the systems. The simulation results show the efficacy of the proposed control scheme.
References
Fig. 1: Change in rotor angle when no control is applied to the system
Fig. 2: Change in rotor angular speed when no control is applied to the system
Fig. 3: Change in rotor angle when control is applied to the system
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Fig. 4: Change in rotor angular speed when control is applied to the system
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