Wide-Area Measurement-Based Nonlinear Robust Control of Power System Considering
Signals’ Delay and Incompleteness
G. L. Yu, B. H. Zhang, Senior Member, IEEE, H. Xie, C. G. Wang
Abstract—With the wide application of synchronized phase measurement unit(PMU) in power system, the wide-area measurement system(WAMS) has enabled the use of a
combination of measured information from remote location for
gglobal control purpose. However, the impact of time delays introduced by remote information’s transmission in WAMS has to be considered, and the closed-loop power systems need be modeled as time-delay nonlinear systems. Moreover, the wide- area information is incomplete when not all generators in power system are equipped with the PMU. In order to eliminate the effects of power system model’s nonlinearity and wide-area information’s uncertainty including time delays and incomplete- ness, a novel approach based on inverse system algorithm and linear matrix inequality(LMI) technique is proposed to design a nonlinear robust integrated controller. Digital simulation demonstrates that the four- machine system under the nonlinear robust integrated control has less settling time, swing times, oscillatory peak value and more critical clear time and better voltae stability performance for various time delays of remote incomplete measured information than that under PID control or nonlinear decentralized integrated control.
Index Terms—Center of inertia reference frame, inverse system method, integrated controller, linear matrix inequality (LMI), power system, wide-area measurement system (WAMS)
I. INTRODUCTION
A
POWER system must be modeled as a nonlinear system for large disturbances. Although power system stability may be broadly defined according to different operating conditions, an important problem frequently considered is the problem of transient stability. It concerns the maintenance of synchronism between generators following a severe transient disturbance. Another important issue of power system control is to maintain steady acceptable voltage under normal operating and disturbed conditions, which is referred as the problem of voltage stability.
As an effective means in enhancing transient stability and voltage stability, the integrated control of the excitation and valve system in a turbogenerator has attracted more attentions
This work was supported in part by National Natural Science Foundation of China under Grant 50595413.
The authors are all with the School of Electrical Engineering, Xi'an Jiaotong University, Xi'an 710049, China. (e-mail: GuangliangYu@mail.xjtu.edu.cn).
from the power system researchers. In recent years, many attempts have been made to design an integrated controller, see e.g., [1]–[7]. With the wide application of synchronized phase measurement unit(PMU) in power system, the wide-area measurement system(WAMS) has enabled the use of a combination of measured signals from remote location for global control purpose. It is found that if remote signals from one or more distant locations of the power system are applied to local controller design, system dynamic performance can be enhanced [8]–[12]. [10] proposes a decentralized/hierarchical damping control structure. Wide-area signals based PSS is used to provide additional damping. A sequential optimization procedure is used to tune the PSS global and local control loops. In [11], a two-level hierarchical structure consists of a local controller for each generator at the first level helped by a multivariable central one at the secondary level. The secondary level controller uses remote signals from all generators to synthesize decoupling control signals that improve the local controllers’ performances. [12] uses multi-agent concepts to coordinate several supervisor PSS(SPSS) based on remote signals while exchanging information with local PSS to improve power systems stability. The SPSS uses a rule-based fuzzy-logic system and robust PSS to deal with uncertainties introduced by nonlinear terms and operating conditions.
However, in fact, the PMUs are gradually placed to form WAMS and not all generators in power system are equipped with the PMUs, so the states of some generators without PMUs can not be synchronously measured and transmitted, such as voltage, current, angle and frequency (speed), which result in the incompleteness of the wide-area information. Moreover, when wide-area or global signals are transmitted in WAMS, the time delay can vary from tens to several hundred milliseconds or more. In the Bonneville Power Administration (BPA) system, the latency of fiber optic digital communication has been reported as approximately 38 ms for one way, while latency using modems via microwave is over 80 ms [13]. Communication systems that entail satellites may have even longer delay. The delay of a signal feedback in a wide-area power system is usually considered to be in the order of 100ms [14]. If routing delay is included, and if a large number of signals are to be routed, there is a potential of experiencing long delays and variability in these delays.
Therefore, WAMS provide the incomplete time-delay wide-
1-4244-1298-6/07/$25.00 ©2007 IEEE.
area signals for power system stability control. How to design an incomplete time-delay wide-area signals based controller has become a major concern in the wide-area control. [15]–[17] only evaluate the effect of time delays on stabilizing control of wide-area power system without considering global signals’ incompleteness.
According to inverse system algorithm and linear matrix inequality(LMI) technique, this paper derives linear uncertain time-delay model of excitation and turbine control system for a turbogenerator under the center of inertia reference frame (COI), and presents a novel nonlinear robust integrated controller based on incomplete time-delay wide-area signals. 2
δ0=(∑Miδi)/MT
ω0=(∑Miωi)/MT
ni=1i=1
n
In this model, only the high pressure steam vessel is under the control of the main inlet control valves, and namely, the output active power of the intermediate or low pressure steam vessels is assumed to be invariable due to a large time constant of the reheater.
The above model (1) can be described by the following nonlinear state equations
The wide-area measurement information only includes the on-line measured center of rotor angle, the center of rotor speed and total unbalanced power. The main characteristic of the controller is that the impact of time delays and incompleteness on the controller performance is slight by robust H∞ control theory and simulation results.
The rest of this paper is organized as follows. In Section II the multimachine control model under the center of inertia reference frame(COI) is linearized by inverse system method. Section III derives a suitable linear uncertain time-delay model of control system. A nonlinear robust controller design methodology based on LMI technique is exposed in Section IV. Four-machine simulation results from validation tests are presented in Section V. The paper ends with a conclusion.
II. FEEDBACK LINEARIZATION BY INVERSE SYSTEM METHOD Under the center of inertia reference frame(COI), multi- machine interactions are equivalent to the interactions between each machine and the center of inertia, so n-machine power system becomes n independent two-machine power system, in which the center of inertia is regarded as a fictitious machine.
Moreover, each equivalent two-machine power system can be changed into a single-machine power system. Thus, the mathematical models for designing a controller become into n single-machine power system models, and the controller can be independently designed in each single-machine power system. A model for the ith generator with both excitation and
turbine control loops can be written as follows under the center
of inertia reference frame [18]–[21] θ&i
=ω%iMiω&%−PMi=PHi+CMLiPm0iei−iMPCOI−Diω%iT (1)
Td′0iE&qi′=−Eqi′−(xdi−x′di)Idi+EfiTHiP&Hi=−PHi+CHiPm0i+CHiUgiwhere θi=δi−δ0 ω%i=ωi−ω0 n
PCOI=∑(Pmi−Pei)
i=1
Pei=E′qi
Iqi−(xdi′−xqi)IdiIqi ⎧⎪⎨X&=f(X,U)=[f1(X),f2(X),f3(X,u1),f4(X,u2
)]T (2) ⎪⎩Y=h(X,U)=[yT
1,y2]
where X=[x1,x2,x3,xT4]=[θi,ω%i,Eqi′,PHi]Tis a state vector, Y=[y1,y2]T=[Vti,θTi]is an output vector, andU=[uT1,u2] =[Efi,Ugi]Tis the control vector.
According to the inverse system method[22], [23], new equation is obtained by making differential of each output component to time t
⎧⎪
y&1=Lf(X,U)h1(X)=α1(X)+β11(X)u1⎪y&⎪2=Lf(X,U)h2(X)=x2⎨y22=Lf(X,U)h⎪
&&2(X)=x&2 (3) ⎪y⎪
&&&
2=L3f(X,U)h2(X)=α2(X)−Dix&2/Mi+⎩ β21(X)u1+β22(X)u2where α121=V[xqiIqiI&qi−(Eqi−xdiIdi)x′diI&di−Eqi(Eqi−xdiIdi)](4.a) tiTd′0iα1M(P12=m0i−Pmi)−P&1COI−[I&
qiEqi
iTHiMTMi −I&qiIdi(xdi−xqi)−I&diIqi(x′di−xqi)−E (4.b)
qiIqiT]d′0i
β=1
11V(Eqi−xdiIdi) (4.c) tiTd′0i
β=−Iqi21M (4.d) iTd′0iβC22=Hi
M (4.e) iTHiThe control input of the original nonlinear system can be expressed from (3). That is
⎡−1⎢u1⎤⎡β11(⎧⎡v1⎤⎡α1(X)⎤⎪⎫⎣u⎥⎦=⎢X) 0⎤⎣β⎪
21(X)β(X)⎥⎦⎨⎪⎩⎢⎣v−2⎥⎦⎢⎣α2(X)⎥⎦⎬ (5) 222⎪⎭
where V=[v1,v2]T=[y&1,&&y&2+Dix
&2/Mi
]T Then the α-th order integral inverse system of the original
system is obtained as follows
&=f{X,W−1(X)[V−p(X)]}⎧X⎪ˆ:⎨ (6) Σα−1⎪⎩U=W(X)[V−p(X)]
3
&(t)=AZ(t)+BV(t)+BV(t−d) (11) Z0d
where
⎡0 0 0 0⎤⎡1 0 0⎤
⎢0 0 1 0⎥⎢0 0 0 ⎥
⎥; B=⎢⎥; A=⎢0
⎢0 0 0 1⎥⎢0 0 0⎥⎢⎥⎢⎥
⎣0 1 0⎦⎣0 0 0−Di/Mi⎦
⎡0 0 0⎤
where
⎡β(x) 0⎤
;p(X)=[α1(X)α2(X)]T W(x)=⎢11⎥
⎣β21(x) β22(x)⎦
The integral inverse system is connected to the original system (2) in series, thus, the state equation of the pseudo-linear system is easily written as
⎧⎪
V&ti=v1⎨⎪&&&
Di⎩
θi=v2−Mx&D2=v2−iMθ&& (7) iiiDefine
z1=Vti,z2=θi,z3=θ&i=ω%i,z4=θ&&i=ω&%i
the normal form of (7) is expressed as follows
⎡⎢z&1⎤⎡0 0 0 0⎤⎡z1⎤⎡1 0⎤⎢z&2⎥⎢⎢z&⎥⎢⎢⎥⎥⎢z⎥=0 0 1 0⎢0 0⎥v3⎥⎢0 0 0 1⎥⎢2⎢z⎥+⎢⎥⎡⎢1⎤3⎥⎢0 0⎥⎥ (8)
⎢⎣z&⎥⎢⎣v2⎦
4⎥⎦⎣0 0 0 −Di/M⎥⎢⎥⎢⎥
i⎦⎢⎣z4⎥⎦⎣0 1⎦
III. LINEAR UNCERTAIN TIME-DELAY MODEL CONSIDERING
INCOMPLETE WIDE-AREA INFORMATION The linear state equation of an excitation and turbine control system by feedback linearization has been expressed as (8). The vector form of (8) is described as
Z
&(t)=AZ(t)+BV(t) (9) The state equation (8) or (9) is derived without considering the wide-area information’s uncertainty included time delays and incompleteness. Because processing and communication delay can be very
important with today PMUs interconnected using fiber-optic or satellite based technology, the power system should be model- ed into differential-algebraic equations with time delays instead of ordinary differential-algebraic equations.
In order to simplify the analysis, it is assumed that all the feedback wide-area signals have the constant time delay d. Moreover, it can be seen from (3), (4), (5) that the imaginary control input V1 only feeds back the local measurement information and the control input V2 includes the wide-area measurement signals besides local information, so the state equation (7) is rewritten as follows
⎧⎪⎨U&ti(t)=v1
(t)⎪⎩θ&&&i(t)=v21(t)+v22(t−d)−Diθ&& (10) i/Mi
where v21 is the local imaginary control without time delays,
v22 represents the global imaginary control with time delay d, and v2(t)=v21(t)+v22(t−d). Define
z1=Vti,z2=θi,z3=θ&i=ω%i,z4=θ&&i=ω&%i
the normal form of (10) is expressed as
⎢B0 0 0 ⎥d=⎢
⎢0 0 0⎥Z=[Uti,θi,ω%i,ω&%i
]T;⎥; V=[v 1,v21,vT⎢22].⎣0 0 1⎥⎦
The state vector Z of (11) includes the on-line center of
rotor angle δ0and the center of rotor speed ω0, which are related to all generators. However, in fact, the PMUs are gradually placed to form WAMS and not all generators are equipped with the PMUs, so the states of some generators without PMUs can not be synchronously measured and transmitted, such as voltage, current, angle and frequency (speed). Thus, the center of rotor angle δ0and the center of rotor speed ω0 can’t be precisely calculated either, so we only obtain the incomplete wide-area measurement information. Assume that the on-line measured center of rotor angle and center of rotor speed are δ0 and ω0, respectively. Then we get
⎧⎪
δ0=δ0+∆δ0
⎨⎪ω0=ω0+∆ω0 (12) ⎩ω&0=ω&0+∆ω&0where ∆δ0,∆ω0,∆ω&0 are the measuring errors due to
wide-area information’s incompleteness.
Thus, the state variables θi,ω%i,ω&%i
can be expressed by (12) as follows
⎧⎪
θi=δi−δ0=δi−δ0−∆δ0=θi−∆δ0⎨⎪ω%i=ωi−ω0=ωi−ω0−∆ω0=ωi−∆ω0 (13) ⎩ω&%i=ω&i−ω&0=ω&i−ω&0−∆ω&0=ω&i−∆ω&0
whereθi,ωi,ω&i
are the state variables based on incomplete wide-area information.
So, the state vector Z of (11) is described as
Z(t)=Z(t)+∆Z0(t) (14)
where
Z=[Uti,θi,ωi,ω&i]T; ∆Z0=[0,∆δ0,∆ω0
,∆ω&0]T Substituting (14) into (11), the linear uncertain time-delay model of power system in consideration of the incomplete wide-area information is expressed ultimately as
Z&(t)=AZ(t)+CW(t)+B0
V(t)+Bd
V(t−d) (15) where
⎡⎢ 0 0⎤CW(t)=∆Z& 0 0⎥∆ω&
0(t)−A∆Z0(t)=⎢
⎢⎥⎡0⎤ ⎢ 0 0⎥⎢⎣∆ω&&0⎥⎢⎣D⎥⎦i/Mi1⎥⎦
IV. DESIGN OF NONLINEAR ROBUST CONTROLLER A. Robust H∞ Control for Linear Time-delay System Using LMI Approach
Consider a linear system with time-varying delays in state and control input (see [24], [25])
⎧⎪x
&(t)=Ax(t)+Adx[t−d1(t)]+B1w(t)+⎪
⎪
B2u(t)+Bdu[t−d2(t)]
⎨⎪z(t)=Cx(t)+Cdx[t−d1(t)]+D11w(t)+ (16) ⎪
D12u(t)+Ddu[t−d2(t)]
⎪⎩x(t)=0,t<0;x(0)=x0
where x∈Rn is the state vector, u∈Rp is the control input vector, w∈Rr
is the disturbance input vector, which belongs to Lz∈Rm
2[0,∞), and is controlled signal output. In here, time-varying delays are satisfied with
0≤di<∞, (d&it)≤mi
<1, i=1,2 (17) Given the scalar γ>0, the design of a robust H∞of the time-delay system (16) is to find a state feedback control
controller law u(t)=Kx(t)K∈Rp×n
(18) such that: 1) the closed-loop system is internally stable; 2) the closed-loop system guarantees, under zero initial conditions, the closed-loop transfer function from wto z ||Gs) ||
zw(∞<γ Theorem [24]: Consider the continuous time-delay system (16). For a given positive constant γ, if there exist positive-
definite matrices Q, S1, S2, and a matrix M such that
⎡T
⎢U1 B1 U2 M Q⎤⎢T⎢B1 －γ2I DT⎥11 0 0⎥⎢UT2 D11 U3 0 0⎥⎥<0 (19) ⎢⎢M 0 0 －S2 0⎥⎢⎣Q 0 0 0 －S⎥1⎥⎦
holds for the time delays (17), then the time-delay system (16) is quadratically stable with a H∞ norm bound γ by the controller (18). In here, some terms are defined as follows:
UT+AQ+MTBT
1=QA2+B2M+
(1−m%)−1AT
+(1−m−1T1dS1Ad%2)BdS2Bd
UTT+(1−m%1T
2=MTD12+QC1)−AdS1Cd+
(1−m
%−1
T
2)BdS2Dd
；
U3=−I+(1−m%1CT+(1−mT
1)−CdS1d%2)−1DdS2Dd;
M=KP−1; Q=P−1; S−1i=Ri,i=1,2
Equation (19) is a LMI form in terms of Q, M, S1 and S2. Therefore, the state feedback gain matrix K can be calculated from M=KP−1 after finding the LMI solutions, Q, M, S1 and S2, from (19).
4
B. Design of Robust Feedback Control Law
According to the above theorem (19), the LMI of the linear uncertain time-delay system (15) can be transformed to the following form:
⎡ UT⎢1 C Q M Q⎤⎢T2
⎥⎢ C －γI 0 0 0⎥
⎢ Q 0 －I 0 0⎥<⎢⎢ M 0 0 －S⎥0 (20) 2 0⎥⎢⎣ Q 0 0 0 －S1⎥⎥⎦where
U=QAT+AQ+MTBTM+BT
10+B0dS2Bd
; Q>0;S1>0;S2>0
The LMI (20) is easy to solve using LMI toolbox in Matlab
so that the feedback gain matrix K can be obtained. Thus, the
feedback control law of the linear uncertain time-delay system (15) is expressed as follows V*=⎡⎢v1*⎤⎣v2*⎥⎦=⎡⎢v1
*⎤
⎣v21*+v⎥=K(Z-Z0) (21) 22*⎦where Z0 is the pre-disturbance initial value.
As far as the continuous control is concerned, the state
vector Zshould return to its post-disturbance value when power system network topology varies. It is a complicated problem to coordinate the relation between control measures
and the post-disturbance stable equilibrium point. Therefore the control object in this paper has to be that the state vector Z returns to its pre-disturbance stable equilibrium point.
Equation (21) represents a feedback control law just for the linear system (15), which must be transformed for the original
nonlinear system. Substituting (21) into (5), the robust H∞ feedback control law U* for the original nonlinear system (2)
can be ultimately achieved as follows
⎧⎪⎪
u1*=v1*−α1⎨
β11⎪v*⎪αβ(v*α) (22) ⎩
u*=2
−2−12
β−21122β11β22Remark 1: The robust control law given in (22) can be decomposed into a local term (which depends only on locally measurable variables) and a global term (which depends on remote signals):
U*=Ul+Ug (23)
There are the center of rotor angleδ0, the center of rotor speed ω0and the total unbalanced powerPCOI in the global term Ug. With the wide application of synchronized phase measurement unit in power system, these global information can be easily obtained by measuring and calculating.
In the local term Ul, some variables including Eqi,Idi,Iqi are difficult to measure. So these variables must be transformed into measurable variables.
When the resistances of stator windings are negligible, the voltage drop in the interior of a generator is approximated by
Eqi−Vti=
Qeixdi (24.a) Vti
5
TABLE I
SYNCHRONOUS MACHINE PARAMETERS
1 2 3 4 Sn 209.5MVA 209.5MVA 138.8MVA 247.5MVA Vn 18kV 18kV 13.8kV 16.5kV ′5.2s 5.2s 4.8s 6s Td0
TH 0.35s 0.35s 0.3s 0.41s xd 1.0836 1.0836 1.552 0.475 xq 1.0836 1.0836 1.552 0.475
0.1198 0.1198 0.1813 0.0608 x′d
50Hz 50Hz 50Hz 50Hz ƒ
M 6.4/2πƒ 6.4/2πƒ 3.01/2πƒ 23.64/2πƒ As far as a turbogenerator is concerned, the effect of
′≈xqi′. transient saliency can be negligible, and namely, xdiThus, we have
Pei≈EqiIqi (24.b)
Moreover
22 (24.c) Idi=Iti−Iqiare transformed into measurable Therefore, Eqi,Idi,Iqi
variables Qei,Pei,Iti by (24.a), (24.b) and (24.c). It should be
noted that the above assumption and approximation are practical and justified in view of the physical features of the turbogenerator.
Remark 2: Note that feedback control law u1* and u2* depend on derivatives ofθi,Iqi,IdiandPCOI. An approximation of the differentiation operation (which filters out very high frequencies) is very often appropriate. To this end, we used the following transfer function:
(s)=2ω2Hrss2+3ω,ωr
=100πrad/s (25) rs+2ω2r5
Remark 3: Since all practical limitations on excitation and valve input have not been taken into account during the design, two scaling gains µ1,µ2and saturation are added to the controller outputs to give
u*=µv1*−α1
11β 0≤u1*≤6
11
(26) u⎡v*−α2β21(v1*−α12*=µ2⎢2⎣β−)⎤
⎥⎦
|u2*|≤122β11β22The parameters µ1,µ2 range from 0 to 1 and can be tuned by the designer considering more problematic contingencies.
V. RESULTS OF CASE STUDIES
A. System Description
A 4-machine WSCC system in Fig. 1 is used to illustrate the nonlinear robust integrated controller (NRIC) described above, and the parameters related to the studied system are listed in Table I and Table II. Moreover, comparisons are made with several different types of integrated controllers.
561G127G2G3PL193810PL211PL34G4Fig. 1. Four-machine WSCC power system
D 0.0041s 0.0041s 0.0019s 0.015s
TABLE II
TRANSFORMERS, LINES AND LOADS PARAMETERS
Bus Bus Resist. React. Suscept. 1 6 0 0.06 0 2 7 0 0.0625 0 3 9 0 0.0586 0 4 11 0 0.0576 0 5 6 0.0597 0.3024 0.0753 5 7 0.0119 0.1008 0.0251 5 9 0.085 0.072 0.0179 7 8 0.032 0.161 0.0367 9 10 0.039 0.17 0.043 8 11 0.01 0.085 0.0211 10 11 0.017 0.092 0.019 5 PL1=2.2+j1.0 8 PL2=2.0+j0.55 10 PL3=2.1+j0.4
Three cases are studied as follows:
Case 1: The generator is equipped with the conventional Proportion-Integral-Differential controller (PIDC);
Case 2: The generator is equipped with the nonlinear decentralized integrated controller (NDIC);
Case 3: The generator is equipped with the proposed nonlinear robust integrated controller (NRIC). GGGGB. All Generators Equipped with PMU without Considering Time Delays
A three-phase short circuit occurs on one of the lines between buses 5 and 6 near bus 5 at 0.0 second for 0.2s, then the faulted line is opened at 0.2s. When all generators are equipped with the controllers, the simulation results are shown in Figs. 2, 3 and 4, which indicate generator G1 rotor angle responses of the three control configurations under different reference frames respectively.
In Fig. 2, generator G4 (with 59.9 percent of total inertia constant) is taken as a reference machine. It can be seen from Fig. 2 that NRIC quickly damps down generator G1 oscillation at 1s in the first swing and the oscillatory peak value is only 65°. This shows that NRIC works much more effectively than PIDC or NDIC for damping system oscillations and decreasing the settling time, the swing times.
The center of inertia is taken as a reference frame in Fig. 3. Since the trend of the rotor angle curves in Fig. 3 is almost the same with that in Fig. 2, the rotor angles can be exactly figured under the center of inertia.
100with PIDCwith NDIC)e80with NRICreged(e60lgna ro40tor 1G200012time(second)3456Fig. 2. Dynamic responses of rotor angles of generator G1 relative to generator G4 under different control 100with PIDCwith NDIC)e80with NRICreged(60elgna ro40tor 1G200012time(second)3456Fig. 3. Dynamic responses of rotor angles of generator G1 relative to the center of inertia (COI) under different control 120with PIDC)100with NDICewith NRICrege80d(elgn60a roto40r 1G200012time(second)3456Fig. 4. Dynamic responses of rotor angles of generator G1 relative to generator G3 under different control
However, as for a reference frame in which some machine is taken as reference machine, the rotor angles can be exactly shown only by choosing the proper reference machine. Otherwise, a false conclusion is likely to be drawn. For example, if generator G3 (with 7.6 percent of total inertia constant) is taken as reference machine in Fig. 4, then we will come to a conclusion that the rotor angle curves under nonlinear decentralized integrated control is nearly the same as that under nonlinear robust integrated control after 1.6s. But in fact this conclusion is wrong.
Fig. 5 shows the time response of terminal voltage of generator G1 with different controller during fault and post-fault regimes. It can be noted that the terminal voltage of generator G1 with NRIC settles down more quickly than that with PIDC or NDIC.
6
1.2With NDIC 1.11).u.p(0.9With PIDC With NRIC Vt0.80.70.6012time(second)3456Fig. 5. Dynamic responses of terminal voltage of generator G1 with different controller 0.450.4 0.394 s)dn0.345 so0.35ces0.3 (0.273 sem0.25i0.224 st ra0.2 elc l0.15aciti0.1 rc0.050 Without Controller With PIDCWith NDICWith NRICFig. 6. Critical clear time of 4-machine system with different controller
)e2.5er|∆δ14|ged(2|∆δ24|el|∆δ34|gna r1.5otro evi1taler fo0.5 srorre0012time(second)3456Fig. 7. Absolute errors of relative rotor angles with NRIC under three-phase short circuit
The critical clear time of the 4-machine system under different control is given in Fig. 6 when a three-phase short circuit occurs on one of the lines between buses 5 and 6 near bus 5. It can be calculated that the critical clear time is increased by 75.9 percent under the proposed nonlinear robust integrated control. Therefore, it is obvious that NRIC greatly improves transient stability of power system.
It is noted that in all simulations unmeasured variables
Eqi,Idi,Iqi
are computed approximately from measurable variables Qei,Pei,Iti by (24.a), (24.b) and (24.c). The errors of
the relative rotor angles owing to the approximation are plotted in Fig. 7 when the above three-phase short circuit occurs. It can be seen from Fig. 7 that all absolute errors are within 2.5°. And thereby the approximation is reasonable and practical.
80)ere60ge(delgnDelay= 200msa40 rDelay= 400mstoor G120Delay= 40msDelay= 20ms0024time(second)681012Fig. 8. Dynamic responses of rotor angles of generator G1 relative to the center of inertia with NRIC for different time delays 1.2Delay= 400ms1.11).Delay= 200msDelay= 20msDelay= 40msu.(p0.9tV0.80.70.6024time(second)681012Fig. 9. Dynamic responses of terminal voltage of generator G1 with NRIC for different time delays
C. All Generators Except the Third Generator Equipped with PMU Considering Time Delays
It is assumed that only the third generator (with 15.9 percent of entire rated power capability) is not equipped with a PMU. When other generators are equipped with NRIC, the time responses of rotor angle and terminal voltage of generator G1 for different time delays under the above three-phase short circuit are shown in Figs. 8 and 9 respectively.
Because a PMU is not installed on the generator G3, the states of generator G3 can’t be synchronously measured and transmitted. Therefore, the center of rotor angle, the center of rotor speed and total unbalanced power can’t be precisely calculated either, and we only obtain the incomplete wide-area measurement signals. Moreover, the processing and communi- cation delay is also unavoidable. These uncertain factors will have an effect on the wide-area information based controller performance. However, as the feedback wide-area signals’ uncertainty including time delays and incompleteness is taken into account, it can be seen from Figs. 8 and 9 that NRIC still ensures sufficient damping of system oscillations and keeps the system stable, which demonstrates that the designed NRIC using inverse algorithm and LMI technique is not sensitive to time delays, since four curves corresponding to different time delays are almost same in Fig.8 or 9.
On the other hand, wide-area signals loss may be not critical with NRIC, since the local control that depends only on locally measurable variables is still present under the exceptional circumstances, thus serving as a fully working backup control.
7
80θOIC60θ1 toθ2 ev40θ34italer 20selgna0r otro-20-40024time(second)681012Fig. 10. Dynamic responses of rotor angles relative to COI for 200-ms time delay when the third generator is without any controller, the others with NRIC 80θOI60Cθ1 oθ2t e40vθ34italer20 selgn0a rot-20ro-40024time(second)681012Fig. 11. Dynamic responses of rotor angles relative to COI for 200-ms time delay when the third generator is with PIDC, the others with NRIC
D. The Third Generator Equipped with PIDC and Others with NRIC
When the third generator is equipped with neither PMU nor controller, and other generators are with NRIC, the rotor angle responses of each generator for 200ms time delays under the above three-phase short circuit are given in Fig. 10.
As shown in Fig. 10, though generators G1, G2, G4 settle down at 2.5s, the generator G3 slowly damps down multi- swing oscillation until 10s.
In order to ensure sufficient damping of oscillations, the generator G3 can be equipped with a PIDC for all practical purposes. Fig. 11 shows the rotor angle responses of each generator for 200ms time delays with combining NRIC and PIDC. It can be seen from Fig. 11 that not only does the generator G3 settles down quickly, but also the others have better performance measures such as the settling time and the swing times.
In the above simulation cases, only the case with 200ms time delays is considered. The other three conditions in the cited 0-400ms delays range were also tested and similar results were obtained.
VI. CONCLUSION
A nonlinear robust integrated controller based on incomplete time-delay wide-area measurement information is presented to improve transient stability and voltage stability of power system. This controller is designed under the derived linear uncertain time-delay model of multimachine control system,
using inverse system algorithm and linear matrix inequality (LMI) technique. It has been verified that the impact of time delays and uncertainty on the controller performance is slight by robust H∞ control theory and four-machine system simulation results. Moreover, the simulation results also confirm that the system with the proposed controller has less settling time, swing times, oscillatory peak value and more critical clear time and better voltage stability performance than that with the conventional PID controller or nonlinear decentralized integrated controller.
As far as some generators without PMUs are concerned, they can be equipped with PID controllers for all practical 8
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VIII. BIOGRAPHIES
Guangliang Yu was born in Jilin province, China, on Nov 9, 1978. He received the B.Eng. degree in information engineering and the M.Eng. degree in electrical engineering from Northeast Dianli University in 2001 and 2004, respectively. He is currently pursuing the Ph.D. degree at Xi’an Jiaotong University. His research interests include nonlinear control, robust control, and their applications to power system based on WAMS.
Baohui Zhang (SM’99) was born in Hebei Province, China, in 1953. He received the M.Eng. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University, Xian, China, in 1982 and 1988, respectively. He has been a Professor in the Electrical Engineering Department at Xi’an Jiaotong University since 1992. His research interest is in the area of power system analysis, control, communication, and protection.
Huan Xie was born in Hunan province, China, on June 16, 1979. He received the B.Eng. and the M.Eng. degrees in electrical engineering from Hehai University in 2001 and 2004, respectively. He is presently completing the requirements for the PhD degree at Xi’an Jiaotong University. His research interests are power system stability control, WAMS and its application in power systems.
Chenggen Wang was born in Anhui province, China, on Nov 19, 1981. He received the B.Eng. degree in automation engineering and the M.Eng. degree in electrical engineering from Northeast Dianli University in 2003 and 2006, respectively. He is currently pursuing the Ph.D. degree at Xi’an Jiaotong University. His research interests are nonlinear control, robust control, and their applications to power system.
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