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Power System Stabilizer Design Based on Structured Singular Values
Copyright © IFAC Control of Power Systems and Power Plants, Beijing, China, 1997
POWER SYSTEM STABILIZER DESIGN BASED STRUCTURED SINGULAR VALUES
ON
T. C. YANG* and J. H. ZHANG** ·School of Engineering, University of Sussex, Brighton, BN1 9QT, UJ( ··North China University of Electric Power (Beijing), Qinghe, Beijing, 100085, P.R. China
Abstract. In this paper, power system stabilizer design for multi-machine power systems is translated into an equivalent problem of decentralized controller design for Multi-Input Multi-Output (MIMO) control systems. Subject to a condition based on the structured singular values, each stabilizer can be designed independently. The robust stability condition for power systems with stabilizers on can be easily stated as to achieve a sufficient interaction margine (Yang and Cimen, 1996) and a sufficient gain and phase margine during each independent design. Within this general framework, the conventional stabilizer design methodology based on the concept of synchronous and damping torques is used to decide the design details of each stabilizer. The suggested design method is applied to a model of a practical 10 machine power system. Copyright © 1998 IFAC Key Words. Power system stabilizers, multi-machine power systems, structured singular values.
sign for multi-machine power system can be conveniently formulated as a large-scale system decentralized stabilization problem. However, probably not enough attention has been paid to the fact that the decentralized stabilizer design problem can be translated into a problem of diagonal feedback controller design for Multi-Input MultiOutput (MIMO) system; and some MIMO system design methods, for example, the sequential loop closure method (Yang, 1996), can be applied to the PSS design. In this paper a new design method based on the translated MIMO system model is presented. In this method each PSS can be designed independently on its own "loop" subject to a condition derived from the structured singular values. The design method and its application to a model of a practical 10 machine 39 bus-bar power system are presented in Section 2 and Section 3 respectively.
1. INTRODUCTION
Power System Stabilizer (PSS) units have long been regarded as an effective way to enhance the damping of ElectroMechanical Oscillations (EMO) in power systems. The linearized models can be used for the study and a conventional power system stabilizer can be 'considered as a single-input single-output feedback controller installed on a generation set (a machine). Since the pioneering work of deMello and Con cordia in 1969, control engineers, as well as power system engineers, have showed great interest and made significant contributions in PSS design and applications. Most PSS design methods developed in recent years are based on well-developed modern control theory. These include: eigenvalue assignment, optimal control, self tuning and adaptive control, variable structure control, internal model control, high gain design, Hoc based design, rulebased and neural network based control, etc. (See for example, the references listed in (Yang, 1996)).
2. A NEW PSS DESIGN METHOD
State space models are commonly used for the analytical study of multi-machine power systems. The design of each individual stabilizer in a multimachine system must be coordinated due to the interactions between machines and controllers. The minimum requirement for the PSS design is that when the open-loop system is closed by the stabilizers, the dampings of the EMO modes concerned should be improved to a satisfactory degree and all system eigenvalues should have negative real parts. It is well known that the PSS de-
2.1 Translate Into An Equivalent Design Problem
In multi-machine power system steady-state stability analysis, a number of state variables is used for the dynamics of each generating set (machine). The dynamics of transmission lines and loads are neglected but their static characteristics provide necessary information for the modelling of each machine and the links between machines. Each machine can be considered as a subsystem in a large-scale system and PS Ss installed on machines
371
as local controllers. A state space model with the consideration for the speed input PSS design can be built up as:'
x yO
+
Ax
=
yes)
G*(s) = C(sI - A)-1B
(2)
COx
From the system stability point of view, an equivalent block diagram (Figure 2) can be drawn to represent the MIMO system in Figure 1, where G(s) [gij(S)]i,j=1 ,2,... 1 -G*(s) . Both systems have the same return ratios.
=
=
y
In Grosdidier and Morar's paper, a transfer function G(s) = [g;j(S)]i ,j=1 ,2,... 1 for a I by I MIMO plant is decomposed into:
=
G(s) = G(s) + G(s)
(7)
=
where G(s) diag[gii(S)]i=1,2 ,... 1 is a diagonal matrix; all diagonal elements in G( s) are zeros and off-diagonal elements in G(s) are equal to those in G(s) .
(3) (4)
=
where u [U1 U2 .. · UI f ; y [Y1 Y2 .. · YI f ; Band Care n by 1 and I by n matrixes respectively. It is worthwhile pointing out that, although the matrixes (vectors) B , C (u, y) in (3)(4) now have reduced dimensions compared with their equivalents in (1)-(2), x and A remain unchanged. The whole system dynamics is therefore unaffected for the design concerned.
=
2.2 Decentralized Controller Design for MIMO systems
After the PSS allocation, those terms in BO u O CO . and yO corresponding to the machines where a PSS has not been allocated can be deleted. This leads to: Bu
=
If all gij (s) (i =1= j) in G( s) were equal to zero, then each stabilizer could be designed independently just as if it were in a single machine system as shown in Figure 3. However, since gij(S) (i =1= j) are not zeros, the following question must be answered . If each fi(s) PSSi(S) (i 1,2, ... /) is designed to form a stable closed-loop system as shown in Figure 3, what are the additional conditions which can guarantee that the global system of Figure 2 is stable and all dampings ofthe EMO modes concerned are improved to a satisfactory degree? The answer to this question is discussed in the next subsection based on the theorem given by Grosdidier and Morar (1986).
A list of I EMO modes whose dampings are to be improved can be found by eigenstructure analysis. Their total number 1 is normally less than the number of machines N in the system. The general practice is to installl stabilizers. Each PSS is allocated to a most suitable machine and each PSS is used mainly to improve the damping of a particular EMO mode. The allocation problem has been studied extensively in the literature, for example in (Yang et al. , 1987), and will not be addressed here. This paper focuses on stabilizer design when a PSS allocation has already been given. The PSS allocation method given in (Yang et al., 1987) is adopted in this paper. This is not only because of its conceptual and computational simplicity, but also because an important theoretical issue, "fixed modes", has been properly addressed in (Yang et al. , 1987) without using any complicated mathematics.
+
(6)
The design of I decentralized local stabilizers now becomes the design of a I by I diagonal matrix F(s) = diag[fi(s)]i=1 ,2,... 1 • Ii(s) is equal to the transfer function of the i-th stabilizer: PSSi(S) as shown in Figure 1, where a negative sign is also put before G*(s) in order to form a negative feedback system.
where x is an n by 1 state vector ; A is an n by n matrix with N diagonal blocks for the dynamics of N machines and off-diagonal blocks for the dynamics of links and interactions; uO is an N by 1 vector [u~ u~ .. . uCJv f and u? corresponds to the i-th machine voltage regulator terminal to which the output of a possible PSS will apply; yO is an N by 1 vector [y~ y~ ... yCJv )T and yp corresponds to the i-th machine shaft speed which can be used as the input to a possible PSS ; BO and CO are appropriated n by Nand N by n matrixes respectively.
Ax Cx
G*(s) = [g:j(S)kj=1,2, ... 1 (5)
can be calculated as:
(1)
BOuO
= G*(s)U(s);
Using the notations:
E(s) = G(s) G- 1 (s)
(8)
B(s) = G(s)F(s)(I + G(s)F(s))-l = diag[hi(s») (9) H(s) = G(s)F(s)(I+ G(s)F(s))-l
(10)
where F(s) = diag[li(s)]i=1,2 ,... 1 is a diagonal transfer function for a decentralized controller in Figure 2; B(s) or H(s) is a closed-loop transfer function matrix for a feedback system consisting of F(s) and G(s) , or F(s) and G(s) respectively, Grosdidier and Morar have proved the following
An I by I transfer function matrix G*(s) linking U(s) = [ U1(S) U2(S) ... UI(S) f and yes) = [ Y1 (s) Y2 (S) .. . YI (S) ]T:
372
theorem:
lowing comments.
The closed-loop system H(s) is stable if:
(1) As pointed by Skogestad and Morar (1986), for ordinary MIMO systems, if G(s) is unstable, condition (c-I) is generally not satisfied. However, if the model G(s) is obtained by the translation from a large scale system , the condition (c-I) is generally satisfied even if G(s) is unstable. The poles of G(s) are given by the roots of det[sI - A] and the elements of G( s) , the diagonal elements m G(s) , can be calculated as: -Ci adj(sI - A] b; 9i;(S) = det[sI _ A] (11)
(c-l) G(s) and G(s) have the same number of Right Half Plane (RHP) poles; (c-2) H(s) is stable; and V w (c-3)* umaz(H(jw)) < j.t-l(E(jw)) where O"maz denotes the maximum singular value of; and j.t denotes Doyle's structured singular value in respect to the decentralized controller structure of F(s) . Since H is a diagonal matrix, condition (c-3)* can be replaced by:
If there is no the same pole-zero cancellation in
Ilhi(jw)11 where 11 11 (c-3)
all 9i;(S), the poles of G(s) are also given by the roots of det[sI - A], the condition (c-I) is naturally satisfied. Even there are some cancellations, so long as they are not at RHP, the condition (c-I) is still satisfied.
denotes the magnitude of.
This theorem gives sufficient conditions for the system H(s) to be stable if the controller is designed based on the fully non-interactive model G(s) , i.e. each fi(S) is designed independently on a Single-Input Single-Output (SISO) model 9ii(S). In particular, condition (c-3) says that the magnitude of the frequency response of SISO closed-loop
.
transfer functIon
hi (S ) =
(2) The stability condition (c-4) is given for the nominal plant G(s) . If power system operating condition changes, the plant model will also change. It is perhaps not possible to establish a clear relationship between the change of power system operating conditions and the change ofvalues involved in condition (c-4). However, generally speaking, the robust stability can be achieved if:
fi(S)9ii(S) • () () must be S 9ii S
1 + Ji
less than a scalar frequency dependent function j.t-l(E(jw)) . Grosdidier and Morar (1986) have also proved that , although (c-3) is a sufficient condition and therefore may have some conservativeness , compared with the other conditions developed from control theory for the independent decoupled design, for example the diagonal dominant condition and the generalised diagonal dominant condition, (c-3) gives the tightest restrictive band and is the least conservative. Since the same restriction j.t-I (E(jw)) is applied to all hie s) in condition (c-3), a modification on this condition can be made to provide more flexibility and therefore to further reduce the possible cbnservativeness caused by the inflexibility in condition (c-3):
(r-I) Condition (c-4) is satisfied with a sufficient margin. This can be checked by plotting IIhi(jw)1I and j.t-l(E(jw)W(jw)) w;(jw) on the same graph and an interaction margin for loop i can be defined as the shortest vertical distance between the two curves. (r-2) There are sufficient gain and phase margins in each SISO loop for the stability. This can also be checked by a Nyquist plot of f; (jW)9ii (jw). (3) The main objective of the PSS design is to improve the dampings of the EMO modes concerned . It is known that a poorly damped oscillation is also characterised by a big resonant peak in the system frequency response. This implies that, for a good design , there should be no big resonant peak in 11 h; (jw )11. In addition, general knowledge of feedback system dynamics and the MIMO system examples given by Grosdidier and Morar (1986) show that increasing the interaction, gain and phase margins as stated in (r-I) and (r-2) will lead to more damped response. It is therefore believed that the required dampings will be provided if the design leads to sufficient interaction and stability margins, and there is no significant resonant peak in 11 hi (jw ) 11 . Furthermore, as showed in the next subsection, the objective of improving the dampings is the main consideration in the design of I;(s), which can be based on the
( c-4)*
O"maz(W-1(jw)H(jw)) < j.t-l(E(jw)W(jw)) Vw whereW(s) = diag[w;(s)];::;:1,2 ,... 1 is a properly chosen diagonal weighting function matrix. (c-4)* can also be replaced by: (c-4) Ilh;(jw)w;l(jw)1I < j.t-l(E(jw)W(jw)) ; Vw (i= 1,2, ... /) From (c-4) , although j.t-l(E(jw)(W(jw)) is still the same for all SISO loops, the restrictions on Ilh;(jw)1I are now different. In fact , (c-3) is a special case of (c-4) with W = I . Before applying the above results developed by Grosdidier and Morar (1986) to our decentralized PSS design problem, it is useful to make the fol-
373
concept of synchronous and damping torques originally developed for the conventional PSS design.
bilizer design and power system voltage stability. In this system , Machine 10 can be considered as a reference machine (Busbar 39 considered as an infinity bus) and its dynamics are not required in the modelling. The dynamics of each of the other 9 machines can be modelled by 6 state variables for the study of PSS design. Therefore, the model established is a large scale system with 9 subsystems and 54 state variables.
(4) Since E(s) = G(s) <;-1(s) , if 9;;(S) (i f j) is "small" in relative to 9;; (s ) (i = j) , E( s) will also be "small" . J.L(E( s)) can therefore be considered as a measure of the interactions for the plant G(s) .
2. 3 PSS Desi9n for Each SISO Loop
The system operating condition given in (Pai, 1989) with a load-flow result was considered as the "worst situation" for the steady-state stability. A 54-state small signal model based on this operating condition ca~ be formed and is considered as a nominal model in this paper. Eigenstructure analysis reveals that the system is stable but there are four poorly damped EMO modes with their damping ratios no more than 0.03 . The eigenvalues for these modes are: (1) -0.1895±i7.4036 (2) -0 .1198±i6.9023 (3) -0 .2662±i8.8400 (4) -0 .1132±i7.8415
Based on the theoretical background and discussions in Section 2.2, PSS can be designed independently so long as sufficient margins stated in (r-1) and (r-2) are obtained. Despite much progress in applying modern control technology to the PSS design with many different suggested PSS structures, many utilities, according to Tse and Tso (1993) , still prefer a conventional PSS structure. The reasons may include ease of on-line tuning and lack of assurance on the stability related to some adaptive or variable structure schemes. The basic form of the transfer function for a speed-input conventional PSS is:
Tw s 1 + Tl s 1 + T3S PSS(s) = K$l + Tw s 1 + T 2 s 1 + T4 s
By applying the PSS allocation method given in (Yang et al. , 1987), it is decided to install four PSSs on machine 2, 3, 4, and 6 to improve the dampings of the above modes (PSS on machine 2 for mode 1: -0 .1895±i7.4036; etc.). These stabilizers (loops) are numbered 1, 2, 3, and 4 respectively. After the PSS allocation , a four by four transfer function matrix G( s) can be obtained.
(12)
where K . is the PSS gain; Tw is the time constant for the wash-out stage which is to prevent steadystate voltage shift ; Tt , T 2 , T3 and T4 are time constants for two phase-lead stages. (In some cases more than two phase-lead stages are required and a filter and an output limiter will be added to PS S( s) in practical applications.)
Figure 6 gives the Bode plot for 911 (jw). The other three Bode plots for 9n(jw), 933(jW) and 944(jW) have a very similar shape and are not given here. It can been seen from Figure 6 that there is a resonance peak in the magnitude response and also a sharp drop of the phase angle around the frequency range of the poorly damped EMO modes. These are due to the characteristics of the EMO modes concerned. The magnitude of the frequency response is quite small since, in the per unit system, the " ratio" between the output (machine speed deviation) and the input (machine reference voltage change) is a very small value.
By using the conventional PSS structure of equation (12) and extending the basic concepts in the conventional method, each stabilizer can be designed in the same way as that in (Larsen and Swann, 1981) . To limit the length of the paper, the details are not repeated here. With the help of the concepts and methods given in the literatures, all PSS parameters can be finally tuned during commissioning.
3. DESIGN EXAMPLE The similarity in the shape of Bode plots for 9ii {jW) (i 1,2,3,4) indicates that similar PSS parameters may be used for all stabilizers and there is no need to introduce a weighting function W(s) to give different restrictions on hieS) with regard to condition (c-4) for the global stability. The time constants in the four stabilizers are simply set as follows:
=
A 10 machine 39 bus and 46 line power system (Figure 5) is used in this paper as a test system. This test system was developed as a representative of the power system in the Northeastern United States and is also called the NEW ENGLAND system. This system has been used for the study of power system stabilities in a number of publications (See references in (Yang, 1996)). These include dynamic equivalents for transient stability, direct analysis of transient stability, energy function analysis for power system stability, steady-state stability analysis, power system sta-
(1) The washout stage time constant is set to Tw = 10 for all four stabilizers ; (2) The two phase lead stages are identical for all
374
four stabilizers (in T4 ) and: Tl = T3 = 0.534, lizer 1; Tl = T3 = 0.573, lizer 2; Tl = T3 = 0.447, lizer 3; Tl = T3 = 0.504, lizer 4.
each stabilizer, Tl = T3; T2
=
T2 = T4 = 0.0534
for stabi-
T2 = T4 = 0.0573
for stabi-
T2 = T4 = 0.0447
for stabi-
7'4 = 0.0504
for stabi-
T2 =
sufficient and there is no resonant peak in all h;(jw) , we can believe that (1) sufficient dampings have been provided by the stabilizers for the EMO modes concerned, and (2) the system robust stability is satisfactory. The simulation results for the m-th (m=2, 3, 4, 6) machine speed-deviation b.w time-responses, when a small step disturbance of 0.05pu (per unit) is applied to the m-th machine voltage regulator terminal, for the system without PSS and with the four PSS on are given in Figure 10 as full-line curves and dotted-line curves respectively.
Based on the system frequency responses, it was found that /{, = 40 is a suitable gain for PS S 1 ( s) . This gives an interaction margin of 7.98dB, a gain margin of 9.81dB and a phase margin of 61.2 0 for loop 1. The Bode plots for Jl-l(E(jw) and Ilhl(jw)11, and the Nyquist plots for PSS l (jW)91l(jW) are given in Figure 7,8 and 9 respectively.
4. CONCLUSIONS A new power system stabilizer design method based on the structured singular values is suggested in this paper for multi-machine power systems. The design method is applied to a model of a practical 10 machine power system.
The plot of Jl- l (E(jw» in Figure 7 shows that there is a minimum value of 7.96dB around the frequency of 7.0 rad/sec. The magnitude of Ilhl(jw)11 at that frequency is about -0.2dB. There is no resonant peak in Ilhl(jw)ll.
5. REFERENCES Grosdidier, G. and M. Morar (1986). Interaction measures for system under decentralized control. Automatica, 22, 309-319. Larsen, E.V. and D.A. Swann (1981) . Applying power system stabilizers (three parts). IEEE Trans., PAS-lOO, 3017-3046. Pai, M.A. (1989) . Energy function analysis for power system stability. Kluwer Academic Publishers. Tse, C.T. and S.K. Tso (1993) . Refinement of conventional PSS design in multimachine system by model analysis. IEEE Trans ., PS-S, 598605. Yang, T.C . (1996) Applying a sequential loopclosure method to power system stabilizer design. Control Engineering Practice, 10, 13711380. Yang, T.C. and H. Cimen (1996). Applying structured singular values to robust decentralized power system load frequency control, Proc. IEEE Int. Conf on Industrial Technology, 1020-1025. Yang, T.C. , N. Munro and A. Brameller (1987). A new decentralised stabilisation method with application to power system stabiliser design for multimachine systems. Int. 1. Electrical Power and Energy Systems, 9, 206-216.
The interaction margin can also be checked from the Nyquist plot in Figure 8, where a small M circle for 8dB is also plotted. Although it appears that the circle is very near to the Nyquist plot, the points around w = 7.0 rad/ sec on the plot are far away from the circle which indicates a sufficient interaction margin. The part of the Nyquist plot near to the critical point -1 is enlarged in Figure 9 to assess the loop gain and phase margins. Similarly, the gains 30, 40 , 40 are chosen for stabilizer 2, 3, 4. Since the EMO frequency concerned for stabilizer 2 design Wo = 6.90 rad/ sec is nearer to the frequency where Jl-l(E(jw» has a minimum value, stabilizer 2 gain is set at a lower value of 30 for a sufficient interaction margin. The margins achieved for each loop are given as follows: interaction margin: 6.81dB, gain margin : 11.7dB, phase margin: 66.4 0 for loop 2; interaction margin: 9.04dB, gain margin : 10.2dB, phase margin: 64.20 for loop 3; interaction margin: 8.75dB, gain margin : 9.84dB , phase margin: 63 .70 for loop 4; The shapes of Bode plots for hj(jw) (i = 2, 3,4) and Nyquist plots for PSS;(jW)9;;(jw) (i = 2,3,4) are similar to those in Figure 7, 8 and 9. These plots are therefore not given here. According to the theorem given in Section 2.2, although all stabilizers are designed independently, the global closed-loop system is stable when all PSSs are connected. Since all margins are quite
375
- G *(5) .
--
~---
.
.
. . .. .
,
. .
~ '~:':' t~' -r'T~ ; '- -' - ~""' :': ';' ~ ~ ..
-~- -::-..: ~~~-,~~~~~~~,~oo~~~~~~'~'~'~;~o,--~--~~~~,O'
,,
---------
Frequency (radlsec)
F (s)
Fig. 7. Bode Plot of /l-l(E(jw)) (solid line) and IIhJ(jw)11 (dashed line)
Fig. 1. Multivariable System Design for PSSs 15r---~-----T-----r----~----T---~
~
Fig. 2. An Equivalent Block Diagram
~
-10
Fig. 3. Independent SISO Feedback System Design -1~5!------!----~-----f::------!C-----=-------!
r---- . . -- From Other Machine. Fig. 8. Nyquist Plot of h(jw)911(jW) With a M Circle for 8 dB O.2·.---...---....----.---....,..--.......
--~--__.--__.
- 0.2
Fig. 4. Torques Act on a Machine
-0."
- 0.8
-, -1.2
0.2
Fig. 9. Nyquist Plot of h (jW)911 (jw) for Stability Margins
x 10'"
Mlc.hlDe 3
l~------------~
Fig. 5. 10 Machine 39 Busbar Test System
lf
Bode plot of opa.l00p lYltem(l00p 1)
·100
x 10'"
MachiDC4
2~---------------,
·15~O·1
I]
s;;;J
,'~L,,----------,~~~---------,O~ ' --------~'o'
SystcmwithoutPSS: _
~SyltCm
with PSS on:·· .
Fig. 10 . Machine Speed Deviation Time Responses in Per Unit
Fig. 6. Bode Plot of 911 (jW)
376
POWER SYSTEM STABILIZER DESIGN BASED STRUCTURED SINGULAR VALUES
ON
T. C. YANG* and J. H. ZHANG** ·School of Engineering, University of Sussex, Brighton, BN1 9QT, UJ( ··North China University of Electric Power (Beijing), Qinghe, Beijing, 100085, P.R. China
Abstract. In this paper, power system stabilizer design for multi-machine power systems is translated into an equivalent problem of decentralized controller design for Multi-Input Multi-Output (MIMO) control systems. Subject to a condition based on the structured singular values, each stabilizer can be designed independently. The robust stability condition for power systems with stabilizers on can be easily stated as to achieve a sufficient interaction margine (Yang and Cimen, 1996) and a sufficient gain and phase margine during each independent design. Within this general framework, the conventional stabilizer design methodology based on the concept of synchronous and damping torques is used to decide the design details of each stabilizer. The suggested design method is applied to a model of a practical 10 machine power system. Copyright © 1998 IFAC Key Words. Power system stabilizers, multi-machine power systems, structured singular values.
sign for multi-machine power system can be conveniently formulated as a large-scale system decentralized stabilization problem. However, probably not enough attention has been paid to the fact that the decentralized stabilizer design problem can be translated into a problem of diagonal feedback controller design for Multi-Input MultiOutput (MIMO) system; and some MIMO system design methods, for example, the sequential loop closure method (Yang, 1996), can be applied to the PSS design. In this paper a new design method based on the translated MIMO system model is presented. In this method each PSS can be designed independently on its own "loop" subject to a condition derived from the structured singular values. The design method and its application to a model of a practical 10 machine 39 bus-bar power system are presented in Section 2 and Section 3 respectively.
1. INTRODUCTION
Power System Stabilizer (PSS) units have long been regarded as an effective way to enhance the damping of ElectroMechanical Oscillations (EMO) in power systems. The linearized models can be used for the study and a conventional power system stabilizer can be 'considered as a single-input single-output feedback controller installed on a generation set (a machine). Since the pioneering work of deMello and Con cordia in 1969, control engineers, as well as power system engineers, have showed great interest and made significant contributions in PSS design and applications. Most PSS design methods developed in recent years are based on well-developed modern control theory. These include: eigenvalue assignment, optimal control, self tuning and adaptive control, variable structure control, internal model control, high gain design, Hoc based design, rulebased and neural network based control, etc. (See for example, the references listed in (Yang, 1996)).
2. A NEW PSS DESIGN METHOD
State space models are commonly used for the analytical study of multi-machine power systems. The design of each individual stabilizer in a multimachine system must be coordinated due to the interactions between machines and controllers. The minimum requirement for the PSS design is that when the open-loop system is closed by the stabilizers, the dampings of the EMO modes concerned should be improved to a satisfactory degree and all system eigenvalues should have negative real parts. It is well known that the PSS de-
2.1 Translate Into An Equivalent Design Problem
In multi-machine power system steady-state stability analysis, a number of state variables is used for the dynamics of each generating set (machine). The dynamics of transmission lines and loads are neglected but their static characteristics provide necessary information for the modelling of each machine and the links between machines. Each machine can be considered as a subsystem in a large-scale system and PS Ss installed on machines
371
as local controllers. A state space model with the consideration for the speed input PSS design can be built up as:'
x yO
+
Ax
=
yes)
G*(s) = C(sI - A)-1B
(2)
COx
From the system stability point of view, an equivalent block diagram (Figure 2) can be drawn to represent the MIMO system in Figure 1, where G(s) [gij(S)]i,j=1 ,2,... 1 -G*(s) . Both systems have the same return ratios.
=
=
y
In Grosdidier and Morar's paper, a transfer function G(s) = [g;j(S)]i ,j=1 ,2,... 1 for a I by I MIMO plant is decomposed into:
=
G(s) = G(s) + G(s)
(7)
=
where G(s) diag[gii(S)]i=1,2 ,... 1 is a diagonal matrix; all diagonal elements in G( s) are zeros and off-diagonal elements in G(s) are equal to those in G(s) .
(3) (4)
=
where u [U1 U2 .. · UI f ; y [Y1 Y2 .. · YI f ; Band Care n by 1 and I by n matrixes respectively. It is worthwhile pointing out that, although the matrixes (vectors) B , C (u, y) in (3)(4) now have reduced dimensions compared with their equivalents in (1)-(2), x and A remain unchanged. The whole system dynamics is therefore unaffected for the design concerned.
=
2.2 Decentralized Controller Design for MIMO systems
After the PSS allocation, those terms in BO u O CO . and yO corresponding to the machines where a PSS has not been allocated can be deleted. This leads to: Bu
=
If all gij (s) (i =1= j) in G( s) were equal to zero, then each stabilizer could be designed independently just as if it were in a single machine system as shown in Figure 3. However, since gij(S) (i =1= j) are not zeros, the following question must be answered . If each fi(s) PSSi(S) (i 1,2, ... /) is designed to form a stable closed-loop system as shown in Figure 3, what are the additional conditions which can guarantee that the global system of Figure 2 is stable and all dampings ofthe EMO modes concerned are improved to a satisfactory degree? The answer to this question is discussed in the next subsection based on the theorem given by Grosdidier and Morar (1986).
A list of I EMO modes whose dampings are to be improved can be found by eigenstructure analysis. Their total number 1 is normally less than the number of machines N in the system. The general practice is to installl stabilizers. Each PSS is allocated to a most suitable machine and each PSS is used mainly to improve the damping of a particular EMO mode. The allocation problem has been studied extensively in the literature, for example in (Yang et al. , 1987), and will not be addressed here. This paper focuses on stabilizer design when a PSS allocation has already been given. The PSS allocation method given in (Yang et al., 1987) is adopted in this paper. This is not only because of its conceptual and computational simplicity, but also because an important theoretical issue, "fixed modes", has been properly addressed in (Yang et al. , 1987) without using any complicated mathematics.
+
(6)
The design of I decentralized local stabilizers now becomes the design of a I by I diagonal matrix F(s) = diag[fi(s)]i=1 ,2,... 1 • Ii(s) is equal to the transfer function of the i-th stabilizer: PSSi(S) as shown in Figure 1, where a negative sign is also put before G*(s) in order to form a negative feedback system.
where x is an n by 1 state vector ; A is an n by n matrix with N diagonal blocks for the dynamics of N machines and off-diagonal blocks for the dynamics of links and interactions; uO is an N by 1 vector [u~ u~ .. . uCJv f and u? corresponds to the i-th machine voltage regulator terminal to which the output of a possible PSS will apply; yO is an N by 1 vector [y~ y~ ... yCJv )T and yp corresponds to the i-th machine shaft speed which can be used as the input to a possible PSS ; BO and CO are appropriated n by Nand N by n matrixes respectively.
Ax Cx
G*(s) = [g:j(S)kj=1,2, ... 1 (5)
can be calculated as:
(1)
BOuO
= G*(s)U(s);
Using the notations:
E(s) = G(s) G- 1 (s)
(8)
B(s) = G(s)F(s)(I + G(s)F(s))-l = diag[hi(s») (9) H(s) = G(s)F(s)(I+ G(s)F(s))-l
(10)
where F(s) = diag[li(s)]i=1,2 ,... 1 is a diagonal transfer function for a decentralized controller in Figure 2; B(s) or H(s) is a closed-loop transfer function matrix for a feedback system consisting of F(s) and G(s) , or F(s) and G(s) respectively, Grosdidier and Morar have proved the following
An I by I transfer function matrix G*(s) linking U(s) = [ U1(S) U2(S) ... UI(S) f and yes) = [ Y1 (s) Y2 (S) .. . YI (S) ]T:
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theorem:
lowing comments.
The closed-loop system H(s) is stable if:
(1) As pointed by Skogestad and Morar (1986), for ordinary MIMO systems, if G(s) is unstable, condition (c-I) is generally not satisfied. However, if the model G(s) is obtained by the translation from a large scale system , the condition (c-I) is generally satisfied even if G(s) is unstable. The poles of G(s) are given by the roots of det[sI - A] and the elements of G( s) , the diagonal elements m G(s) , can be calculated as: -Ci adj(sI - A] b; 9i;(S) = det[sI _ A] (11)
(c-l) G(s) and G(s) have the same number of Right Half Plane (RHP) poles; (c-2) H(s) is stable; and V w (c-3)* umaz(H(jw)) < j.t-l(E(jw)) where O"maz denotes the maximum singular value of; and j.t denotes Doyle's structured singular value in respect to the decentralized controller structure of F(s) . Since H is a diagonal matrix, condition (c-3)* can be replaced by:
If there is no the same pole-zero cancellation in
Ilhi(jw)11 where 11 11 (c-3)
all 9i;(S), the poles of G(s) are also given by the roots of det[sI - A], the condition (c-I) is naturally satisfied. Even there are some cancellations, so long as they are not at RHP, the condition (c-I) is still satisfied.
denotes the magnitude of.
This theorem gives sufficient conditions for the system H(s) to be stable if the controller is designed based on the fully non-interactive model G(s) , i.e. each fi(S) is designed independently on a Single-Input Single-Output (SISO) model 9ii(S). In particular, condition (c-3) says that the magnitude of the frequency response of SISO closed-loop
.
transfer functIon
hi (S ) =
(2) The stability condition (c-4) is given for the nominal plant G(s) . If power system operating condition changes, the plant model will also change. It is perhaps not possible to establish a clear relationship between the change of power system operating conditions and the change ofvalues involved in condition (c-4). However, generally speaking, the robust stability can be achieved if:
fi(S)9ii(S) • () () must be S 9ii S
1 + Ji
less than a scalar frequency dependent function j.t-l(E(jw)) . Grosdidier and Morar (1986) have also proved that , although (c-3) is a sufficient condition and therefore may have some conservativeness , compared with the other conditions developed from control theory for the independent decoupled design, for example the diagonal dominant condition and the generalised diagonal dominant condition, (c-3) gives the tightest restrictive band and is the least conservative. Since the same restriction j.t-I (E(jw)) is applied to all hie s) in condition (c-3), a modification on this condition can be made to provide more flexibility and therefore to further reduce the possible cbnservativeness caused by the inflexibility in condition (c-3):
(r-I) Condition (c-4) is satisfied with a sufficient margin. This can be checked by plotting IIhi(jw)1I and j.t-l(E(jw)W(jw)) w;(jw) on the same graph and an interaction margin for loop i can be defined as the shortest vertical distance between the two curves. (r-2) There are sufficient gain and phase margins in each SISO loop for the stability. This can also be checked by a Nyquist plot of f; (jW)9ii (jw). (3) The main objective of the PSS design is to improve the dampings of the EMO modes concerned . It is known that a poorly damped oscillation is also characterised by a big resonant peak in the system frequency response. This implies that, for a good design , there should be no big resonant peak in 11 h; (jw )11. In addition, general knowledge of feedback system dynamics and the MIMO system examples given by Grosdidier and Morar (1986) show that increasing the interaction, gain and phase margins as stated in (r-I) and (r-2) will lead to more damped response. It is therefore believed that the required dampings will be provided if the design leads to sufficient interaction and stability margins, and there is no significant resonant peak in 11 hi (jw ) 11 . Furthermore, as showed in the next subsection, the objective of improving the dampings is the main consideration in the design of I;(s), which can be based on the
( c-4)*
O"maz(W-1(jw)H(jw)) < j.t-l(E(jw)W(jw)) Vw whereW(s) = diag[w;(s)];::;:1,2 ,... 1 is a properly chosen diagonal weighting function matrix. (c-4)* can also be replaced by: (c-4) Ilh;(jw)w;l(jw)1I < j.t-l(E(jw)W(jw)) ; Vw (i= 1,2, ... /) From (c-4) , although j.t-l(E(jw)(W(jw)) is still the same for all SISO loops, the restrictions on Ilh;(jw)1I are now different. In fact , (c-3) is a special case of (c-4) with W = I . Before applying the above results developed by Grosdidier and Morar (1986) to our decentralized PSS design problem, it is useful to make the fol-
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concept of synchronous and damping torques originally developed for the conventional PSS design.
bilizer design and power system voltage stability. In this system , Machine 10 can be considered as a reference machine (Busbar 39 considered as an infinity bus) and its dynamics are not required in the modelling. The dynamics of each of the other 9 machines can be modelled by 6 state variables for the study of PSS design. Therefore, the model established is a large scale system with 9 subsystems and 54 state variables.
(4) Since E(s) = G(s) <;-1(s) , if 9;;(S) (i f j) is "small" in relative to 9;; (s ) (i = j) , E( s) will also be "small" . J.L(E( s)) can therefore be considered as a measure of the interactions for the plant G(s) .
2. 3 PSS Desi9n for Each SISO Loop
The system operating condition given in (Pai, 1989) with a load-flow result was considered as the "worst situation" for the steady-state stability. A 54-state small signal model based on this operating condition ca~ be formed and is considered as a nominal model in this paper. Eigenstructure analysis reveals that the system is stable but there are four poorly damped EMO modes with their damping ratios no more than 0.03 . The eigenvalues for these modes are: (1) -0.1895±i7.4036 (2) -0 .1198±i6.9023 (3) -0 .2662±i8.8400 (4) -0 .1132±i7.8415
Based on the theoretical background and discussions in Section 2.2, PSS can be designed independently so long as sufficient margins stated in (r-1) and (r-2) are obtained. Despite much progress in applying modern control technology to the PSS design with many different suggested PSS structures, many utilities, according to Tse and Tso (1993) , still prefer a conventional PSS structure. The reasons may include ease of on-line tuning and lack of assurance on the stability related to some adaptive or variable structure schemes. The basic form of the transfer function for a speed-input conventional PSS is:
Tw s 1 + Tl s 1 + T3S PSS(s) = K$l + Tw s 1 + T 2 s 1 + T4 s
By applying the PSS allocation method given in (Yang et al. , 1987), it is decided to install four PSSs on machine 2, 3, 4, and 6 to improve the dampings of the above modes (PSS on machine 2 for mode 1: -0 .1895±i7.4036; etc.). These stabilizers (loops) are numbered 1, 2, 3, and 4 respectively. After the PSS allocation , a four by four transfer function matrix G( s) can be obtained.
(12)
where K . is the PSS gain; Tw is the time constant for the wash-out stage which is to prevent steadystate voltage shift ; Tt , T 2 , T3 and T4 are time constants for two phase-lead stages. (In some cases more than two phase-lead stages are required and a filter and an output limiter will be added to PS S( s) in practical applications.)
Figure 6 gives the Bode plot for 911 (jw). The other three Bode plots for 9n(jw), 933(jW) and 944(jW) have a very similar shape and are not given here. It can been seen from Figure 6 that there is a resonance peak in the magnitude response and also a sharp drop of the phase angle around the frequency range of the poorly damped EMO modes. These are due to the characteristics of the EMO modes concerned. The magnitude of the frequency response is quite small since, in the per unit system, the " ratio" between the output (machine speed deviation) and the input (machine reference voltage change) is a very small value.
By using the conventional PSS structure of equation (12) and extending the basic concepts in the conventional method, each stabilizer can be designed in the same way as that in (Larsen and Swann, 1981) . To limit the length of the paper, the details are not repeated here. With the help of the concepts and methods given in the literatures, all PSS parameters can be finally tuned during commissioning.
3. DESIGN EXAMPLE The similarity in the shape of Bode plots for 9ii {jW) (i 1,2,3,4) indicates that similar PSS parameters may be used for all stabilizers and there is no need to introduce a weighting function W(s) to give different restrictions on hieS) with regard to condition (c-4) for the global stability. The time constants in the four stabilizers are simply set as follows:
=
A 10 machine 39 bus and 46 line power system (Figure 5) is used in this paper as a test system. This test system was developed as a representative of the power system in the Northeastern United States and is also called the NEW ENGLAND system. This system has been used for the study of power system stabilities in a number of publications (See references in (Yang, 1996)). These include dynamic equivalents for transient stability, direct analysis of transient stability, energy function analysis for power system stability, steady-state stability analysis, power system sta-
(1) The washout stage time constant is set to Tw = 10 for all four stabilizers ; (2) The two phase lead stages are identical for all
374
four stabilizers (in T4 ) and: Tl = T3 = 0.534, lizer 1; Tl = T3 = 0.573, lizer 2; Tl = T3 = 0.447, lizer 3; Tl = T3 = 0.504, lizer 4.
each stabilizer, Tl = T3; T2
=
T2 = T4 = 0.0534
for stabi-
T2 = T4 = 0.0573
for stabi-
T2 = T4 = 0.0447
for stabi-
7'4 = 0.0504
for stabi-
T2 =
sufficient and there is no resonant peak in all h;(jw) , we can believe that (1) sufficient dampings have been provided by the stabilizers for the EMO modes concerned, and (2) the system robust stability is satisfactory. The simulation results for the m-th (m=2, 3, 4, 6) machine speed-deviation b.w time-responses, when a small step disturbance of 0.05pu (per unit) is applied to the m-th machine voltage regulator terminal, for the system without PSS and with the four PSS on are given in Figure 10 as full-line curves and dotted-line curves respectively.
Based on the system frequency responses, it was found that /{, = 40 is a suitable gain for PS S 1 ( s) . This gives an interaction margin of 7.98dB, a gain margin of 9.81dB and a phase margin of 61.2 0 for loop 1. The Bode plots for Jl-l(E(jw) and Ilhl(jw)11, and the Nyquist plots for PSS l (jW)91l(jW) are given in Figure 7,8 and 9 respectively.
4. CONCLUSIONS A new power system stabilizer design method based on the structured singular values is suggested in this paper for multi-machine power systems. The design method is applied to a model of a practical 10 machine power system.
The plot of Jl- l (E(jw» in Figure 7 shows that there is a minimum value of 7.96dB around the frequency of 7.0 rad/sec. The magnitude of Ilhl(jw)11 at that frequency is about -0.2dB. There is no resonant peak in Ilhl(jw)ll.
5. REFERENCES Grosdidier, G. and M. Morar (1986). Interaction measures for system under decentralized control. Automatica, 22, 309-319. Larsen, E.V. and D.A. Swann (1981) . Applying power system stabilizers (three parts). IEEE Trans., PAS-lOO, 3017-3046. Pai, M.A. (1989) . Energy function analysis for power system stability. Kluwer Academic Publishers. Tse, C.T. and S.K. Tso (1993) . Refinement of conventional PSS design in multimachine system by model analysis. IEEE Trans ., PS-S, 598605. Yang, T.C . (1996) Applying a sequential loopclosure method to power system stabilizer design. Control Engineering Practice, 10, 13711380. Yang, T.C. and H. Cimen (1996). Applying structured singular values to robust decentralized power system load frequency control, Proc. IEEE Int. Conf on Industrial Technology, 1020-1025. Yang, T.C. , N. Munro and A. Brameller (1987). A new decentralised stabilisation method with application to power system stabiliser design for multimachine systems. Int. 1. Electrical Power and Energy Systems, 9, 206-216.
The interaction margin can also be checked from the Nyquist plot in Figure 8, where a small M circle for 8dB is also plotted. Although it appears that the circle is very near to the Nyquist plot, the points around w = 7.0 rad/ sec on the plot are far away from the circle which indicates a sufficient interaction margin. The part of the Nyquist plot near to the critical point -1 is enlarged in Figure 9 to assess the loop gain and phase margins. Similarly, the gains 30, 40 , 40 are chosen for stabilizer 2, 3, 4. Since the EMO frequency concerned for stabilizer 2 design Wo = 6.90 rad/ sec is nearer to the frequency where Jl-l(E(jw» has a minimum value, stabilizer 2 gain is set at a lower value of 30 for a sufficient interaction margin. The margins achieved for each loop are given as follows: interaction margin: 6.81dB, gain margin : 11.7dB, phase margin: 66.4 0 for loop 2; interaction margin: 9.04dB, gain margin : 10.2dB, phase margin: 64.20 for loop 3; interaction margin: 8.75dB, gain margin : 9.84dB , phase margin: 63 .70 for loop 4; The shapes of Bode plots for hj(jw) (i = 2, 3,4) and Nyquist plots for PSS;(jW)9;;(jw) (i = 2,3,4) are similar to those in Figure 7, 8 and 9. These plots are therefore not given here. According to the theorem given in Section 2.2, although all stabilizers are designed independently, the global closed-loop system is stable when all PSSs are connected. Since all margins are quite
375
- G *(5) .
--
~---
.
.
. . .. .
,
. .
~ '~:':' t~' -r'T~ ; '- -' - ~""' :': ';' ~ ~ ..
-~- -::-..: ~~~-,~~~~~~~,~oo~~~~~~'~'~'~;~o,--~--~~~~,O'
,,
---------
Frequency (radlsec)
F (s)
Fig. 7. Bode Plot of /l-l(E(jw)) (solid line) and IIhJ(jw)11 (dashed line)
Fig. 1. Multivariable System Design for PSSs 15r---~-----T-----r----~----T---~
~
Fig. 2. An Equivalent Block Diagram
~
-10
Fig. 3. Independent SISO Feedback System Design -1~5!------!----~-----f::------!C-----=-------!
r---- . . -- From Other Machine. Fig. 8. Nyquist Plot of h(jw)911(jW) With a M Circle for 8 dB O.2·.---...---....----.---....,..--.......
--~--__.--__.
- 0.2
Fig. 4. Torques Act on a Machine
-0."
- 0.8
-, -1.2
0.2
Fig. 9. Nyquist Plot of h (jW)911 (jw) for Stability Margins
x 10'"
Mlc.hlDe 3
l~------------~
Fig. 5. 10 Machine 39 Busbar Test System
lf
Bode plot of opa.l00p lYltem(l00p 1)
·100
x 10'"
MachiDC4
2~---------------,
·15~O·1
I]
s;;;J
,'~L,,----------,~~~---------,O~ ' --------~'o'
SystcmwithoutPSS: _
~SyltCm
with PSS on:·· .
Fig. 10 . Machine Speed Deviation Time Responses in Per Unit
Fig. 6. Bode Plot of 911 (jW)
376