Decentralized Robust Control for Damping Inter-area Oscillations in Power Systems

0.51 MB·

by  Jianming Lian

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Decentralized Robust Control for Damping
Inter-area Oscillations in Power Systems
Jianming Lian, Shaobu Wang, Ruisheng Diao and Zhenyu Huang
Abstract—Aspowersystemsbecomemoreandmoreintercon- on the generator rotor. However, the classical design of the
nected,theinter-areaoscillationshasbecomeaseriousfactorlim- PSS, which relies on the linearization of a single-machine
iting large power transfer among different areas. Underdamped
infinite-bus(SMIB)model,hastwomajorshortcomings.First,
(Undamped) inter-area oscillations may cause system breakup
theSMIBmodelisareduced-ordermodelbyneglectingthose
and even lead to large-scale blackout. Traditional damping con-
7 trollers include Power System Stabilizer (PSS) and Flexible AC importantdynamicsassociatedwithnetworkinterconnections.
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Transmission System (FACTS) controller, which adds additional It cannot adequately describe the inter-area oscillation modes
0
damping to the inter-area oscillation modes by affecting the real due to network interconnections. Hence, the classical PSS is
2
power in an indirect manner. However, the effectiveness of these
effective in damping local oscillation, but it may not provide
n controllers is restricted to the neighborhood of a prescribed
enough damping to the inter-area oscillations unless carefully
a set of operating conditions. In this paper, decentralized robust
J controllers are developed to improve the damping ratios of the tuned. Furthermore, the local controller design is independent

8 inter-area oscillation modes by directly affecting the real power of one another without any proper coordination among them.
through the turbine governing system. The proposed control Second, the linearization based controller design greatly re-

] strategyrequiresonlylocalsignalsandisrobusttothevariations stricts the validity of the controller to the neighborhood of
Y in operation condition and system topology. The effectiveness of
the operating point under consideration. When the system
S the proposed robust controllers is illustrated by detailed case
studies on two different test systems. loading or network topology drastically changes during the
.
s normal system operation, the resulting PSS may no longer
c Index Terms—Small signal stability, inter-area oscillation, os-
yield satisfactory damping effect.
[ cillation mitigation, decentralized robust control.
Another approach of damping the inter-area oscillations is

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to introduce various supplementary modulation controllers to
v I. INTRODUCTION flexible AC transmission system (FACTS) devices (see, for
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3 The inter-area oscillations of low frequency are inherent example, [5]–[8]). Although FACTS controllers can achieve
0 phenomenabetweensynchronousgeneratorsthatareintercon- satisfactory damping effect for inter-area oscillations [9], it is
2 nected by transmission systems. It has been pointed out in [1] notcost-effectivetousethemforthesolepurposeofdamping
0 thattheseoscillationsoftenbecomepoorlydampedwithheavy control. On the other hand, FACTS controllers share the same
.
1 power transfer between different areas over weak connections shortcoming with the classical PSS due to the linearization
0 of long distance. Because sustained oscillations could cause basedcontrollerdesign.Furthermore,theinteractionsbetween
7 systembreakupandevenleadtolarge-scaleblackout,itisex- FACTS controllers and the PSSs could even potentially de-
1
tremelyimportanttomaintainthestabilityoftheseoscillations grade the damping without proper coordination [10].
:
v for the system security. However, environmental constraints Recently, there have been a lot of efforts into the develop-
Xi and economic pressures often push power systems close to mentofcoordinatedandrobustPSSsforpowersystems[11]–
their operational limits [2], which substantially reduces the [14]. In [11], robust tuning based on eigenvalue analysis was
r
a stabilitymarginofthenormaloperation.Oneoftheprominent proposed to select the parameters of PSSs over a prescribed
eventscausedbyundampedoscillationsisthenotablebreakup set of operating conditions. In [12], a novel method based on
andblackoutinthewesternNorthAmericanPowerSystemon modal decomposition was proposed to eliminate the interac-
August 10, 1996 [3]. To ensure the secure system operation, tionsamongdifferentmodesfortuningPSSs.In[13]and[14],
the amount of power transfer on tie-lines between different robust PSSs were designed by utilizing robust control tech-
areas are often limited due to the inter-area oscillations that niquessuchasµ-synthesisandlinearmatrixinequality(LMI).
are poorly damped. Hence, new control strategies that can In addition to robust PSSs, robust excitation control strategies
improvethedampingoftheseinter-areaoscillationsareneeded havealsobeendevelopedfordampingcontrolandtheresulting
to increase the transmission capacity for a better utilization of controllers have different structure from that of the classical
the existing transmission network. PSS. In [15], a novel methodology was proposed to design
The well-known damping controller is the power system robust excitation controller based on the polytopic model
stabilizer (PSS) [4]. It adds a supplementary control signal for improved robustness over varying operating conditions.
into the generator excitation system through the automatic Although these newly developed robust damping controllers
voltage regulator (AVR) to apply an additional electric torque aremorerobustthantheclassicalPSS,theirrobustnessisstill
limited to a finite number of operating conditions.
J.Lian,S.Wang,R.DiaoandZ.HuangarewithPacificNorthwestNational
There have been also persistent efforts on determining
Laboratory,Richland,WA99352,USA(e-mail:{jianming.lian,shaobu.wang,
ruisheng.diao,zhengyu.huang}@pnnl.gov). appropriate global signals as the control input of classical
2
PSSs and FACTS controllers as in [16] and [17]. Because the into the turbine governing system through the governor to
feedback of global signals actually establishes the correlation apply an additional mechanical torque on the generator rotor.
among independent local controllers, it is more effective than This principle is similar to that of the PSS, whereas the
the feedback of local signals only. These efforts have been PSS provides an additional electric torque on the generator
particularly facilitated by the technology of wide area mea- rotor through the generator excitation system. The proposed
surementsystem(WAMS)enabledbythedevelopmentofPha- control strategy has several appealing advantages ascribed
sor Measurement Unit (PMU). Many WAMS-based damping to the underlying coordinated controller design. First, it is
controllershavebeendeveloped(see,forexample,[18]–[21]). robust to the variations in operating condition and system
However,therobustnessofWAMS-baseddampingcontrollers topology. This is because the corresponding controller design
to the variations in operating condition and system topology does not require system linearization over a specific operating
still requires further investigated. Moreover, the destabilizing condition, nor does it depend on a specific network topology
factors of communication network such as time-varying delay either. Such robustness is extremely crucial for the secure
and random packet dropout has to be explicitly taken into operation of power systems, which constantly experience
accountforthereliabilitypurpose[22],whichgreatlycompli- changes in operating conditions and transmission networks.
cates the corresponding controller design. Second, it has no destructive interference with existing PSSs
In most cases, the inter-area oscillations are usually caused inthesystemregardingdampingcontrol,whicheliminatesthe
bytransferringrealpowerbetweeninterconnectedareas.Thus, need of proper coordination. This is because the turbine gov-
it is a straightforward way to control the real power in the erning system is weakly coupled with the excitation system.
systemfortheoscillationmitigation.However,themajorityof Third, it is decentralized requiring only local signals that are
existing damping controllers introduce additional damping to easily obtained for controller implementation, which greatly
the inter-area oscillation modes through either generation ex- improves the reliability of the controller by eliminating the
citationsystem(e.g.,PSS)ortransmissionline(e.g.,FACTS), use of communication. Lastly, it has simple controller design
which actually affects the real power in an indirect manner. of low complexity and does not involve additional costly
Some of the work that directly affects the real power to equipments such as power electronics and energy storages,
damp the inter-area oscillations has been reported in [23]– whichmakeiteasyforpracticalimplementation.Detailedcase
[25]. In [23], robust damping controllers were designed for studies are performed on both small and medium-sized test
superconducting magnetic energy storages to enhance the systems to illustrate the effectiveness and robustness of the
damping of multiple inter-area modes by injecting real power proposed robust controllers in improving the damping ratios
intothesystem.Itisshownthatrealpowermodulationcanbe of the inter-area oscillation modes.
an effective way to damp power flow oscillation, although the The remainder of this paper is organized as follows. In
resulting controller therein is only robust to the prescribed Section II, the dynamic model of multi-area power systems
set of operating conditions. In [24], two damping control is given. In Section III, the design of decentralized robust
systems using ultracapacitor based energy storages for real controllers is described with the discussion of practical im-
power injection were designed for areas expected to oscillate plementation. Then detailed case studies are presented in
against one another. Because the frequency measurement of Section IV. Conclusions are found in Section V.
one area is required by the controller in the other area, the
communication effects could greatly degrade the expected II. SYSTEMMODELING
controller performance. In [25], a Modal Analysis for Grid
The multi-area power systems usually consist of a number
Operations (MANGO) procedure was established to enable
of generators that are interconnected through a transmission
grid operators to immediately mitigate the inter-area oscilla-
network.Inthispaper,robustcontrollersareproposedforgen-
tionsbyadjustingoperatingconditions,wherethedetectionof
eratorswithsteamvalvegovernors.Althoughonlysteamvalve
lowdamping isachievedby themodal analysis withreal-time
governorsareconsideredherein,theproposedcontrolstrategy
phasor measurements. The MANGO procedure can enact as
can be extended to other types of governors such as hydraulic
part of a remedial action scheme. However, the interactions
governors. However, the resulting controller structures could
among different oscillation modes make it challenging to
varyduetodifferentturbinegoverningsystemdynamicstobe
determine universal dispatch patterns that are effective for all
considered, which is left for future research. As illustrated in
the inter-area oscillation modes.
Fig. 1, the generator dynamics with steam valve governor can
In this paper, the decentralized robust control strategy
be described by the following differential equations:
developed in [26] is adopted herein to improve the damping
Rotor Dynamics:
ratiosoftheinter-areaoscillationmodes.In[26],adistributed
hierarchical control architecture was proposed for large-scale δ˙ =ω =ω −ω , (1)
i ri i o
power systems to improve the transient stability and fre- D ω
ω˙ =− i ω + o (P −P ); (2)
quency restoration. However, the capability of this control ri 2H ri 2H mi ei
i i
strategy in enhancing the small signal stability of multi-
Governor Control:
area power systems has not been analyzed yet. Hence, it is
K 1 1
the main focus of this paper to show the effectiveness of X˙ =− ei ω − X + P , (3)
ei T R ω ri T ei T ci
this control strategy in mitigating the inter-area oscillations. ei i o ei ei
This robust controller introduces an auxiliary control signal 0≤X ≤1;
ei
3
P
ei
P
ei where A , B and G can be derived from (1) through (5)
P ω δ i i i
mi − ω0 ri 1 i and are given by
PmPiei−+ Dωi0+2Hisωri 1 s δi
+ Di+2His s  
Pmωωirωrii+−ri RKRKiωeiRωie0Ki0iωeDi(0ai)+PωRP2c0oHi+−ctoi+PirsT−cmdi+yinnTω=aTmrmmTi−inamix1c=as=xT−=1m11+1a1s1x1+T=e1Ti1se1i+sδ1TiXeiXseiei Xei Ai =0000 −−TmTKmi−TKeieiT21eiDTeHiT3iTi5iRi3iTiRi4ωii0ω0 −2ωH00T015ii TT−m5iiT00−Tm1mTi4ii T4TiTmT(eTmiiTe−00ii5T−TieT3Tiie3ii),
− 0 − Kei 0 0 − 1
Tmin=−1 TeiRiω0 Tei
(b) Governorcontrol
(cid:104) (cid:105)(cid:62)
Bi = 0 0 T3iT4i T3i 1 ,
XXeiei 1+1T+3iTs3isXmXimi1+T14+isT4iPsmiPmi TmiTeiT5i TmiTei Tei
Xei 111+++T1TTm+3miisTsis(mc)iXsTumrbiine11dy++1n+TTa45mTii1ss5i+cissTP5ismi Gi =(cid:104)0 −2ωH0i 0 0 0(cid:105)(cid:62).
Foreachgenerator,theoriginalpowercontrolinputPref will
Fig.1. Generatordynamicswithsteamvalvegovernorcontrol[26]. ci
be augmented with a local decentralized control u so that
i
P =Pref +u .
ci ci i
Turbine Dynamics:
This additional control input u is defined as
i
K T 1
X˙mi =− TmiTeeiiR3iiω0ωri− TmiXmi ui =ki(cid:0)xi−xdi(cid:1), (7)
(cid:18) (cid:19)
1 T T
+ 1− 3i X + 3i P , (4) where xd =[δd ωd Pd Xd Xd](cid:62) is an arbitrary operating
T T ei T T ci i i ri mi mi ei
mi ei mi ei point, and k is the linear feedback gain matrix. In practice,
i
P˙mi =− TmKiTeeiiTT35iiTR4iiω0ωri− T15iPmi xthdie csyanstebme ablewfoaryesacnhyosdeinstuarsbathnececsu.rrent equilibrium state of
(cid:18) (cid:19)
+ 1 1− T4i X + T3iT4i P Remark 1: The state space model (6) is derived without
T T mi T T T ci linearizationsinceallthenonlinearitiesresultedfromnetwork
5i mi mi ei 5i
+ T4i (cid:18)1− T3i(cid:19)X ; (5) interconnections are encompassed by the unknown input Pei.
T T T 1 ei In the following controller design, the variations in Pei will
mi 5i ei1 betreatedasexternaldisturbancestoindividualgeneratorsand
1
Intheabovedynamics,δi isthegeneratorrotorangle(rad),ωi then rejected by the proposed controllers. The determination
is the generator rotor speed (rad/sec), ωri is the relative rotor of controller gain ki in (7) is independent of any specific
speed (rad/sec), ω0 is the nominal rotor speed (rad/sec), Pmi operating condition or network topology in order to guarantee
is the mechanical power (p.u.) provided by turbine, Pei is the the controller robustness. The operating point xd in (7) only
i
electricalpower(p.u.)resultedfromnetworkinterconnections, servesasthereferencefromwhichthecontrolinputshouldbe
Xei isthevalveopening(p.u.),andPci ispowercontrolinput calculated. If the proposed controllers are activated during the
(p.u.).Forthenormaloperationofpowersystems,Pci =Pcrief, normaloperation,xdi willbethecurrentequilibriumoperating
which is the prescribed generator reference power received point. If they are activated during the disrupted operation,
from economical dispatch. xd will be the last equilibrium operating point before the
i
When power systems do not have enough damping, any disruption occurs. Once activated in the system, the proposed
variations in Pei resulted from the changes in either loading controllers will continue to work regardless of the operating
condition or network topology can cause poorly damped condition and network topology.
oscillations of low frequency. In this work, the proposed
decentralized robust controllers will modulate the mechanical
A. Controller Design
power P to introduce additional damping to the poorly
mi
damped oscillation modes. In the following section, the con- Let xe = [δe ωe Pe Xe Xe](cid:62) denote the equilibrium
i i ri mi mi ei
troller design will be briefly presented. The interested readers statethesystemreachesafterdisturbances.Itfollowsfrom(6)
are referred to [26] for more details. and (7) that xe satisfies the following algebraic equation,
i
(cid:16) (cid:17)
A xe+B Pref +ue +G Pe =0, (8)
i i i ci i i ei
III. DECENTRALIZEDROBUSTCONTROL
where ue = k (xe − xd). To study the stability of each
To proceed, suppose there are N generators in the system. i i i i
Let x = [δ ω P X X ](cid:62) and x = [x ... x ](cid:62). generator (6) driven by the controller (7) after disturbances,
i i ri mi mi ei 1 N
the perturbed system dynamics about the equilibrium state
Then the dynamics of the i-th generator can be represented
xe is considered. Let ∆x = [∆x(cid:62) ··· ∆x(cid:62)](cid:62), where
by the following state space model, i 1 N
∆x =x −xe isthevector ofthedeviationsof δ , ω , P ,
i i i i ri mi
x˙ =A x +B P +G P , (6) X ,andX ,respectively,fromtheirequilibriumvaluesafter
i i i i ci i ei mi ei
4
disturbances. Then by subtracting (8) from (6), the perturbed that local decentralized controllers can stabilize the perturbed
dynamics of the i-th generator can be derived as system (9) if the following optimization problem is feasible,
∆x˙i =Ai∆xi+Bi∆ui+Gihi(∆x), (9) minimize (cid:80)Ni=1(γi+κYi +κLi)
subject to Y >0
D
where ∆ui = ui −uei = ki∆xi and hi(∆x) = Pei −Peei.  W G Y H(cid:62) ··· Y H(cid:62) 
The unknown input hi(∆x) = Pei −Peei represents external  G(cid:62)D −DI DO 1 ··· DO N 
disturbances resulted from network interconnections.  D 
 H1YD O −γ1I ··· O <0
Given the two-axis generator model described in [27], the  
disturbance h (∆x) can be equivalently represented as  ... ... ... ... ... 
i
H Y O O ··· −γ I
hi(∆x)=hqiq(∆x)+hqid(∆x)+hdiq(∆x)+hdid(∆x) (10) (cid:20) −κNLiID L(cid:62)i (cid:21)<0,(cid:20) Yi IN (cid:21)>0
with Li −I I κYiI
γ − 1 <0, i=1,...,N,
N i β¯2
hqq(∆x)=(cid:88)E(cid:48) E(cid:48) (G C +B S ), i (12)
i qi qj ij ij ij ij where γ = 1/β , β¯ ≥ 1, W = A Y + Y A(cid:62) +
j=1 B L i+ L(cid:62)B(cid:62)i wiith A =Ddiag(cid:2)AD D··· AD (cid:3)Dand
N D D D D D 1 N
hqid(∆x)=(cid:88)Eq(cid:48)iEd(cid:48)j(BijCij −GijSij), BD =diag(cid:2)B1 ··· BN(cid:3),andκYi andκLi areprescribed
positive limits imposed on the magnitude of L and Y .
j=1 i i
Once the above optimization problem is solved, the gain
N
(cid:88)
hdq(∆x)= E(cid:48) E(cid:48) (−B C +G S ), matricesoflocaldecentralizedcontrollerscanbecalculatedas
i di qj ij ij ij ij K = L Y−1, where K = diag(cid:2)k ··· k (cid:3). When
j=1 D D D D 1 N
solvingtheaboveoptimizationproblem,thesystem’stolerance
N
(cid:88)
hdd(∆x)= E(cid:48) E(cid:48) (G C +B S ), to interconnection uncertainties is maximized.
i di dj ij ij ij ij
j=1
B. Practical Implementation
where E(cid:48) is the q-axis transient EMF (p.u.), E(cid:48) is the dq-
qi di
axis transient EMF (p.u.), C (cid:44) cosδ −cosδe, and S (cid:44) There are two of many possible ways to practically imple-
ij ij ij ij
sinδ −sinδe withδ =δ −δ andδe =δe−δe.Applying menttheproposedrobustcontrollersfordampingtheinter-area
ij ij ij i j ij i j
the same argument as in [28], it gives oscillations. One way is to keep them activated in the system
asapre-emptivestrategytoimprovethedampingratiosofthe
hqq(cid:62)(∆x)hqq(∆x)≤y(cid:62) qqy inter-area oscillation modes. The other way is to install them
i i i Di i
in the system without activation until poorly damped inter-
(cid:2) (cid:3)(cid:62)
where y = y ··· y with
i i1 iN area oscillations are detected, which can be achieved by the
yij = δij −2 δiej = ∆δi−2 ∆δj, athpipsrwoaacyh,tohferaecatli-vtiamtioenmoofdteheessetipmraet-iionnstaalslepdrorpoobsuesdtcinon[t2ro9l]l.eIrns
becomes part of the remedial action scheme as an emergency
and qq =diag(cid:2)dqq ··· dqq(cid:3) with
Di i1 iN measure to prevent potential cascading power failures. The
following simulation results confirm the effectiveness under
N
dqq =4E¯(cid:48)2E¯(cid:48) (cid:0)G2 +B2(cid:1)12 (cid:88)E¯(cid:48) (cid:0)G2 +B2 (cid:1)12 , both ways of controller implementation.
ij qi qj ij ij qk ik ik
k=1
where E¯ represents the allowable maximum absolute value of IV. CASESTUDIES
In this section, the effectiveness of the proposed decentral-
transient EMF (p.u.). Similarly, the same arguments can be
applied,asabove,tohqd(∆x),hdq(∆x)andhdd(∆x).Then, ized robust control in improving the small signal stability is
i i i
first demonstrated under both normal and disrupted system
it follows from (10) that
operations.Inparticular,therobustnessoftheproposedcontrol
h(cid:62)i (∆x)hi(∆x)≤4y(cid:62)i Diyi, (11) with respect to the changes in the operating condition and
systemtopologyisillustratedthroughcomparativecasestudies
where = qq+ qd+ dq+ dd. As shown in [28], the
Di Di Di Di Di that are simulated by the Power System Toolbox (PST) [30].
inequality (11) can be further represented as
Then the proposed control strategy is also investigated for the
h(cid:62)(∆x)h (∆x)≤∆x(cid:62) (cid:62) ∆x, case when the power system has only a very limited number
i i Hi Hi
of generators equipped with steam valve governors.
or, more generally, Case 1: In this case, the performance of the proposed
h(cid:62)(∆x)h (∆x)≤β2∆x(cid:62) (cid:62) ∆x, robust controllers is compared to that of PSSs under different
i i i Hi Hi normal operations by varying the operating conditions. The
where β ≥1 and is directly determined from . selected test system is an IEEE 2-area, 4-machine test system
i i i
H D
Now, the linear feedback gain matrix k can be obtained asshowninFig.2,whoseprototypewasoriginallyintroduced
i
by solving a LMI-based optimization problem as formulated in [4]. Two PSSs are installed on generators G2 and G3, re-
in [28]. It can be shown using similar arguments as in [28] spectively, and a static VAR compensator (SVC) is connected
5
9.5
9
0.60 Hz
8.5
%) 8
atio (7.5
g r 7
n
pi6.5
m
da 6
5.5
5
4.5
250 300 350 400 450 500
tie−line flow (MW)
(a) Withoutrobustcontrollers
Fig.2. Single-linediagramofIEEE2-area,4-machinetestsystem[30].
28 G1, G2, G3 and G4
TABLEI 26 G1 and G4
OSCILLATIONMODEOFMINIMUMDAMPINGRATIOINBASECASEFOR 1.30 Hz G2 and G3
IEEE2-AREA,4-MACHINETESTSYSTEM %)24
Robustcontrollers none G1-G4 G1,G4 G2,G3 atio (22 1.20 Hz
Oscillationmode inter-area local local inter-area g r20
n
Naturalfrequency 0.60Hz 1.30Hz 1.20Hz 0.66Hz mpi18
Dampingratio 6.96% 23.08% 18.20% 16.09% da16
0.66 Hz
14
tobus101.Thebasecasehasa400MWtie-lineflowfromthe 1225 0 300 tie3−5l0ine flow (4M00W) 450 500
left area to the right area, where the real power of two loads
(b) Withrobustcontrollers
at bus 4 and bus 14 is 976 MW and 1757 MW, respectively.
The modal analysis performed in the PST indicates that the Fig.3. Correlationoftheminimumdampingratiowiththetie-lineflowfor
theIEEE2-area,4-machinetestsystem.Theredsolidlinein(a)represents
base case with PSSs only has a major inter-area oscillation
the5%dampingmargin.
mode around 0.60 Hz with a damping ratio of 6.96%. This
inter-area oscillation mode has the minimum damping ratio
among all the oscillation modes of the base case with PSSs are considered by varying the real and reactive power of two
only. Since all the generators are equipped with steam valve loads at buses 4 and 14 with the same percentage. At the
governors, the proposed decentralized robust controllers can same time, the power output of generators G1 to G4 is also
be designed for all the generators. When robust controllers changed accordingly by the same percentage. This adjustment
are implemented on generators G1 to G4, there is an inter- pattern creates a number of operating points with different
areaoscillationmodearound0.70Hzwithagreatlyimproved tie-line flows, and it also minimizes the locational effect
damping ratio of 40.16%. However, the oscillation mode of of generation and load. The corresponding variations of the
minimumdampingratioisalocalmodeassociatedwithgener- minimum damping ratio with and without robust controllers
atorsG1andG2.Whenrobustcontrollersareimplementedon are shown in Fig. 3. When the system has PSSs only, the
generators G1 and G4, there is an inter-area oscillation mode damping ratio of the major inter-area oscillation mode is very
around 0.65 Hz with a greatly improved damping ratio of sensitivetothesystemstresslevel,anddecreasessignificantly
24.10%. However, the oscillation mode of minimum damping with the increase of tie-line flow as shown in Fig. 3(a).
ratio is again a local mode associated with generators G1 and However, it can be seen from Fig. 3(b) that the minimum
G2. When robust controllers are implemented on generators damping ratio of the system is not only greatly improved but
G2andG3,theoscillationmodeofminimumdampingratiois alsomuchlesssensitivetothesystemstresslevel,whichleads
an inter-area oscillation mode around 0.66 Hz with a greatly to a higher power transfer capacity. Thus, the effectiveness of
improved damping ratio of 16.09%. The minimum damping the proposed control in improving the small signal stability is
ratio among all the oscillation modes, which is the most robusttovaryingoperatingconditionsbecausenolinearization
used index for evaluating the small signal stability [31], is is involved in the proposed controller design.
listedinTableIforeachimplementationofrobustcontrollers. Case 2: In this case, the performance of the proposed
Therefore, different controller implementations will lead to control is compared to that of the PSS under the disrupted
differentimprovementsinthesmallsignalstability.Thegreat- system operation by changing the system topology, where the
est improvement of the minimum system damping occurs same test system used in Case 1 is considered again. The
whenrobustcontrollersareimplementedonallthegenerators. power demand of two loads and the power output of four
generatorsareincreasedby5.58%overthebasecase.Thatis,
In order to examine how the minimum damping ratio is the real power of two loads at bus 4 and bus 14 is increased
correlated with the tie-line flow, various system stress levels to 1030.4 MW and 1855 MW. As perviously determined, this
6
test system with PSSs only has a major inter-area oscillation
−10
PSSs only
mode around 0.55 Hz with a damping ratio of 6.34%. Since G1, G2, G3 and G4
−20 G1 and G4
the damping ratio is larger than the 5% damping margin, any G2 and G3
oscillations resulted from small load fluctuations are expected −30
to be well damped. However, the impact of the topology eg)−40
d
changes due to contingencies on the test system with PSSs e (
gl−50
only remains to be examined. Thus, in this case, the system an
topology is changed by tripping a single line between buses −60
3 and 101 without fault. After removing one of the lines 3- −70
101, the test system with PSSs only is identified to have a
−80
majoroscillationmodearound0.44Hzwithadampingratioof 0 2 4 6 8time1 (0sec)12 14 16 18 20
0.46%.Itturnsoutthattheselectedtopologychangehasavery
(a) Robustcontrollersactivatedallthetime
large impact on the test system with PSSs only, which causes
a 5.88% damping reduction. Although there are still PSSs on
−10
generators G2 and G3, the resulting system is expected to PSSs only
G1, G2, G3 and G4
exhibitahighlyoscillatoryresponse,whichisdangeroussince −20 G1 and G4
G2 and G3
it may lead to system breakup and large-scale blackouts. −30
Nowconsiderthetestsystemwithrobustcontrollers.When g)
e−40
d
robust controllers are implemented on G1 to G4, the test e (
gl−50
system after topology change has a local oscillation mode n
a
around 1.30 Hz with the minimum damping ratio of 23.24%. −60
When robust controllers are implemented on generators G1
−70
and G4, the test system after topology change has a local
−80
oscillation mode around 1.18 Hz with the minimum damping 0 2 4 6 8 10 12 14 16 18 20
time (sec)
ratio of 18.20%. When robust controllers are implemented on
(b) Robustcontrollersactivatedat10sec
generators G2 and G3, the test system after topology change
has an inter-area oscillation mode around 0.48 Hz with the Fig.4.684Relativeangleδ3-δ1 inresponsetothetopologychange. IEEETRANSACTIONSONPOWERSYSTEMS,VOL.23,NO.2,MAY2008
minimum damping ratio of 16.90%. Unlike the PSSs, the
proposed control maintains adequate system damping even
whenthesystemtopologychanges.Inordertobetterillustrate
the robustness of the proposed control with respect to the
topology changes, the dynamic response of the test system
is simulated, respectively, with and without robust controllers.
In the first dynamical simulation, the robust controllers are
assumed to be activated in the test system all the time, and
thecorrespondingresponseoftherelativerotoranglebetween
generators G3 and G1 is shown in Fig. 4(a). In the second
dynamical simulation, the robust controllers are not online
initially.Theyareonlyactivatedafter10secasthereactionto
the detection of sustained oscillatory system response. It can
be seen from Fig. 4(b) that the robust controllers can quickly Fig.4. Powerspectrumofrelativevoltageangleswithhighobservabilityof
0.318-Hzand0.422-Hzmodes.
damp out the oscillations once they are activated.
Case 3: In this case, the performance of the proposed
controlisinvestigatedwhenthepowersystemhasonlyavery systemstates.Fig.4showsthepowerspectrumoffourrelative
phaseanglesignalsestimatedfroma60-minsimulationusing
small number of generators with steam valve governors. In
Fig.3. Examplesimulationsystem. periodogram averaging. The spectrums show strong observ-
suchacase,theeffectivenessoftheproposedcontrolcouldbe Fig.5. Single-linediagramof17-machinetestsystem[32].
ability of the 0.318-Hz and 0.422-Hz modes making these
limited due to the restrictive placement of robust controllers. signalsgoodcandidatesforuseinthemodemeteralgorithms.
TABLEI
The selected test system is the 17-machine system developed INTERAREAMODESOFEXAMPLE17-MACHINESYSTEM The power spectrum of the same signals for the 16-machine
in [32], which is a modified version of the one originally are equipped with hydraulic governors. Generator G17 iscoandition are very similar except the peaks at 0.318 Hz and
0.422Hzcombineintoasinglepeakat0.361Hz.
developed in [33] as a simplified model of the western North base generator without speed governor. All the generators are
Thekeytoselectingasignalformode-meteranalysisisstrong
American power grid. This test system as shown in Fig. 5 equipped with fast voltage regulators and PSSs.
observabilityofthedesiredmode.Thisisreflectedinthespec-
consists of generator buses 17 through 24 and 45, and load The modal analysis indicates that this test system wtirtahlplotasadominantpeak.Theidealsignalhasaverylarge
buses 31 through 41. All the generators except generator PSSs only has four major inter-area oscillation modes aroupnedakatthemode(s)ofinterestandverylowenergyatallother
G17 are set to work in pairs. Each pair consists of a base 0.30 Hz, 0.40 Hz, 0.60 HzTAaBnLdE0II.80 Hz, where the 0.40 fHrezquencies.ThesignalsinFig.4reflectthesedesiredqualities
INTERAREAMODESOFEXAMPLE16-MACHINESYSTEM toanextenttypicalinpowersystems.
generator (G1 to G8) without speed governor and a load- mode has the minimum damping ratio. In order to investigate
following generator (G9 to G16) with speed governor. Only the correlation between the damping ratios of these inter-VII. DATAPREPROCESSINGANDALGORITHMTUNING
four of the load-following generators (G11, G12, G13 and area oscillation modes with the system stress level, the poweTroobtainoptimalperformancefromagivenmode-meteral-
G15) are equipped with steam valve governors, while the rest output of generator G2 and the power demand at bus g3o5rithm,thedatamustbeproperlypreprocessedandanalysis
parametersmustbeproperlyselected.Datapreprocessingstan-
andreactiveloadsisobtainedbypassingindependentGaussian dards and optimal analysis parameters were selected by con-
whitenoisethrougha filter.Ithasbeenhypothesizedthat ducting, many Monte Carlo simulations on the previous test
suchafilterisappropriateforloadmodeling. system [17], results in [5]–[9], and throughextensiveexperi-
Two primary system operating conditions are used with encewithactualsystemdata.
the simulations that follow. With the first condition, termed Properpreprocessingofthedataiscriticaltoalgorithmper-
the 17-machine condition, all generators are connected to formance.In order ofoperation,datapreprocessingincludes:
thesystem.Underthiscondition,themostdominantinterarea removingoutliersandmissingdata;detrending;normalization;
modesareshowninTableI.Withthesecondcondition,gen- andzero-phaseanti-aliasingfilteringanddownsampling.
erator 17 is disconnected from the system; this condition is To use each of the algorithms, several analysis parameters
termed the 16-machine condition. Under this condition, the mustbeselected.Thisincludesalltheparametersin(1)–(12).
dominant interarea modes are shown in Table II. The modes Thefollowingparameterssettingsarerecommendedbasedupon
shown in Tables I and II were calculated by conducting an extensivetesting[5]–[9],[17]: ; forYW
eigenalaysis of the entire system’s small-signal model under andYWS; forN4SID; forYWandYWS;
nominal steady-state operating conditions. The eigenanalysis forN4SID; ;
wasconductedusingthemethodologyin[10]. (amountofdatatouseforfftinYWS).Thesignalschosenfor
Forthecasesthatfollow,thegoalistoestimatethemodes analysisarethefourinFig.4.
at0.318Hzand0.422Hzforthe17-machinecondition,and
themodeat0.361Hzforthe16-machinecondition.Atypical VIII. PERFORMANCE
time-domain simulation consists of driving the small-signal Inthissection,theperformanceoftheYW,YWS,andN4SID
form of the system with the random load variations. The algorithmsareevaluatedinthecontextoftherequirementsout-
system’s response consists of small random variations in the linedinSectionII.Specifically,weconsiderestimationaccuracy
7
14
PSSs only
G11, G12, G13 and G15
12
2
%)10
o (
ati 8
g r
pin 6
m
da 4
2
0
55 60 65 70 75 80 85 90 95 100 105
stress level (%)
Fig. 6. Correlation of the damping ratio of the 0.40 Hz mode with the
systemstresslevelforthe17-machinetestsystem.
are adjusted by the same percentage, which is 100% for
Fig.7. Single-linediagramofmodified17-machinetestsystem.
the base case. It can be seen from Fig. 6 that there exists
a consistent correlation between the damping ratio of the
0.40 Hz mode and the system stress level when there are 14 PSSs only
G11, G12, G13, G15 and G18
only PSSs in the system. Similar correlation also exists for all 12
theotherinter-areaoscillationmodes.Whenrobustcontrollers
%)10
are implemented on generators G11, G12, G13 and G15, o (
the damping ratios of the 0.30 Hz, 0.60 Hz and 0.80 Hz ati 8
g r
modes are greatly improved and maintained above the 5% pin 6
m
damping margin at various system stress levels. However, it da 4
can be seen from Fig. 6 that the effectiveness of the proposed
2
control in improving the damping ratio of the 0.40 Hz mode
is not that significant because the required damping margin 0
55 60 65 70 75 80 85 90 95 100 105
stress level (%)
cannot be achieved when the system is highly stressed. This
ismainlybecausetherearenorobustcontrollersimplemented
Fig. 8. Correlation of the damping ratio of the 0.40 Hz mode with the
in the areas interconnected by the tie-lines between buses 17 systemstresslevelforthemodified17-machinetestsystem.
and 18. The low damping ratio of the 0.4 Hz mode is the
result of large power flow over the line 17-18. Therefore,
the effectiveness of the proposed control requires that robust With the modified 17-machine system in Fig. 7, the impact
controllers be implemented in the areas that contribute to the of topology changes resulted from N −1 contingencies on
poorly damped inter-area oscillation modes. Since it is not the damping ratio of the 0.40 Hz mode for the base case is
feasible for the given 17-machine test system to satisfy this also evaluated. Total 51 topology changes are considered with
requirement, other damping controllers should be deployed in eachassociatedwiththetrippingofonetransmissionline.The
the areas around buses 17 and 18 to complement the existing resultingdampingratioofthe0.40Hzmodeaftercontingency
four robust controllers in improving the small signal stability. is shown in Table II, where there are 15 cases with converged
For example, selective FACTS controllers can be considered power flow after simple line tripping. It is confirmed again
to be installed on the line 17-18. thattheeffectivenessoftheproposedcontrolinimprovingthe
small signal stability is robust to the topology change.
In order to verify the above implementation requirement
for the guaranteed effectiveness of the proposed control, the
dampingratioofthe0.40Hzmodeisexaminedforamodified V. CONCLUSIONS
17-machinesystemasshowninFig.7.Inthismodifiedsystem, In this paper, decentralized robust controllers have been
theoriginalgeneratorG2inFig.5issplitintotwogenerators, developed for generators with steam valve governors to im-
G2 and G18. The new G2 is still equipped with the hydraulic provethedampingratiosoftheinter-areaoscillationmodesby
governor,whilethenewG18isequippedwiththesteamvalve directly affecting the real power in the system. The proposed
governor.Theirmachineparametersarecarefullyselectedsuch dampingcontrolstrategyintroducesanauxiliarycontrolsignal
that the damping ratio of the 0.4 Hz mode for the base case into the governor, creating an additional mechanical torque
keeps the same after the system modification. As shown in on the generator rotor. It has several important advantages
Fig. 8, when an additional robust controller implemented on over the existing damping strategies. The most valuable ad-
generator G18, the damping ratio of 0.4 Hz mode is greatly vantage is the robustness with respect to different operating
improved and the required damping margin is guaranteed at conditions and system topologies. The controller performance
various system stress levels. It confirms again the robustness has been demonstrated by detailed case studies on both small
of the proposed control to varying operating conditions. and medium-sized test systems. It has been shown that the
8
TABLEII
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