Contents lists available at SciVerse ScienceDirect
Electric Power Systems Research
jou rn al h om epa ge: www.elsev ier .com/ locate /epsr
reventive reactive power management for improving voltage stability margin
. Alizadeh Mousavi ∗, M. Bozorg, R. Cherkaouicole Polytechnique Fédérale de Lausanne, EPFL STI-DEC/GR-SCI, ELL 139, Station 11, 1015 Lausanne, Switzerland
r t i c l e i n f o
rticle history:eceived 28 November 2011eceived in revised form 16 August 2012ccepted 10 October 2012vailable online 5 December 2012
eywords:
a b s t r a c t
Voltage stability imposes important limitations on the power systems operation. Adequate voltage sta-bility margin needs to be obtained through the appropriate scheduling of the reactive power resources.The main countermeasures against voltage instability could be distinctly classified into preventive andcorrective control actions. This paper proposes a preventive countermeasure to improve the voltage sta-bility margin through the management of the reactive power and its reserve. The voltage and reactivepower management is studied from the generator’s point of view to maximize effective generator reac-
V-curveoad reactive power reserve (LRPR)enerator reactive power reserve (GRPR)oltage stability margin (VSM)omplementarity constraint
tive power reserve (EGRPR). Detailed model of the generators including the armature and field currentlimits, as well as the switch mode between the voltage control and the reactive power limitations areconsidered to maximize the reactive power capability of the generators in emergency states. One-stageand two-stage optimization approaches are utilized to find the optimum solution. The proposed opti-mization procedure is applied on a 6-bus system and the New England 39-bus system to illustrate theeffectiveness of the method.
. Introduction
The voltage and reactive power management has been a con-ern for power system operators, especially after the restructuringf the power industry. In the restructured environment, the oper-tion of the system is constrained by strict economic constraints.s a result, the network is frequently operated under stress andloser to its operating limits. The evidence of these circumstances isidespread blackouts in the recent two decades. Insufficient volt-
ge and reactive power support was an origin or a factor in theajor power outages worldwide [1].In the context of the electricity market, the voltage and reactive
ower control service is classified as one of the ancillary ser-ices. Until now the system operator is the sole responsible for theanagement of this critical ancillary service to ensure secure and
eliable operation of the system.Sufficient voltage stability margin (VSM) should be provided
o preserve the security of the bulk power system against thehort- and long-term instabilities and subsequent voltage degra-ation and collapse. For this purpose, appropriate control actionshould be continuously acquired, deployed and maintained fromhe control resources. These control actions comprise reactive
ower reserve (RPR) and emergency countermeasures that cane considered, respectively, as preventive and corrective controlctions. The corrective actions include load tap changer blocking,
capacitor switching, voltage and reactive power rescheduling, thenactive power rescheduling, and as the last resort load shedding [2].The main preventive actions against voltage instability are (1) man-agement of reactive power resources through load tap changing,capacitor switching, and (2) implementation of hierarchical or cen-tralized voltage and reactive power control schemes, which bothof them affect the RPR. Also, the active power rescheduling can beincluded in the preventive actions which is not taken into consid-eration in this paper [3]. Here, the focus is only on the managementof the reactive power resources as the most important preventiveaction.
In order to provide RPR appropriately, both reactive power gen-eration and its reserve should be considered simultaneously in theprocurement and the scheduling of the reactive power resources.The RPR can be taken into account from the load or the genera-tor point of view which is called LRPR and GRPR, respectively. Theliterature paid more attention to LRPR than GRPR and so more inves-tigation is needed for the latter. Moreover, the system operatorusually has to manage its reactive power resources for a specifiedactive power dispatch obtained from the active power market. Forthis purpose, it is assumed that the management of the active andreactive power is decoupled. Furthermore, the increasing interestfor the setup of a reactive power market, raise the interest for RPRanalysis from the generators’ side, since they are the main providersof this service. As a result, this paper focuses particularly on the
GRPR.
In this paper, an optimization procedure is proposed for reactivepower management considering an operating point correlated to avoltage collapse point to improve the VSM. The aim of the proposed
O. Alizadeh Mousavi et al. / Electric Power Systems Research 96 (2013) 36– 46 37
RPR fo
sgLmstaa
2
ssctga
silmrfibitgoaictvit
aopnRr
Fig. 1. LRPR, TGRPR, and EG
cheme is to distinguish and to improve the effective RPR of theenerators. To deal with it, in Section 2, fundamentals of GRPR andRPR, are discussed more in depth. The proposed reactive poweranagement method regarding VSM is presented based on one-
tage and two-stage optimization approaches in Section 3. Finally,he proposed method is applied and tested on a 6-bus test systemnd on the 39-bus New England system. The simulation results andnalysis are given in Section 4.
. Fundamentals on reactive power reserve
The RPR is a spare reactive power capability available in theystem to assist the voltage control. This capability should be con-idered to respond to unforeseen events that lead to a suddenhange of reactive power requirement. The system operator needso assign sufficient RPR on the best response resources. Thus, theenerators are commonly the main resource of RPR which they arelso referred as spinning RPR.
The RPR can be viewed from the load’s and the generator’s per-pective. The two bus test system, shown in Fig. 1a, is used tollustrate the various viewpoints of the RPR. A generator and aoad are connected to bus 1 and bus 2, respectively. The QV-curve
ethod, for which more details are given in [4], is used to obtain theeactive power margin to a voltage collapse point. For this purposectitious reactive power supports Qf’s are connected to certain loaduses referred as pilot nodes. Here, the term “pilot node” is explic-
tly used for this purpose. The QV-curve, shown in Fig. 1c, expresseshe relationship between the reactive power support (Qf) at theiven bus and the voltage (V) at that bus [2]. The minimum pointf the QV-curve shows the reactive power margin until the volt-ge instability. This point is called voltage collapse point and it isndicated by the white circle. The current operating point withoutompensation (Qf = 0) is indicated by the black circle. The genera-or reactive power output of the current operating point and theoltage collapse point are shown on the generator capability curven Fig. 1b. In this paper, the optimal power flow is used to calculatehe reactive power margin to the voltage collapse point [5].
The load RPR (LRPR), shown in Fig. 1c, is defined as the minimummount of the reactive load increase for which the system loses itsperability. According to the literature, it is also referred as reactive
ower margin. The generator RPR (GRPR) focuses on the effective-ess of the provided RPR by each generator. Technical generatorPR (TGRPR), is defined as the difference between the maximumeactive power capability of the generator and its reactive power
r the two bus test system.
generation at the current operating point. This quantity may notrepresent the useful quantity of the GRPR since at the collapse pointall the amount of the TGRPR cannot be utilized. Effective generatorRPR (EGRPR), as achievable representative of the GRPR, is definedas the difference between the generator’s reactive power output atthe voltage collapse point and the generator’s reactive power out-put at the current operating point. The TGRPR is an upper bound forthe EGRPR. The LRPR, the TGRPR, and the EGRPR for the two bus testsystem are shown in Fig. 1c and b.
The system operator defines the set-points of the voltage andreactive power controllers by using different criteria such as mini-mization of reactive power injection (or maximization of TGRPR),minimization of voltage profile deviation, and minimization oftransmission losses. These different objectives would result intodifferent amount of RPR and consequently different security mar-gins. Nevertheless, the RPRs should be appropriately managed fromthe available resources to enhance the VSM.
Improving the VSM has been considered in the literature in dif-ferent ways. The proposed VAR scheduling methods in [6–8] add apenalty factor to the OPF to maximize the VSM. The penalty factor isderived from the eigenvectors and/or the generators’ participationfactors related to the Jacobian matrix.
RPR provision is widely proposed in literature based on: (a)security constrained OPF (SCOPF) to assess the RPR with differ-ent constraints [3,9] and (b) voltage stability constrained OPF(VSCOPF) to determine preventive [10,11] and corrective [10] con-trols considering voltage stability.
Regarding the literatures on LRPR [12], defines a reactive reserveas the sum of the exhausted reactive reserves at the minimumpoint of the QV-curve. The RPR-based contingency constrained OPF(RCCOPF) presented in [3] utilizes a decomposition method to solvethe preventive voltage control in normal state while consideringthe active power margin of post-contingency states. The proposedRPR management in [5] utilizes a two level Benders decomposition,including a base case and stressed cases, to ensure the feasibility ofthe stressed cases.
Most of the studies on GRPR like in [13] and [14] are performedon TGRPR since it can be calculated easily regardless stability analy-sis. On the other hand, EGRPR depends on the generators capabilitycurve and the network characteristics [15]. That means the max-
imization of TGRPR does not imply necessarily the maximizationof EGRPR all the times. The GRPR is studied from the EGRPR pointof view more in depth in [15] and [16]. The EGRPR for a bus or anarea is determined in [17] as the weighted sum of the individual
3 Powe
Ragr
RRtst
trv
ppoTiip
dcmr
3m
rsssmctuQmt(
Q
Q
wvfis(aemot
vbe
8 O. Alizadeh Mousavi et al. / Electric
PR of generators at the minimum of the QV-curve. The proposedpproach in [18], determines the minimum RPR to face a contin-ency, while stressing the system in its pre-contingency state, untileaching an unacceptable post-contingency response.
The proposed two-step approach in [9] determines the requiredPR first by finding, using a SCOPF, the minimum overall neededPR from the generators for postulated contingencies. Then, addi-ional RPR requirements are determined to account for the dynamicystem behavior and to ensure the voltage stability of certain con-ingencies.
Reference [19] investigates the correlative relationship betweenhe GRPR and the system VSMs for on-line monitoring. A nonlinearelationship between the GRPR and the VSMs and the voltage limitsiolations is investigated in [20].
This classification of RPR is necessary to distinguish the pur-ose of each study and to avoid mixing different concepts. In thisaper, we concentrate on GRPR and specifically the maximizationf EGRPR as the main preventive action against voltage instability.his optimization determines the reactive power generation andts reserve for each generator such that maximum voltage stabil-ty can be attained for the system. Note that this optimization iserformed for a given active power operating point.
This preventive action by increasing the security margin canecrease or even remove the necessity of the corrective action inase of contingency. Furthermore, application of the proposed opti-ization method can be utilized as the objective of tertiary voltage
egulation (TVR), to improve VSMs.
. Proposed method for reactive power reserveanagement
The management of the reactive power generation and itseserve is a correlated task for ensuring the voltage stability of theystem that strongly depends on the generators and transmissionystem capabilities. The power supplied by a generator is con-trained by its capability curve. For a given active power output, theaximum reactive power support of a generator is obtained while
onsidering the limitation of the field current (Qri), the limitation ofhe armature current (Qai) and the under-excitation limit [21]. Thender-excitation limit is considered by the inequality constrainti > Q min
i, where Q min
iis negative and represents the generator
inimum reactive power output. The maximum produced reac-ive power regarding the field and armature limitations is given by1) and (2), respectively.
maxi = min
⎧⎨⎩Qri = −V2
i
Xsi+
√V2
iI2fi
X2si
− P2i
⎫⎬⎭ (1)
maxi = min
{Qai =
√V2
i· I2
ai− P2
i(2)
here i is the index of the generators, Vi is the generator terminaloltage, Pi is the generator active power output, Ifi is the maximumeld current, Iai is the maximum armature current, and Xsi is theynchronous reactance. Q max
iis defined as the minimum of (1) and
2) to avoid an additional constraint on the upper limit of Qi andlso additional variables for complementarity constraints as it isxplained in the next paragraph and Section 3.1. These detailedodels of the generator operating limits must be considered in
rder to utilize the maximum reactive power capability and to meethe reactive power demands during emergency states [5].
Moreover, three modes of generator operation, namely withinoltage control range, over-excitation and under-excitation shoulde considered in the evaluation of the VSM and RPR. Over/underxcitation is considered when the maximum reactive power limit
r Systems Research 96 (2013) 36– 46
is reached [9]. The generator switch between the constant terminalvoltage and the constant reactive power output is handled by thefollowing complementarity problem:
0 ≤ (Q Ci − Q min
i ) ⊥ Vuei ≥ 0 (3)
0 ≤ (Q maxi − Q C
i ) ⊥ Voei ≥ 0 (4)
VCi = V∗
i + Vuei − Voe
i (5)
where the operator ⊥ denotes the complementarity of two quanti-ties. The voltage magnitudes at the collapse point (VC
i) are defined
as the sum of the voltage at the operating point (V∗i
) plus the under-excitation voltage (Vue
i) and minus the over-excitation voltage (Voe
i)
[22]. In this formulation (Q Ci
) is the generator reactive power out-put at the collapse point. Q max
iis the maximum reactive power
output obtained from (1) and (2). Q mini
is the minimum reactivepower output that represents the under-excitation limit. The com-plementarity model allows the voltage levels to be changed whengenerators reach reactive power limits. These complementarityconstraints (3) and (4) could be, respectively, taken into consid-eration by the following nonlinear constraints:
(Q Ci − Q min
i ) · Vuei ≤ 0 (i ∈ NG : PV nodes) (6)
(Q maxi − Q C
i ) · Voei ≤ 0 (i ∈ NG : PV nodes) (7)
In order to prevent a strict complementarity constraint and therelated problems [22], the righthand sides’ zeros of (6) and (7) arereplaced by a small positive number (ε = 10−7).
For the ith generator, TGRPR and EGRPR are defined by the fol-lowing equations, as shown in Fig. 1c:
TGRPRi = Q maxi − Qi (8)
EGRPRi = Q Ci − Qi (9)
where Qi is the generator reactive power output at the operatingpoint. Note that in the definition of TGRPR given in (8), the first term,Q max
i, depends on the voltage magnitude of the generators.
The objective of the proposed method for the reactive powergeneration and reserve management is to maximize EGRPR and asa consequence to improve the VSMs at the pilot nodes. The pilotnodes are chosen in such a way that by maintaining their voltagesat a given level, the voltages of the whole region buses are kept ata desirable level. As a result, increasing the VSM of the pilot nodesinherently improves the voltage stability of the system.
Therefore, the maximization of the EGRPR, given in (9), can alsobe formulated as follows. Similar formulation could be presentedfor the maximization of TGRPR.
max EGRPR = max∑i ∈ NG
{Q Ci − Qi}
↔ min∑i ∈ NG
{Qi − Q Ci }
(10)
This optimization consists in the minimization of the differencebetween the sum of the generators reactive power output at theoperating (�Qi) and the collapse point (˙Q C
i). The vector of the
control variables (u) includes the voltage of PV generators and thereactive power output of PQ generators. The control variables couldbe considered as the complicating variables since they are presentin the current operating point and the voltage collapse point. There-fore, this optimization problem could be solved according to twodifferent ways: one-stage (simultaneous) or two-stage [22]. In thispaper, these two optimization approaches are investigated to solve
the proposed optimization.
The overall structures and general formulation of the twoapproaches are illustrated in Fig. 2. In the presented general for-mulation f(x,xC,u,s) is maximization of EGRPR. x indicates the vector
O. Alizadeh Mousavi et al. / Electric Power Systems Research 96 (2013) 36– 46 39
ge an
othtAz(u
ttmaepalwpa
mofomo
3a
cmrcza
-
Fig. 2. Structure of one-sta
f the state variables and s is the vector of injected fictitious reac-ive power for pilot nodes at the voltage collapse point. g(x,u) and(x,u) correspond to equality and inequality constraints, respec-ively. Superscript “C” represents the variables at the collapse point.dditionally for two-stage optimization, e(xC,u,s) is the maximi-ation of the generators reactive power output at the collapse point˙Q C
i). u* stands for the values of the control variables which are
sed as the input for this optimization.Before explaining the details of these two approaches, it is worth
o mention that the proposed optimization looks like similar to theraditional voltage and reactive power management based on the
aximum loading margin [22], since both of them consider thenalysis of a current operating point and a collapse point. How-ver, the specifications of the objective function and the collapseoint for the maximization of the loading margin and the EGRPRre quite different. The collapse point for the maximization of theoading margin is obtained by increasing the loading level of the
hole system linearly in one direction until reaching a bifurcationoint whereas the collapse point in case of EGRPR maximization isttained based on the reactive power margins at the pilot nodes.
In comparison to the contingency constrained optimizationethods [(3), (5) and (9)], the proposed approach takes into account
nly one voltage emergency state that is the collapse point. There-ore, the proposed method does not need any contingency selectionr analysis of several contingencies. However, the proposed opti-ization could be developed to provide appropriate EGRPR for a set
f postulated contingencies.
.1. Simultaneous optimization of EGRPR for current operatingnd collapse points
This approach assumes that the state variables at both of theurrent operating point and the voltage collapse point are the opti-ization variables in addition to the control variables. Hence, the
elationship between the current operating point and the voltageollapse point is taken into consideration. Therefore, the maximi-
ation of EGRPR given by (10) is subjected to the following equalitynd inequality constraints:
here the variables without and with superscript C represent theariables related to the operating point and the collapse point,espectively. In this formulation NB, NL, NG and NP are the num-er of buses, the number of lines, the number of generators, andhe number of pilot nodes, respectively. n and m are the index ofuses, l is the index of lines, i is the index of generators, d is the
ndex of demands and p is the index of pilot nodes. Vn is the voltageagnitude of bus n, and �nm is the voltage angle difference between
he buses n and m. Gnm and Bnm are the real and imaginary part ofhe {n,m} element of the admittance matrices. The active powernd the reactive power are shown by P and Q, respectively. Theeactive power capacity limits of each generator are specified bymini
and Q maxi
. The limits of voltages at bus n are Vminn and Vmax
n .he maximum transfer capability of the transmission lines is giveny Imax
l.
At the current operating point (resp. voltage collapse point) thective and reactive power balance equality constraints are given by11) and (12) (resp. (16) and (17)). Note that the reactive power bal-nce equality constraint given by (17) is different from (12). In (17)he variable Qfp is added that demonstrates the fictitious reactiveower injection (load) at the pilot nodes. The positive and nega-ive values of Qfp are devoted to the reactive power consumptionnd generation, respectively. The transmission lines flow limit isonsidered in (13) (resp. (18)). The limits of the generators reactiveower and the voltage magnitude at each bus are considered in14) and (15), respectively. Note that the generators limits are con-idered with their capability curves obtained from (1) and (2). Athe voltage collapse point, the limitations of the generators reactiveower output and the voltage of the buses are given by (19)–(24).
V∗i
and Q ∗i
in (20) and (24) are the voltage and reactive powerf the PV and PQ generators at the operating point, respectively.hese two equality constraints in addition to the inequality con-traints (21) and (22) correlate the current operating point and theollapse point while considering the complementarity constraintsentioned by (5)–(7). It should be noted that at the collapse point,
he generators switching to the under-excited mode is not takennto consideration. In fact, in response to the increase of the ficti-ious reactive power loads at the pilot nodes, the generators needather to switch from the voltage control mode to the over-excitedode in order to increase their reactive power support at the col-
apse point. Besides the fact that the PQ generators reactive powerutput are the same at the current operating point and at the volt-ge collapse point (according to (24)), they can participate in theptimization process.
.2. Two-stage optimization for the current operating point andhe collapse point
The maximization of EGRPR given by (10) can be solved inwo stages through decomposing the problem into two smallerroblems. The first stage determines an operating point and theecond stage calculates the collapse point based on the results ofhe first stage. These two smaller problems are generally solved
uch simpler than a single larger one [22]. However, a drawbackf this approach is that the solution of the two-stage optimizations obtained iteratively. Several ways can be proposed to correlatehese two stages and to fix the operating point [22].
Ref. [16] decomposes the problem in (10) using the Bender’s
ecomposition method with a master problem (operating point)nd a sub problem (collapse point). The proposed method in16] does not consider the generators switch between con-tant terminal voltage and constant reactive power output. The
r Systems Research 96 (2013) 36– 46
Benders’ decomposition provides acceptable results in the absenceof the complementarity constraints (5)–(7) as demonstrated in [16].However, it is not effective whenever these constraints are takeninto considerations. Here, to overcome this problem a Subgradientmethod is used as a bi-level optimization method.
Subgradient methods are iterative first-order methods whichcan be applied to a variety of problems [23]. They are simple toimplement and their computational burden is small. However, theirprogress to the optimum is slow and oscillatory [24]. In this paper,a subgradient method iteratively updates the control variables asfollow (see Fig. 2):
u(k+1) = u(k) + ˛k · �(k)
||�(k)|| (25)
Here u(k) is the vector of the control variable at kth iteration, �(k)
is the subgradient of the objective function (e(k)) with respect tou(k), and ˛k > 0 is the kth step size. ||�(k)|| is the Euclidean normof �(k) which normalize the magnitude of the subgradients for theset of PV and PQ generators separately. The subgradients �(k) areobtained from the solution of the optimization at the voltage col-lapse point (sub-problem). For the voltage at PV nodes and thereactive power at PQ nodes (the control variables) �(k) compo-nents are the Lagrangian multipliers of (20) and (24), respectively.Many different types of step size schemes can be used [23]. In thispaper, the so called non summable but square summable step sizeis selected. It is given as follows:
˛k = a
b + k(26)
where a and b are positive constant scalars. Here, 1.5 ≤ a ≤ 2.5 andb = 20. They should be chosen for each specific problem. Note thatwhenever the obtained control variables in (25) in kth iterationgo beyond their upper or lower limits, their values are set at thecorresponding limits as shown in Fig. 2.
Then, a power flow is performed to evaluate the EGRPR in eachiteration k and to determine, respectively, the voltage and reactivepower at PQ and PV generators. If the power flow solution demon-strates that voltages of PQ generators are beyond the limits, thecorresponding control variables are iteratively reduced by 2.5%.This procedure iterates till obtaining the state variables (voltagemagnitudes and angles) within allowed limits.
Since the subgradient method is not a descent (resp. ascent)method in minimization (resp. maximization), the best objectivevalue found so far (f (k)
best) should be kept for tracking the optimum.
f (k)best
= max{f (u(1)), . . . , f (u(k))} (27)
The algorithm converges when limk→∞
f (u(k)) − f ∗ ≤ ı where f*
denotes the optimum solution of the problem obtained fromone-stage approach and ı is a small number. More sophisticatedstopping criteria are presented in [23].
In the two-stage optimization, the appropriate selection of thestarting point (u(1)) is an important subject. The initial voltage setpoints equal to 1 is not always an appropriate starting point. Theauthors found that, the results of the TGRPR maximization problem,given by (8), is an appropriate starting point; because its result leadsto wider feasible region in the EGRPR maximization problem.
3.3. Pilot node selection
The pilot nodes are the most voltage sensitive nodes that reflectthe state of the voltage in a control zone. The optimal selection of
the pilot nodes for a zonal or a secondary voltage control is stud-ied in different literatures [25]. They could be selected for a set ofcontingencies [25] or random reactive power disturbances [26,27].In this paper, the term pilot node is explicitly used for the buses
O. Alizadeh Mousavi et al. / Electric Power Systems Research 96 (2013) 36– 46 41
Table 1The 6-bus system generators data.
Bus P0g (MW) Pmax
g (MW) Capability curve
If (pu) Ia (pu) Xs (pu)
wIn
4
oeobeIo3wc
(((
cTti
“
4
36ec
stage approach. The maximum difference between their solutionsis lower than 1.2%.
Besides, in the case with PV generators only (3 PV generators),the amounts of EGRPR for the three approaches are close to each
Table 2The 6-bus system bus data.
Gen 1 1 – 200Gen 2 2 50 150Gen 3 3 60 180
here the fictitious reactive power injections (Qf) are connected to.t is assumed that these pilot nodes are the most voltage sensitiveodes.
. Test case study
The proposed method for the maximization of EGRPR is testedn 6-bus system for two different loading levels to demonstrate thevolution of EGRPR of each generator. The simulations are carriedut for the case with PV generators only and the case includingoth PV and PQ generators. Then, the mechanism according whichach generator increases its participation in EGRPR is described.n addition, the effectiveness of the proposed methods in the casef a larger power system is investigated using the New England9-bus system. For each case study, the optimization is performedith three different approaches listed below and the results are
ompared.
1) minimization of the generated reactive power (min. Q)2) maximization of the EGRPR in one-stage (max. EGRPR Os)3) maximization of the EGRPR in two-stage (max. EGRPR Ts)
The first approach is taken into consideration as the referencease that aims at minimizing the total generated reactive power.his objective could be interpreted as the minimization of the reac-ive power cost. The last two approaches (max. EGRPR), as describedn Section 3, have the same objective.
The OPF model is a nonlinear problem which is solved usingfmincon” with interior-point algorithm [28] in MATLAB R2011a.
.1. 6-Bus study case
The 6-bus test system is shown in Fig. 3. The system contains
generators, 3 loads, and 11 transmission lines. The data for the-bus system are provided in Tables 1–3. Table 1 shows the gen-rators’ data including the current active power dispatch and theapability curves according to (1) and (2). The buses data are given
Fig. 3. One line diagram of 6-bus system.
1.37 2.20 0.201.35 1.65 0.251.36 1.98 0.23
in Table 2, including each bus type and its active and reactive powerload. Table 3 provides the branches data, including the two endsbus numbers, the series resistances (R), the series inductances (XL),the shunt conductance (XG), and the maximum transfer capabilities(Imax). The voltage deviation of all buses is acceptable within ±10%of the nominal voltage. In this study case, bus 5 is selected as thepilot node since it is directly connected to all generators.
In one simulation, all generators are assumed as PV generators(G1 is the swing generator and G2 and G3 are PV generators) andin another one, G3 is considered as PQ generator. For the swinggenerator, PG1 is considered as an unknown variable to compensatethe active power losses. The change of the active power losses is dueto different scheduling of voltage and reactive power resources andthus different reactive power flows. The simulation is performed fortwo load level; the load level mentioned in Table 1 (low loading)and a 20% increase of that load (high loading).
The simulation results with different approaches for the high(low) loading level are shown in Fig. 4 and Tables 4 and 5 (Fig. 5and Tables 6 and 7). In Tables 4–7, the obtained solutions for thecontrol variables are given in bold.
Figs. 4 and 5 illustrate the sum of all generators EGRPR for the 3different approaches for high and low loading levels, respectively.The horizontal axes are devoted to the number of iterations, sincethe two-stage approach is iterative. The comparison of Figs. 4 and 5shows that, the EGRPR is lower for high loading level than for lowloading level for all approaches. Likewise, the EGRPR of the two-stage approach approximately converges to the amount of the one-
Bus Typea Pd (MW) Qd (MVAR)
1 0 0 02 1 0 03 1 0 04 2 70 705 3 70 706 2 70 70
a The numbers stand for the bus types are as follow: 0 for swing node, 1 for PVnode, 2 for PQ node, and 3 for Pilot node.
he obtained solutions for the control variables are given in bold.
ther. The reason is that the generators’ switch between the differ-nt operation modes (given by (5)–(7)) allows them to change theiroltages in order to increase their reactive power output at the col-apse point. If the generator mode switch at the collapse point is not
aken into consideration, as in the case in [16], the EGRPR becomesarger for max EGRPR than min Q. As shown in Tables 4 and 6, thisssue is more evident for the high loading level than for the lowoading level because the system reaches its limits.
able 6ptimization results of 6-bus system for low loading level with 3 PV generators.
Objective G1
Min Q Vi (pu) 1.1000
Voei
(pu) 0.0000
Qi (MVAR) 17.32
Q Ci
(MVAR) 71.46
EGRPR (MVAR) 54.14
Max EGRPR one stage Vi (pu) 1.0941
Voei
(pu) 0.0000
Qi (MVAR) 54.69
Q Ci
(MVAR) 89.39
EGRPR (MVAR) 34.69
Max EGRPR two stages Vi (pu) 1.1000
Voei
(pu) 0.0000
Qi (MVAR) 52.23
Q Ci
(MVAR) 91.01
EGRPR (MVAR) 38.78
he obtained solutions for the control variables are given in bold.
51.14 – 91.43
Furthermore, in the case with PQ generator, although the gen-erator switch mode is considered, max EGRPR noticeably increasethe amount of EGRPR in comparison to min Q. It is due to the factthat min Q aims to minimize the sum of the generators reactive
power outputs at the operating point and this value for the PQ gen-erators is kept constant for the collapse point according to (23).But for max EGRPR, the reactive power outputs of the PQ gen-erators are adjusted in such a way it allows the PV generators
a Here, load shedding is the case and it is executed with constant power factor.
o increase their EGRPR. As shown in Table 5 (resp. in Table 7),hese results demonstrate 17.83 MVAR (20.97 MVAR) increase inGRPR, and 21.93 MVAR (29.29 MVAR) increase in the generatorseactive power outputs at the voltage collapse point, with only.1 MVAR (8.32 MVAR) increase in generation reactive power at theurrent operating point. The reactive power output of G3 increasesrom 88.09 MVAR (64.94 MVAR) to 120.35 MVAR (121.57 MVAR), toeduce the required reactive power of other generators at the cur-ent operating point. It helps G1 and G2 to decrease their reactive
ower generation at the current operating point, and to improveheir reactive power output at the voltage collapse point, and con-equently increase their EGRPRs, as shown in Table 5 (Table 7).omparing the cases with three and two PV generators, the system
EGRPR decreases from 101.28 MVAR (141.54 MVAR) to 91.85 MVAR(132.068 MVAR) because the number of voltage control generatorsis lower.
In the optimization process, the EGRPR of each generatorincreases by different ways as depicted in Fig. 6a–c. In these fig-ures, the EGRPR for min Q and max EGRPR are given by EGRPR0
iand EGRPR∗
i, respectively. In the first case, as shown in Fig. 6a, a
small increase of Qi, leads to a higher increase of Q Ci
. In the secondcase, as shown in Fig. 6b, with a large decrease of Qi, Q C
idecreases
slightly. Finally in Fig. 6c, since Q Ci
reaches the maximum capabilityof the generator (Q max
Operating Point - Befor Optimiza tion Operating Point - After Optimiza tion Voltage Coll aps Point - Befor Optimiza tion Voltage Coll aps Point - After Optimiza tion
a)
Pi
Qi
EGRPRi0
Operating Point - Befor Optimiza tion Operating Point - After Optimiza tion Voltage Coll aps Point - Befor Optimiza tion Voltage Coll aps Point - After Optimiza tion
b)
EGRPRi*
Pi
Qi
EGRPRi0
Operating Point - Befor Optimiza tion Operating Point - After Optimiza tion Voltage Coll aps Point - Befor Optimiza tion Voltage Coll aps Point - After Optimiza tion
EGRPRi*
c)
EGRPRi0
EGRPRi*
a)
EGRPRi0
b)
EGRPRi*
EGRPRi0
EGRPRi*
c)
Operating Poin t – Min . QOperating Poin t –Ma x. EGR PRVoltage Col lapse Po int – Min . QVoltage Col lapse Po int –Ma x. E GRP R
Operating Poin t – Min . QOperating Poin t –Ma x. EGR PRVoltage Col lapse Po int – Min . QVoltage Col lapse Po int –Ma x. E GRP R
Operating Poin t – Min . QOperating Poin t –Ma x. EGR PRVoltage Col lapse Po int – Min . QVoltage Col lapse Po int –Ma x. E GRP R
FiQ
rtltcFo
ig. 6. Increase of EGRPR by different ways: (a) low increase of Qi with higherncrease of QCi , (b) big descend of Qi with lower decrease of QCi , and (c) drop in
i while QCi is reached to the generator’s maximum output (Q maxi
).
The effectiveness of the proposed optimization methodsegarding the transmission system contingencies is investigated inhe case of 6-bus system with 2 PV and 1 PQ generators for the highoading level. As shown in Table 8, three contingencies are selected
o illustrate three different types of post-contingency states. Theontingencies are studied in two cases, min Q and max EGRPR Os.or each contingency, the VSM as well as the type and the amountf the corrective actions for unstable cases are given. In this respect,
r Systems Research 96 (2013) 36– 46
the VSM is assumed to be the sum of the LRPR for all load buses. Forevery load bus an OPF is solved to calculate VSM from QV-curve.The negative (resp. positive) values of the VSM indicate the marginto instability (resp. stability). Note that the voltage and the reactivepower rescheduling, as well as the active power rescheduling andthe load shedding are considered as corrective actions, respectively.
In the case of contingency L1, the system is stable for min Qand max EGRPR, but the VSM is much higher in the latter. For con-tingency L3, the system is unstable for min Q while it remainsstable in max EGRPR. For min Q, the required corrective actionwas the voltage and reactive power rescheduling. The proposedoptimization decreases the number of such unstable contingenciesand consequently the necessity of corrective actions. This situa-tion is more observed for the high loading level. For contingencyL8, the system is unstable in min Q and max EGRPR since the OPFsfor obtaining the VSMs do not converge. Indeed, the voltage andreactive power rescheduling is not effective and corrective actionssuch as the active power rescheduling or load shedding shouldbe taken. Thus, the amount of the corrective action (here loadshedding) is the same in min Q and max EGRPR. It is worth to men-tion that different OPFs are developed to calculate the correctiveactions.
4.2. New England 39-bus system
The effectiveness of the proposed method is demonstrated aswell using New England 39-bus system as depicted in Fig. 7. Thedata of the generators for the New England 39-bus system is pro-vided in Table 9. The data of the buses and the branches can befound in literature. The voltage of each bus is acceptable between0.94 and 1.06 pu In this simulation, G6 and G8 are assumed as PQgenerators.
As it is mentioned in Section 3.3 the results of the proposedmethod depends on the selection of the pilot nodes. Thus, buses 4,7, 8, 12, 20, 21, 23, 24, 25 and 28 are selected as the most voltagesensitive buses based on the presented results in [27].
The simulation results for New England 39-bus system are givenin Figs. 8–10. The system EGRPR increases by the proposed one-and two-stage reactive power scheduling as illustrated in Fig. 8.The 18.07 MVAR increase in the reactive power output of genera-tors is the additional cost for gaining 86.06 MVAR increase in thegenerator’s reactive power support at the voltage collapse point.The share of each generator’s in the system EGRPR by differentoptimization approaches is shown in Fig. 9. The generators reac-tive power outputs at the current operating point and the voltagecollapse point with different optimization approaches is demon-strated in Fig. 10. The EGRPRs of G1 and G5 significantly increase,since they are the most important generators for the voltage andreactive power control. On the other hand, the EGRPRs of G4, G9,and G10 decrease because their RPRs are not effective for the volt-age and reactive power control. Also the EGRPRs of G2, G3, and G7remain approximately unchanged. Note that the EGRPR of G6 andG8, as PQ generators, is equal to zero. As illustrated in Fig. 10, theyplay a role in the optimization by increasing their reactive poweroutput.
As Fig. 10 shows, the reactive power output of all generators atthe operating point increase, except G1 and G5. These generatorsreactive power outputs at the operating point are reduced signif-icantly, while their reactive power outputs at the voltage collapsepoint remain unchanged, and as a result, the support of G1 andG5 for voltage control (EGRPR1 and EGRPR5) drastically increases.Moreover, as shown in Fig. 10, the reactive power output of G7 and
G10 at the voltage collapse point increase. For the rest of the gen-erators, the increase of the reactive power output at the voltagecollapse point is less than 1%.
O. Alizadeh Mousavi et al. / Electric Power Systems Research 96 (2013) 36– 46 45
Table 9The New England 39-bus system generators data.
Fig. 7. One line diagram of the New England 39-bus system.
10 20 30 40 502400
2500
2600
2700
2800
Iteration
EG
RPR
(M
VA
R)
Min. QMax. EGRPR one-stageMax. EGRPR two-stage
Fig. 9. EGRPR’s of each generator for different optimization approaches for New
Fig. 8. Sum of all generators’ EGRPR for 39-bus system.
England 39-bus system.
46 O. Alizadeh Mousavi et al. / Electric Powe
Fig. 10. Generated reactive power of each generator at the current operatingpE
4
toctcsfcop
5
aamsalvtr
ltstE
aHfmmgwturmga
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
(2010) 938–951.[27] S. Mei, X. Zhang, M. Cao, Power Grid Complexity, Springer, Beijing, 2011.
oint and the voltage collapse point for different optimization approaches for Newngland 39-bus system.
.3. Discussion on the convergence of two-stage method
The main specification of the two-stage approach in comparisono the one-stage approach is that it deals with an adjustment of theperating point and a smaller optimization problem to derive theollapse points in an iterative manner. The problem solution relatedo the collapse point is more time consuming since it refers to theomplementarity constraints. The simulation results in the 6-busystem demonstrate that the solution of the two-stage approach isaster than one-stage approach, while in the 39-bus system it is theontrary. As a result in this optimization, the number of iterationsffsets the advantage of the decomposition, particularly when theower system size increases.
. Conclusion
The voltage and reactive power scheduling along with RPR man-gement are proposed as a convenient preventive countermeasuregainst voltage instability. The suggested optimization methodaximizes EGRPR to improve the VSM at the pilot nodes using one-
tage or two-stage approaches. For the small systems, two-stagepproach is more efficient than one-stage, but it is not effective inarger study cases, since it iteratively solves a large problem at theoltage collapse point. The solutions are compared to the result ofhe minimization of the generators reactive power output as theeference case.
Owing to the detailed modeling of the generators reactive powerimits and the switch mode that allows them to increase their reac-ive power output at the collapse point, the EGRPR does not increaseignificantly in the system with PV generators only. But in the sys-em with PQ generators, this optimization effectively increases theGRPR.
Contingency analysis demonstrates that the necessity of voltagend reactive power rescheduling as corrective actions are reduced.owever, some contingencies need the corrective actions in the
orm of active power scheduling or load shedding because the opti-ization is not effective in these cases. Therefore, the proposedethod ensures the maximum attainable preventive security mar-
in from the available voltage and reactive power control resourcesithout any change in the active power schedule. This can dis-
inguish the effective RPR of generators in the system that can betilized in the realization of reactive power markets and unbundledeactive power support services. Furthermore, since this opti-
ization method takes into account the role of both PV and PQ
enerators, it can be extended for the systems with other voltagend reactive power control devices.
[
r Systems Research 96 (2013) 36– 46
Acknowledgements
The authors thankfully acknowledge Swiss Electric Research asthe results reported in this paper have been carried out within theframework of the research project “Security of Multi-Area PowerSystems (MARS)”.
References
[1] O. Alizadeh Mousavi, R. Cherkaoui, “Literature survey on fundamental issuesof voltage and reactive power control”, October 2011 [Online]. Available at:http://mars.ethz.ch/en/researh-and-publications/publications.html
[2] T.V. Cutsem, C. Vournas, Voltage Stability of Electric Power Systems, KluwerAcad. Publ., Boston/London/Dordrecht, 1998.
[3] H. Song, B. Lee, S.H. Kwon, V. Ajjarapu, Reactive reserve-based contingencyconstrained optimal power flow (RCCOPF) for enhancement of voltage stabilitymargins, IEEE Transactions on Power Systems 18 (4) (2003) 1538–1546.
[4] B.H. Chowdhury, C.W. Taylor, Voltage stability analysis: V–Q power flow simu-lation versus dynamic simulation, IEEE Transactions on Power Systems 15 (4)(2000) 1354–1359.
[5] D. Feng, B.H. Chowdhury, M.L. Crow, L. Acar, Improving voltage stability byreactive power reserve management, IEEE Transactions on Power Systems 20(1) (2005) 338–345.
[6] A. Rabiee, MVAR management using generator participation factors forimproving voltage stability margin, Journal of applied sciences 9 (11) (2009)2123–2129.
[7] T.V. Menezes, L.C.P. da Silva, V.F. da Costa, Dynamic VAR sources scheduling forimproving voltage stability margin, IEEE Transactions on Power Systems 18 (2)(2003) 969–971.
[8] T.V. Menezes, L.C.P. da Silva, C.M. Affonso, V.F. da Costa, MVAR manage-ment on the pre-dispatch problem for improving voltage stability margin, IEEProceedings Generation, Transmission & Distribution 151 (6) (2004) 665–672.
[9] F. Capitanescu, Assessing reactive power reserves with respect to operatingconstraints and voltage stability, IEEE Transactions on Power Systems 26 (4)(2011) 2224–2234.
10] B. Lee, Y.H. Moon, H. Song, Reactive optimal power flow incorporatingmargin enhancement constraints with nonlinear interior point method, IEEProceedings Generation, Transmission & Distribution 152 (6) (2005) 961–968.
11] F. Capitanescu, T. Van Cutsem, Preventive control of voltage security margins:a multi-contingency sensitivity-based approach, IEEE Transactions on PowerSystems 17 (2) (2002) 358–364.
12] R.A. Schlueter, A voltage stability security assessment method, IEEE Transac-tions on Power Systems 13 (4) (1998) 1423–1438.
13] M. Zima, D. Ernst, On multi-area control in electric power systems, in: Proc. ofPSCC, 2005.
14] Y. Phulpin, M. Begovic, M. Petit, J.B. Heyberger, D. Ernst, Evaluation of net-work equivalents for voltage optimization in multi-area power systems, IEEETransactions on Power Systems 24 (2) (2009) 729–743.
15] P.A. Ruiz, P.W. Sauer, Reactive power reserve issues, in: Proceeding of NorthAmerican Power Symposium, Carbondale, 2006, pp. 539–545.
16] O. Alizadeh Mousavi, M. Bozorg, A. Ahmadi-Khatir, R. Cherkaoui, Reactivepower reserve management: preventive countermeasure for improving volt-age stability margin, in: IEEE Power Eng. Soc. Gen. Meet., 2012.
17] R. Ramanathan, C.W. Taylor, BPA reactive power monitoring and control fol-lowing the August 10, 1996 power failure, in: Proceeding of VI-th SEPOPEConference, 1998.
18] F. Capitanescu, T. Van Cutsem, Evaluation of reactive power reserves withrespect to contingencies, in: Bulk Power Syst. Dyn. and Control V, 2001.
19] L. Bao, Z. Huang, W. Xu, On-line voltage stability monitoring using var reserves,in: IEEE Power Eng. Soc. Gen. Meet., 2003, p. 1754.
20] B. Leonardi, V. Ajjarapu, Investigation of various generator reactive powerreserve (GRPR) definitions for online voltage stability/security assessment, in:Power and Energy Soc. Gen. Meet. – Convers. and Deliv. of Electr. Energy in the21st Century, 2008, pp. 1–7.
21] A.E. Efthymiadis, Y.I. Guo, Generator reactive power limits and voltage stabil-ity, Power System Control, Operation and Management. Fourth InternationalConference on Power System Control and Management (1996) 196–199.
22] W. Rosehart, C. Roman, A. Schellenberg, Optimal power flow with complemen-tarity constraints, IEEE Transactions on Power Systems 20 (2) (2005) 822–843.
23] S. Boyd, A. Mutapcic, Subgradient Methods Notes for EE364b, Stanford Univer-sity, 2008, April.
24] A.J. Conejo, E. Castillo, R. Minquez, Decomposition techniques in mathematicalprogramming, Engineering and Science Applications (2006).
25] T. Amraee, A. Soroudi, A.M. Ranjbar, Probabilistic determination of pilot pointsfor zonal voltage control, IET Generation, Transmission & Distribution 6 (1)(2012) 1–10.
26] T. Amraee, A.M. Ranjbar, R. Feuillet, Immune-based selection of pilot nodes forsecondary voltage control, European Transactions on Electrical Power 20 (7)
28] G.L. Torres, V.H. Quintana, An interior point method for nonlinear optimalpower flow using voltage rectangular coordinates, IEEE Transactions on PowerSystems 13/4 (1998) 1211–1218.
