1

The Generation of ZIP-V Curves for Tracing Power System Steady State Stationary Behavior Due to Load and Generation Variations

Abstract — The P-V Curve, Q-V Curve, or P-Q-V Curve have beenwidely used to analyze power system behaviors under varying loadingconditions. These curves have been generated under the condition thatthe constant P-Q load component of a bus (or a collection of buses) var-ies, with the constant current load and constant impedance load beingkept indeed “constant”. As such, the physical meaning of these curvescan be easily explained. Motivated by the facts that load models haveprofound impacts on power system behaviors and that the nonlinearload model, ZIP-model, is popular in modeling nonlinear behaviors ofloads, this paper proposes a new class of curves, called ZIP-V curves, tobetter trace power system steady-state stationary behavior due to loadand generation variations. The ZIP-V curves encompass the traditionalP-V, Q-V, P-Q-V curves (constant P-Q load), I-V curve (constant cur-rent load), Z-V curve (constant impedance load), or generalized curvessuch as IP-V (constant current and constant power load), ZP-V (con-stant impedance and constant power load) or IZ-V (constant currentand constant impedance load) curve when the values of correspondingcomponents are kept constant. A tool based on the Continuation PowerFlow (CPFLOW) method useful for generating the ZIP-V curves isdeveloped and its application to generate ZIP-V curves of a 4561-businterconnected power system is illustrated.

1. INTRODUCTION

The P-V Curve, Q-V Curve, or P-Q-V Curve have beenwidely used to analyze power system behavior under varyingloading conditions. Voltage stability analysis and loadabilityanalysis are examples of the application of these curves inpower system analysis. These curves are obtained by Contin-uation Power Flow (CPFLOW) [1] as the constant P-Q loadcomponent of a bus (or a collection of buses) varies, with theconstant current load and constant impedance load beingkept indeed “constant”. As such, the physical meaning ofthese curves can be easily explained and these curves servetheir purpose well - on the conservative side in predictingseveral system limits such as voltage stability limit or load-ability limit.

Load models are known to have profound impacts onpower system behaviors. It is well-recognized that the con-stant P-Q load model gives unsatisfactory results for powersystem voltage stability analysis. The nonlinear load model,ZIP-model which is a combination of constant current, con-stant power and constant impedance, is popular in modelingthe nonlinear behaviors of loads. The purpose of this paper is

not to evaluate the assumption of fixing the values of con-stant current loads and constant impedance loads in generat-ing the P-V Curve, Q-V Curve, or P-Q-V curves; instead itspurpose is to show that the assumption can be relaxed and anew class of curves, called ZIP-V curves can be generated.ZIP-V curves represent power system quasi-steady-statebehaviors as all the values of constant Z, I, P-Q (ZIP) varyand their variations are uniform. The ZIP-V curve canbecome an I-V, Z-V, IP-V, ZP-V or IZ-V curve when thevalues of the corresponding component are kept constant. Anew set of parameterized power flow equations is proposedin the paper. A tool based on the CPFLOW method is devel-oped and its application to generate ZIP-V curves for a 4561-bus interconnected power system is illustrated.

2. CPFLOW METHOD

The CPLFOW method is used in this paper to obtain aclass of ZIP-V curves. We next briefly discuss the funda-mental idea of the CPFLOW method.

Let and . The lowercase rep-resents generation and the lowercase represents loaddemand. The set of power flow equations can be representedin compact form as

(1)

Now one can investigate the steady-state behavior of thepower system under slow variation of both loading condi-tions and real power redispatches. For example, if one needsto trace the power system state from the base-case load-gen-eration condition to a new load-generationcondition , then one can parameterize the set ofpower flow equations as such

(2)

where the load-generation vector b is

(3)

It follows that the parameterized power flow equationsbecome the base-case power flow equations when ,

(4)

and when , the power system is at the new load-gen-eration condition and can be described by

Pi Pgi Pdi–≡ Qi Qgi Qdi–≡ gd

f x( ) P x( ) P–

Q x( ) Q–≡ 0 , where x V θ( , )==

Pd0

Qd0

Pg0, ,[ ]

Pd1

Qd1

Pg1, ,[ ]

F x λ( , ) f x( ) λb–≡ 0=

b P1

P0

–

Q1

Q0

–

≡

λ 0=

F x 0( , ) P x( ) P0

–

Q x( ) Q0

–

0= =

λ 1=Pd

1Qd

1Pg

1, ,[ ]

Hua Li Hsiao-Dong Chiang Member IEEE Fellow

School of Electrical EngineeringCornell UniversityIthaca, NY 14853

Hirotaka YoshidaTechnical Research Center

The Kansai Electric Power Co. Inc.3-11-20, Nakoji, Amagasaki

Hyogo, 661 Japan

Yoshikazu Fukuyama, Yosuke NakanishiPower Engineering Development Lab.

Fuji Electric Corporate R&D, Ltd.No.1, Fuji-machi, Hino-city

Tokyo, 191 Japan

2

(5)

As shown in the above procedure, one can investigate theeffects of varying real power generations as well as varyingload demands on power system steady-state behaviors byanalyzing the set of parameterized power flow equations (2).In fact, one can parameterize any change in PQ loads in con-junction with any change in P generations by selecting anappropriate vector b. One can thus compute the desired P-V,Q-V, P-Q-V curves based on equation (2) by varying theparameter . However this approach fails when the set ofparameterized power flow equations approaches its bifurca-tion point (nose point) due to the numerical ill-condition ofequation (2). Special indirect methods have been constructedto overcome this difficulty[1][2][3][4], such as continuationmethods. CPLFOW is one of them and overcomes thisnumerical difficulty as follows:

1. Treat the as another state variable

(6)

2. Introduce a new parameter s, the mathematical meaning of s isthe arclength on the solution curve, which is defined as follows:

(7)

It can be shown that the new set of equations (2) and (7)is numerically well-conditioned, even at the ‘nose’ point.CPFLOW solves these augmented power flow equations toobtain a solution curve passing through the ‘nose’ pointwithout encountering the numerical difficulty of ill-condi-tioning.

The implementation of CPFLOW is shown in Figure 1.The major loop of CPFLOW consists of two parts: a predic-tor and a corrector. The predictor finds an approximationpoint for the next solution on the solution curve and the cor-rector computes the exact solution based on the combinedequation set (2) and (7).

3. ZIP LOAD VARIATION

Traditionally, when one applies CPFLOW to computeP-V curves, only the constant PQ portion of a load varies,while the constant current portion and constant impedanceportion are kept indeed “constant”. As such, the physicalmeaning of these curves can be easily explained but the loadmodel may not be accurate. Load models are known to havesignificant impacts on simulating power system behaviors.In order to accurately characterize the voltage behavior of apower system, it is necessary to use accurate load models inthe computation of P-V, Q-V curves. For a power system ofn buses, the equation (2) of i-th bus can be expressed as fol-lows:

(8)

where are the base-case real and reactive powerinjection at the bus i respectively, : is voltage magnitude atbus i. are the real part and imaginary part of the net-work admittance between bus i and bus j respectively. isthe angle difference between bus i and bus j. is the pro-posed real generation variation at bus i and. is the pro-posed reactive generation variation at bus i. arethe proposed real and reactive load variations at bus i

Without losing generality, equation (8) can be rewrittenin the following form:

(9)

where are the set of base-case powerflow equations. is a proposed real generation vari-ation. are the proposed real and reactive loadvariations respectively.

Assuming that the load variation can be expressed as acomposition of ZIP (constant impedance, constant currentand constant PQ) format and this composition keeps con-stant, can be expressed as follows:

(10)

where: are diagonal coeffi-cient matrices and respectively represent the constant imped-ance, constant current and constant PQ portion of the loadvariation and is the voltage magnitude vector.

F x 1( , ) f x( ) b– P x( ) P1

–

Q x( ) Q1

–

0= = =

λ

λ

Xn 1+ λ=

xj xj s( )–( )2

j 1=

n 1+

∑ s∆( )2=

Figure 1 An overview of CPFLOW

compute the base-case power flow

Parameterize the load and generation variation

Predicator

Corrector

The whole P-V curve obtained?

No

Yes

P0i vivj Gij δcos ij Bij δijsin+( ) λ Pgi ∆Pl i–∆( )=0+j 1=

n

∑–

Q0 i vivj G ij δsin ij Bi j– δi jcos( ) λ Qgi ∆Q l i–∆( )=0+j 1=

n

∑–

P0 i Q0 i,vi

Gi j Bij,δij

Pgi∆Qgi∆

Pli Ql i∇,∆

fp δ v,( ) λ Gp λ– Lp∆∆+ 0=

fq δ v,( ) λ Lq∆– 0=

fp δ v,( ) fq δ v,( ) Rn∈,

Gp∆ Rn∈

L∆ p Lq R2n∈∆,

L∆

Lp∆ a1pv2

a2pv a3p+ +=

Lq∆ a1qv2 a2qv a3q+ +=

a1p a2p a3p a1q a2q a3q Rn n×∈, , , , ,

v Rn∈

3

Substituting (10) into (9), we have the following newset of parameterized power flow equations with ZIP varia-tions

(11)

If only the constant PQ load component varies, then thecoefficient matrices and become zero matri-ces and the new set of parameterized power flow equations(11) is reduced to the traditional set of parameterized powerflow equations (2).

4. CPFLOW WITH ZIP LOAD VARIATION

Combining equation (11) and (7), one gets the followingaugmented (parameterized) power flow equations:

(12)

where: , and ,

,

Solving equations (12) give rise to the desired ZIP-Vcurves. Next we will show how to modify the originalCPFLOW method described in [1] to solve the new set ofparameterized power flow equations (12).

PredictorThe predictor in CPFLOW is used to find the approxi-

mation of the next point in the solution curves. CPFLOWused two predictor approaches: secant and tangent. Thesecant method, extrapolating the next solution point basedon the previous two solution points in the solution curve isnot affected by the load model used in the computation. Thetangent method, using the tangent at the current point to pre-dict the next solution point, is affected by the load modelused.

We define the power flow state variable vector X as fol-lows:

(13)

Differentiating (13) with respect to the arclength s, we have:

(14)

where the arclength s is defined as follows:

(15)

Differentiating both side of the equation (12), it follows:

(16)

(17)

where

and . is a diago-

nal matrix of voltage magnitude at each bus.The following equation is required to ensure that the

scalar s is the arclength on the solution curve

(18)

The tangent vector along the solution curve is obtained by

solving the set of equation (16), (17) and (18).

The predicted vector at the next solution point by the

tangent method becomes:

(19)

where is the step size of arclength s.

CorrectorThe corrector solves the augmented equations (12) with

the predicted point, say (19) as the initial guess. In principle,any effective numerical procedure for solving a set of non-linear algebraic equations can be used for a corrector. InCPFLOW, we used the Newton-Raphson method in whichthe system Jacobian matrix plays an important role.

In order to accommodate the ZIP load model in the loadvariation, we must derive the Jacobian matrix consideringZIP load variation which can be expressed as follows:

(20)

D is given in (17) and

(21)

where is the i-th solution in the solution curve. Note thatthe required modifications only appear on the diagonal ele-ments and the arclength s related elements.

5. NUMERICAL STUDY

We have implemented the ZIP load variation model inthe CPFLOW and applied the modified CPFLOW to a realsystem to generate a family of ZIP-V curves for differentload variations and different load compositions.

Network ComponentsBuses: 4561, swing bus: 1, Generator: 1225, Loads:

2754, fixed shunts:916, switchable shunts: 186, Lines: 7512,Fixed transformer: 755, Fixed phase shifter:2 ULTC Trans-former: 294, ULTC phase shifter: 13, Areas:22.

fp δ v,( ) λ Gp λ a1pv2

a2pv a3p+ +( )–∇+ 0=

fq δ v,( ) λ a1qv2

a2qv a3q+ +( )– 0=

a1p a1q, a2p a2q,

f δ v,( ) λ∆G λ a1v2

a2v a3+ +( ) 0=–+

xj xj s( )–( )2

j 1=

n 1+

∑ s∆( )2=

a1a1p

a1q

= a2a2p

a2q

= a3a3p

a3q

=

f δ v,( )fp δ v,( )

fq δ v,( )= G Gp

0=

X δ1… δn v1

… vn λ, , , , , ,[ ]T=

dXds-------

sd

dδ1 …sd

dδn

sd

dv1 …sd

dvn

sddλ, , , , , ,=

X X∆⟨ ⟩T•∆ s∆( )2=

D dXds-------• 0=

D ∂f∂δ------ B C=

Bv∂

∂f 2a1V a2+( )λ–= C ∆G a1V2

a2V a3+ +( )–=

∂f∂v------ ∂f

∂v1-------- … ∂f

∂vn--------= ∂f

∂δ------ ∂f

∂δ1-------- … ∂f

∂δn--------= V R

n n×∈

dXds------- dX

ds-------

T• 1=

dXds-------

Xi 1+

Xi 1+ Xi hdX i

ds--------+=

h

J D

E=

E 2 X X s( )–( )T=

Xi

4

Simulation ResultsDue to space limitation, only one numerical simulation

is presented. The real power demands at two load buses inarea 63 increases 2000 MW, which is supported by theswing bus. The reactive load demands increases proportion-ally in order to maintain constant power factors. We assumethat the load increase is uniformly between these two loadsbased on their base-case load demands and nominal voltagelevel (say 1.0). For example, if the base-case load demandsfor the two buses 37886, 38280 are 10MW and 20MWrespectively, a uniform load increase of a total 30 MW addsanother 10 MW (at voltage 1.0) to bus 37886 and another 20MW (voltage 1.0) to bus 38280.

We investigate four different load compositions in theproposed load variation: (I) constant P-Q load; (II) constantcurrent load, (III) constant impedance load and (IV) 20%constant impedance load, 20% constant current load and60% constant P-Q load. Bus 38280 at 138kv, one of the twoload buses that are subject to load variations, is monitored.The generated P-V, I-V, Z-V, and ZIP-V curves for the fourdifferent load compositions are shown in Figure 2, Figure 3,Figure 4, and Figure 5, respectively.

The load margins limited by the nose points for eachload composition are listed in Table I.

Among the four different load compositions, the con-stant P-Q load model has the minimal amount of load mar-gin, which is 2405MW, and the constant impedance loadmodel carries the maximal amount of load margin, which is2566MW and 6.7% more than that of the constant P-Q load.The constant current loads can carry more load than the con-stant P-Q load and the constant impedance loads is able tocarry most load among the four studied load models, whichhas been observed in our extensive numerical studies.

This numerical results reconfirm the traditional assump-tion that the constant P-Q load model serves its purpose well- on the conservative side in predicting several system limitssuch as voltage stability limit or loadability limit. Howeverthe numerical study has demonstrated the significant impactof load model on the load margin of a power system. If onecan relax this assumption and take a more accurate loadmodel into consideration, one is able to take an aggressivestep to bear more load and achieve more financial benefit. Inabove example, the constant impedance loads can bear about6% more load than that of the constant PQ loads.

6. CONCLUSION

In this paper, we have developed a tool to generate ZIP-V curves for tracing power system steady state stationarybehavior due to both real generation and load variations. Wehave developed a new set of parameterized power flow equa-tions with ZIP load variations. We have implemented thisnew function in CPFLOW. The numerical study on a 4561-bus real power system shows that (i) the traditional assump-tion that only constant P-Q load varies makes the predictedresults always on the conservative side; (ii) the aboveassumption can be relaxed and a new class of curves, calledZIP-V curves, can be generated; and (iii) the load marginbased on the ZIP-V curve is generally greater than that basedon the traditional P-V curve.

Figure 2 P-V curve at bus 38280 for load composition (i)

Figure 3 I-V curve at bus 38280 for load composition (ii)

Figure 4 Z-V curve at bus 38280 for load composition (iii)

Total P(MW)

Total P(MW)

Total P(MW)

Figure 5 ZIV-V curve at bus 38280 for load composition (iv)

Table I Load margin to the nose point for different load models.

Load Model I II III IV

Load Margin 2405 2516 2566 2472

Load margin increase against the constant P-Q load

0% 4.62% 6.7% 2.79%

Total P(MW)

5

7. REFERENCE

[1] Hsiao-Dong Chiang, Alexander J.Flueck, Kirit S.Shah, NealBalu “CPFLOW: A Practical Tool for Tracing Power SystemSteady State Stationary Behavior Due to Load and GenerationVariations” IEEE Transactions on Power Systems, Vol.10,No.2, May 1995 pp623-pp630.

[2] K.lba, H. Suzuki, M. Egawa, T.Watanabe, “Calculation of theCritical Loading Condition with Nose Curve Using HomotopyContinuation Method,” IEEE Trans. On Power Systems, Vol.6,No.2 , May 1991, pp.584-593

[3] V.A. Ajjarapu and C.Christy, “The continuation Power Flow: Atool for Steady State Voltage Stability Analysis”, IEEE Trans.on Power Systems, Vol.7, No.1, Feb. 1992, pp416-423

[4] C.A. and F.L. Alvarado, “Point of Collapse andContinuation Methods for Large AC/DC Systems:, IEEEETrans. on Power Systems, Vol. 8, No. 1, Feb. 1993, pp.1-8.

[5] P.W.Sauer and B.C. Lesieutre, “Power System LoadModeling,” System and control Theory for Power Systems(J.H.Chow et al., Eds.) Springer-Verlag, 64, 1995.

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