3644 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 33, NO. 4, JULY 2018

An Optimal Thevenin Equivalent Estimation Methodand its Application to the Voltage Stability

Analysis of a Wind HubStephen M. Burchett , Senior Member, IEEE, Daniel Douglas, Student Member, IEEE,

Scott G. Ghiocel , Member, IEEE, Maximilian W. A. Liehr, Student Member, IEEE,Joe H. Chow , Life Fellow, IEEE, Dmitry Kosterev, Anthony Faris, Eric Heredia, and Gordon H. Matthews

Abstract—A simplified voltage stability analysis method is to usea static Thevenin equivalent to represent the electrical connectionto a load center. Assuming fixed values of the Thevenin voltage andreactance, a power-voltage (P V ) curve analysis can be performedto find the voltage collapse point and stability margin. This paperproposes a method to compute the static Thevenin equivalent volt-age and reactance of a power system using measured data. Themethod is validated with simulated PMU data and with 24-hourSCADA data at a wind hub on a medium voltage transmissionsystem in the US western system. For the wind hub, the Theveninequivalent parameters are used to compute the maximum powertransfer capability of the wind hub as constrained by voltage sta-bility. Then the method is extended to online application by usingshort data windows and forgetting factors.

Index Terms—Voltage stability, stability margin, thevenin equiv-alent, AQ bus, PV curve.

I. INTRODUCTION

THEVENIN equivalent is a simple circuit analysis tool tocalculate the voltage and current supplying a load or sub-

system. It is often incorporated in the computation of voltagestability margins, as shown by the system in Fig. 1.

In Fig. 1, the load bus is connected to a fixed voltage sourceEth via a fixed reactance Xth . As the active and reactive powerload (P,Q) increases, the voltage V at the load bus decreases.This behavior is illustrated in Fig. 2 with the well-known PV

Manuscript received December 5, 2016; revised April 26, 2017 and Septem-ber 12, 2017; accepted November 4, 2017. Date of publication November 24,2017; date of current version June 18, 2018. This work was supported in part bythe Bonneville Power Administration TIP 348, in part by the DOE CERTS pro-gram from Lawrence Berkeley National Laboratory, in part by the EngineeringResearch Center Program of the NSF and the DOE under NSF Award EEC-1041877, in part by the CURENT Industry Partnership Program, and in partby NSF Award EEC-1550029. Paper no. TPWRS-01821-2016. (Correspondingauthor: Stephen M. Burchett.)

S. Burchett, D. Douglas, M. Liehr, and J. Chow are with the Departmentof Electrical, Computer, and Systems Engineering, Rensselear Polytechnic In-stitute, Troy, NY 12180 USA (e-mail: burchs4@rpi.edu; dougld2@rpi.edu;liehrm@rpi.edu; chowj@rpi.edu).

S. Ghiocel is with Mitsubishi Electric Power Products, Inc., Rexford, NY12148 USA (e-mail: Scott.Ghiocel@meppi.com).

D. Kosterev, A. Faris, R. Heredia, and G. Matthews are with Bonneville PowerAdministration, Vancouver, WA 98666 USA (e-mail: dnkosterev@bpa.gov;ajfaris@bpa.gov; emheredia@bpa.gov; ghmatthews@bpa.gov).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2017.2776741

Fig. 1. Thevenin equivalent model.

Fig. 2. P V curves for a stiff-source single-load system with various constantpower factor loads.

curves for several constant power factor loads. The maximumpower transfer is achieved at the tip of the PV curve, also knownas the voltage collapse point.

The basic assumption in a PV -curve analysis is that theThevenin voltage source Eth and reactance Xth are constantduring the analysis process [1]–[3]. For real power systems thisassumption would be true for some short period of time, as longas the system operating condition does not change drastically.This paper is interested in calculating such Thevenin equivalentsusing measured data only, without the need for power systemline and generator data.

This paper demonstrates the systematic computation ofThevenin equivalent model using simulated PMU data and mea-sured SCADA data. The key contribution is a simple methodto reliably calculate the Thevenin voltage source Eth and

0885-8950 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

BURCHETT et al.: OPTIMAL THEVENIN EQUIVALENT ESTIMATION METHOD AND ITS APPLICATION 3645

Fig. 3. Thevenin reactance calculation with each point on the horizontal axisrepresenting 5 minutes.

reactance Xth from measured power network data. Previousmeasurement-based Thevenin equivalent parameter computa-tion methods (see for example, [4]–[6]) are mostly based onfitting the measured data to a PV curve using methods suchas least-squares approximation. The challenge with these ap-proaches is that there may not be enough variations in the(P,Q, V ) measurement set to cover a significant portion ofthe PV curve to yield reliable results. For example, considerthe situation shown in Fig. 2 in which the measured data iscontained in a small circle on the PV curve. This approachmay be acceptable using simulated data with no measurementnoise. However, with data measured from real power systems,during periods where the (P,Q, V ) values are relatively con-stant, the presence of measurement and quantization noise mayresult in these methods yielding rather erratic results. Fig. 3shows the Thevenin equivalent reactance values estimated us-ing 5 min (P,Q, V ) data windows, over a twenty-four period,for the wind hub system described in Section IV. For a practi-cal power system, it would be unreasonable to expect that theequivalent reactance would vary so substantially over such shortspans of time.

The proposed optimal Thevenin equivalent parameter calcu-lation method is discussed in Section II. It is applied to simulatedPMU data from a test system in Section III. The method is thenapplied to day-long SCADA data at the wind hub on a me-dian voltage transmission system in the US western system inSection IV. In addition, Section IV also contains a voltage sta-bility analysis using the AQ-bus method [10]. Section V extendsthe algorithm proposed in Section II to an online method usingshort data windows augmented by forgetting factors.

II. THEVENIN EQUIVALENT COMPUTATION FROM

SIMULATED/MEASURED DATA

For a Thevenin equivalent representation of a power systemin Fig. 1, given the measured P,Q consumption of the load, thevoltage of the bus computed from the equivalent must matchclosely the measured V for significant variations of P,Q. Theload variations include all static load types. This property of-fers a simple and reliable approach to compute the Theveninequivalent voltage and reactance values.

Fig. 4. A power system consisting of a small system (System 1) and a largesystem (System 2).

To motivate the approach, consider a test system consisting ofa large synchronous generator in System 1 and a small generatorin System 2. The System 1 generator is modeled as a classicalmachine. The System 2 generator is modeled with subtransientdynamics and includes excitation and governor models. Thevoltage V on the boundary bus and the (P,Q) flow cominginto the boundary bus are measured. Alternatively, the currentmeasurem(ent I∠φ can be used instead of (P,Q).

The large generator in System 1 is connected to the boundarybus via a transmission line of reactance 0.2 pu. The small gen-erator in System 2 is connected to the boundary bus via a trans-former with reactance 0.15 pu. The transient and subtransientreactances of the generator are X ′

d = 0.42 pu and X ′′d = 0.31

pu, respectively, and the voltage regulator gain is KA = 141.Disturbances (or control actions) are applied to both systems,

but not simultaneously, to change the (P,Q) flow and the voltageV . For a disturbance in System 2, one can compute the Theveninequivalent for System 1, and vice versa.

Suppose a disturbance is applied in System 2 such that thevalues of (P,Q, V ) would vary. If System 1 is represented by aThevenin voltage Eth and a Thevenin reactance Xth , then therelationship

Eth = V + jXth(I∠φ) = V + jXth(P + jQ)∗/V (1)

holds for all (P,Q, V ) values, where Eth is a voltage phasor,with Eth as its amplitude. If the correct Xth is found, Eth is aconstant value.

Thus a Thevenin equivalent computation method is to assumea range of reasonable values for Xth (as Xth is not known aheadof time), and use them to calculate Eth . Then the value of aparticular Xth yielding a mostly constant value for Eth can beselected as the Thevenin equivalent parameters (Eth ,Xth ).

To illustrate this concept, consider the simulation of the two-system model in Fig. 4 in which disturbances are applied toSystem 2 to induce voltage steps in V . Fig. 5 shows the simulated(P,Q, V ) responses.

Using a range of values 0.05, 0.1, 0.15, 0.2, and 0.3 pu forXth , the resulting Eth responses are shown in Fig. 6. Note thatfor smaller values of Xth , the calculated Eth follows closelythe response of the measured voltage V . For the largest valueXth = 0.3 pu, the calculated voltage diverges from V , that is, ifV drops, Eth increases. However, if Xth = 0.2 pu is selected,Eth is mostly constant, except at the instants of the step changesin V .

3646 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 33, NO. 4, JULY 2018

Fig. 5. Values of (P, Q, V ) for the voltage step disturbances.

Fig. 6. Values of Eth for various values of Xth .

This process of selecting an optimal value of Xth such thatEth will be mostly constant is summarized in the proposedalgorithm:

Optimal Thevenin equivalent estimation algorithm:1) Obtain values of (P,Q, V ) by either simulation or

measurement2) Select a range of reactance values for Xth , as Xthi , i =

1, 2, ..., n3) For each selected value of Xthi , calculate Ethi(k), where

k denotes the data point time index.4) For each Ethi , calculate its average

Ethi =

(N∑

k=1

Ethi(k)

)/N (2)

where N is the number of sample points, and the root-mean-square error as

εi =

√∑Nk=1(Ethi(k) − Ethi)2

N(3)

5) Select Ethi with the lowest value of εi and its two adjacentneighboring points Eth(i−1) and Eth(i+1) . Apply a 3-point

quadratic fit algorithm in [7] to obtain the optimal valueof Xth .

6) Calculate Eth(k) using Xth .7) Calculate Eth as the average of Eth(k)

Eth =

(N∑

k=1

Eth(k)

)/N (4)

It should be noted that a similar estimation idea is used toestimate Xth by iteratively computing the Thevenin equivalentvoltage Eth [8]. This approach has the extra complexity ofhaving to estimate the angle of the Thevenin voltage.

III. THEVENIN EQUIVALENT COMPUTED USING

SIMULATED DATA

The optimal estimation algorithm is applied to three simu-lation cases, in which the (P,Q, V ) data on the boundary busis sampled at 30 points per second. The first two cases involvedisturbances in System 1, to obtain the Thevenin equivalent pa-rameters for System 2. The third case involves a disturbance inSystem 2, allowing the computation of the Thevenin equivalentparameters for System 1. In all these cases, Gaussian noise hasbeen added to the simulated response.

A. Voltage Step Disturbances

Applying the optimal estimation algorithm to the voltage stepdisturbances data in Fig. 5 yields Xth = 0.2002 pu and Eth =1.0804 pu. The results are consistent with System 1 data thatXth is the sum of the transmission line reactance (0.2 pu) and themachine transient reactance which is small when scaled by thelarge machine base, and Eth is essentially the large generator ter-minal bus voltage (1.08 pu). As a further verification, a sequenceof 2-bus power flow solutions is obtained with the large genera-tor bus voltage set at Eth = 1.0804 pu, the boundary (load) buspower set at the simulated (P,Q) values, and Xth = 0.2002 puused as the reactance connecting the two buses. The calculatedload bus voltage Vcal is shown with the simulated voltage Vsimin the lower plot of Fig. 5. The root-mean-square (RMS) errorin the Thevenin voltage approximation is 0.000532 pu.

B. Active-Power Step Disturbances

Next an active power step disturbance case is considered.Fig. 7 shows the simulated (P,Q, V ) data at the boundary busfor a sequence of active-power step disturbances in System 2,with System 1 at the same operating condition as the voltagestep disturbances.

Applying the optimal estimation algorithm results in Xth =0.2018 pu and Eth = 1.0807 pu, which for practical purposes,are identical to those obtained in the voltage step disturbancesfor estimating the Thevenin equivalent of System 1. To verifythe result, again a sequence of 2-bus power flow solutions isused to obtain the load bus voltage Vcal , which is shown withthe simulated voltage Vsim in the lower plot of Fig. 7. The RMSerror in the Thevenin voltage approximation is 0.000530.

BURCHETT et al.: OPTIMAL THEVENIN EQUIVALENT ESTIMATION METHOD AND ITS APPLICATION 3647

Fig. 7. Values of (P, Q, V ) for the active-power step disturbances.

Fig. 8. Values of (P, Q, V ) for the voltage sag disturbances.

C. Voltage Sag Disturbances

For the third case, a voltage sag disturbance is applied inSystem 1 to compute the Thevenin-equivalent for System 2.Fig. 8 shows the simulated (P,Q, V ) data at the boundary busfor a sequence of voltage-sag disturbances in System 1, withSystem 2 also responding to this disturbance.

Applying the optimal estimation algorithm results in Xth =0.1852 pu and Eth = 1.0768 pu. The results are consistent withthe parameters in System 2. The Thevenin reactance Xth is thesum of the transformer reactance and a fraction of the transientreactance as impacted by the voltage regulator gain, as the ex-citer is regulating the generator terminal bus. Also Xth is higherthan the terminal bus voltage (1.0 pu. This analysis shows thatthe fixed voltage point is inside the synchronous machine. Toverify the result, a sequence of 2-bus power flow solutions isused to obtain the load bus voltage Vcal , which is shown withthe simulated voltage Vsim in the lower plot of Fig. 8. The RMSerror in the Thevenin voltage approximation is 0.0023. Notethat this RMS error is higher than those obtained for System 1,because Vcal , computed from a static model, does not reproducethe oscillations occurring during the switching instants.

IV. VOLTAGE STABILITY ANALYSIS OF A WIND HUB

In this section, 2-sec SCADA data is used to derive twoThevenin equivalents for a wind hub shown in Fig. 9. Ten daysof 24-hour data were available for analysis, with each dataset consisting of 43,200 points. The wind hub (WH) bus is a230 kV bus serving as a wind hub/point of connection for sixwind farms [9]. The wind farms consist of 4 type-2 and 2 type-3 wind turbines, with a combined maximum output of about600 MW. The voltage at the wind farms is supported by two28.8 MVAR shunt capacitors in the substation and a wide arrayof shunt capacitors and reactors, and two STATCOMs insidethe wind farms. The WH bus is connected to the west by a long160-mile line (reactance is 0.0861 pu) which is eventually con-nected back to the 500 kV system. It is also connected to theeast via a 48-mile line (reactance is 0.02732 pu) to an area withseveral large generators and to the 500 kV system. Thus the westconnection is regarded as weak, whereas the east connection isregarded as strong.

The objective is to assess the voltage stability margin to ensuresafe transfer of the wind energy to the west and to the east. Atfirst glance, a voltage stability analysis would require accountingfor all the status of the reactive power supply at the WH Bus,which would be quite complicated as the statuses of the shuntcapacitors/reactors at the WH Substation and the wind farmsare not available from the SCADA data. However, with validThevenin equivalent models, the WH Bus can simply be viewedas a PQ bus, with a negative load.

The proposed voltage stability analysis approach is to firstrepresent the east and west systems each with a Thevenin equiv-alent, as shown in Fig. 9, using the (P,Q, V ) SCADA data onthe east and west connections. The second step is to analyze thevoltage stability margin of the 3-bus system consisting of theWH bus, the West Thevenin bus, and the East Thevenin bus.

A. West Thevenin Equivalent

An advantage of estimating a Thevenin equivalent for a windhub is that the power generation from the wind turbines is notconstant in a daily cycle. The variations in the (P,Q) flow wouldprovide more accurate Thevenin equivalent models. Fig. 10show a one-day data of the measured voltage at the WH Busand the (P,Q) power flow to the west system.

Fig. 11 shows the calculated voltage Vth at the west systemusing values of Xth from 0.05 to 0.3 pu. The calculated voltagestays mostly constant for a value of Xth = 0.2 pu. The opti-mal estimation algorithm yields Xth = 0.1835 pu and Eth =1.1181 pu. To verify the solution, the WH Bus voltage is com-puted using the Thevenin model and the (P,Q) flow values.Fig. 12 shows that the resulting voltage matches the measuredvoltage at WH Bus, with a RMS error of 0.0025 pu.

The average Thevenin equivalent parameters for 9 days areshown in the top plot of Fig. 13, indicating good consistency ofresults between Days 3 to 6. Note that there were not enough(P,Q, V ) variations in the Day 8 data to compute the optimalparameters.

3648 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 33, NO. 4, JULY 2018

Fig. 9. Wind hub substation layout.

Fig. 10. 24-hour SCADA voltage at WH Bus and the (P, Q) flow to the westsystem.

Fig. 11. Calculated voltage Vth at west system using various values of Xth .

Fig. 12. Comparison of measured and west system Thevenin equivalentcalculated WH Bus voltage.

Fig. 13. Optimal Thevenin equivalent parameters obtained from the SCADAdata (∇ denotes a voltage data point, and × denotes a reactance value).

BURCHETT et al.: OPTIMAL THEVENIN EQUIVALENT ESTIMATION METHOD AND ITS APPLICATION 3649

Fig. 14. 24-hour SCADA voltage at WH Bus and the (P, Q) flow to the eastsystem.

Fig. 15. Calculated voltage Vth at east system using various values of Xth .

B. East Thevenin Equivalent

The optimal Thevenin equivalent estimation is repeated forthe east system. Fig. 14 shows the measured voltage at the WHBus, and the power flow to the east system.

Fig. 15 shows the calculated voltage Vth at the west systemusing values of Xth from 0.03 to 0.11 pu. Here there is nofixed value of Xth that would result in an approximately fixedvoltage Eth for the whole 24 hours. This reflects the situationthat the terminal voltages of the generators near the east systemwere adjusted during the day to accommodate load changes. Inparticular, there is a noticeable voltage rise between Hours 7 to 8(denoted by A in Fig. 15) and a voltage drop between Hours 21 to22 (denoted by B). As a result, the optimal Thevenin equivalentestimation is performed only for Hours 9 to 20, yielding theoptimal parameters Xth = 0.0498 pu and Eth = 1.0704 pu. Toverify the solution, the WH Bus voltage is computed usingthe Thevenin model and the (P,Q) flow values. Fig. 16 showsthat the calculated WH Bus voltage matches well the measuredvalues from Hours 9 to 20. The RMS error of the Thevenin

Fig. 16. Comparison of measured and east system Thevenin equivalent cal-culated WH Bus voltage.

voltage during this Hours 9 to 20 period is 0.0016 pu. There isalso a divergence between Hours 0 to 8 and 22 to 24.

The optimal parameters obtained for nine days of SCADAdata for the east system using Hours 9 to 20 measured data alsoshow good consistency between Day 2 to Day 6, as shown in thelower plot of Fig. 13. Note that there were not enough (P,Q, V )variations in the Day 8 data to compute the optimal parameters.

C. Voltage Stability Analysis

The AQ-bus method in [10] is used for developing the PVcurve and computing the voltage stability margin, using the eastand west system Thevenin model parameters. In this method,the bus that is vulnerable to voltage collapse is modeled as anAQ bus, where the bus voltage angle and reactive power arespecified. In the wind hub system, the wind hub is selected asthe AQ bus, even though it is supplying power. However, withlimited reactive power support, it encounters similar voltagecollapse problems like a load bus. The power flow computationin the AQ-bus method uses a Jacobian matrix with one fewerdimension than a normal Jacobian matrix, because there are twobuses with fixed angles, allowing it to avoid the Jacobian matrixfrom being singular at the collapse point.

To conduct the voltage stability analysis, the angle separationbetween the wind hub (AQ bus) and the swing bus is increasedto allow for the power exported by the wind hub to the eastand west systems. To solve for the power flow, the split of thewind power exporting to the west and east systems needs to beknown. Here the incremental power transfer is approximated bya linear relationship, shown in Fig. 17.

The east system is chosen as the swing bus because the powerabsorbed varies more than the west system. The west system ismodeled as a PV bus with its active power adjusted in tan-dem with the east system based on the approximated split. Thevoltages at these two buses are set to their Thevenin equivalentvalues.

The PV -curve analysis was performed considering the twoshunt capacitors at the wind hub, which results in PV curvesfor three levels of fixed shunt reactive power compensation: no

3650 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 33, NO. 4, JULY 2018

Fig. 17. Wind hub power export to east and west systems.

Fig. 18. Wind hub P V curves.

TABLE IMAXIMUM WIND HUB POWER TRANSFER

Shunt compensation Pm ax Vcrit

none 805 MW 0.879 p.u.S1 813 MW 0.890 p.u.S1 and S2 822 MW 0.902 p.u.

compensation, S1 only, and both S1 and S2 as shown in Fig. 18.Note that the PV curve does not reach zero voltage when thewind hub power is zero because of power transfers from eastto west through the wind hub bus. Table I shows the maximumpower transfer levels for the various shunt compensation.

A voltage stability analysis for the wind hub system with theline to the East System disconnected, resulting in wind curtail-ment, can be found in [11].

V. AN ONLINE ESTIMATION ALGORITHM

As noted earlier, for significant changes in power system oper-ating conditions, including line and shunt switchings, generator

reference voltage adjustments, and control equipment reachingreactive power output limits, it would be necessary to re-estimatethe Thevenin equivalent parameters. In this section, the devel-opment of online algorithms is considered. In contrast to theuse of day-long data records to estimate the average Theveninequivalent parameters, the objective of an online algorithm isto estimate the time varying Thevenin parameters by means ofmuch shorter periods of measured data. Such an algorithm willrepeatedly compute the Thevenin parameters at regular intervalsor on demand when new measured data is available. The maindifficulty is that the measured (P,Q, V ) data may not have ad-equate variations to provide reliable results. Thus two strategiesare proposed to improve the reliability of the results. The coreof the method will incorporate the optimal Thevenin equivalentparameter estimation method developed earlier.

The first strategy is to take into account not just the newdata, but also incorporate the most recent data in the estimationprocess. There are many such methods such as overlappingwindows in which the new data is used with an equal or longerduration of immediately past data. The method proposed herewill make use of forgetting factors [12].

The measured data is divided into periods of length T . LetXk

th be the estimated Thevenin reactance at period k using theoptimal Thevenin parameter method and Xk

th be the estimatedvalue using the forgetting factor method. Then the proposedestimation scheme is

Xkth = α0X

kth +

−1∑j=−n

αkXk+jth ,

0∑k=−n

αk = 1, (5)

where Xkth is a weighted sum of the current estimate Xk

th andn immediately past optimal Thevenin reactance values. In ad-dition, the forgetting factors αk satisfy

α−n < α−(n−1) < . . . < α0 . (6)

Once Xth has been determined, Vth is computed using steps(6)–(7) of the algorithm discussed in Section II.

In the second strategy, there are some instances that the varia-tion in the measured data in the new data window is so small thatit would be more meaningful to keep the Thevenin parametersfrom the previous data window. From Steps (4) and (5) of theoptimal estimation algorithm, define

εmax = maxi

{εi}, εmin = mini{εi} (7)

and

δ = εmax − εmin (8)

If δ is less than a certain tolerance, new values of Theveninparameters will not be calculated and the previous Theveninparameter values will be kept.

The online algorithm is demonstrated using the 2-secondSCADA data of east WH system for the same 24-hour mea-surement period in Fig. 16. The length of the data window Tand the choice of n have to be chosen carefully, often basedon the measured data. Ideally a short data window should beused, but the resulting estimation can be noisy. Also, in order toget fast updates, n should not be too large. For this illustration,

BURCHETT et al.: OPTIMAL THEVENIN EQUIVALENT ESTIMATION METHOD AND ITS APPLICATION 3651

Fig. 19. Optimal Thevenin reactance values obtained using a 5-minutexbrkwindow and n = 4.

Fig. 20. Optimal Thevenin voltage values obtained using a 5-minute windowand n = 4.

the following parameters were used in the Thevenin equivalentestimation algorithm:

1) The duration of each period is T = 5 mins2) The number of previous periods used is n = 43) Linear varying forgetting factors are applied with

αk = {0.06, 0.13, 0.20, 0.26, 0.33}4) The threshold δ = 0.003 puThe results of the time-varying optimal Thevenin parameters

are shown in Figs. 19 and 20. The average values of the Theveninparameters are also shown in these figures. The online esti-mated Xth without forgetting factors is shown in Fig. 19. Notethat without the forgetting factors, Xth varies between 0.001 to0.13 pu. Using the forgetting factors, the variation of Xth is re-duced to 0.03 to 0.09 pu. Note also that for several periods, therewere not enough variation in 5 minutes for the algorithm to com-pute a new reactance. As mentioned earlier, the voltage setpointsof the generators in the East System are raised during peak loadhours and lowered during the off-peak hours. The estimated Vthshown in Fig. 20 captures this operating characteristic.

Fig. 21. Comparison of measured and east system Thevenin equivalentcalculated WH Bus voltage using online estimation algorithm.

To verify the online algorithm’s performance, the voltageat the boundary bus is computed using the optimal Theveninpair of each measurement window. Fig. 21 shows the improvedmatch with the measured data of the bus voltage throughout the24-hour period.

The merits of the online method, as compared to analysisbased on detailed power system models, include (1) requiringonly measured data of (P,Q, V ) for a load center without need-ing system data and state estimator solutions, (2) accountingfor the impact of changes in the voltage setpoints of gener-ators and the operation of the excitation systems, instead ofassuming the generator terminal bus voltages to be constant,and (3) providing voltage sensitivity analysis of load busesdue to shunt reactor/capacitor bank switching. On the otherhand, the Thevenin model provides only a local view of voltagestability, not an overall system view, and would be only appli-cable to buses exhibiting large (P,Q, V ) variations.

VI. CONCLUSION

An optimal Thevenin model parameter estimation method hasbeen proposed. The method is based on the assumption that theThevenin voltage and reactance are constant during some timeperiods. It is a reliable method as it does not solve a set of nonlin-ear equations involving data with measurement and quantizationnoise. In addition to calculating voltage stability margins, theThevenin parameters can also be used for analysis of voltagerelated operational issues such as substation voltage sensitivitieswith respect to shunt reactor/capacitor bank switchings.

The methods proposed in this paper can be applied to bothoff-line and online application. The techniques have been suc-cessfully applied to simulated data as well as measured data ata wind hub.

It should also be emphasized that the proposed measurement-based methods are applicable to load buses that experiencesufficient variations in active and reactive power consumptionand voltage amplitude. Load buses with voltage and power thatremain mostly constant for long periods of time are not goodcandidates for the proposed methods.

3652 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 33, NO. 4, JULY 2018

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[12] K. J. Astrom and B. Wittenmark, Adaptive Control. Reading, MA, USA:Addison-Wesley, 1989.

Stephen M. Burchett (SM’16) received the B.S. degree in mechanical engi-neering from Union College, Schenectady, NY, USA, in 2005 and the M.S.degree in mechanical engineering from Union Graduate College, Schenectady,NY, USA, in 2006. He is currently working toward the Ph.D. degree in electricalengineering from Rensselaer Polytechnic Institute, Troy, NY, USA. From 2006to 2010, he was with PSEG Power, Newark, NJ, USA. From 2010 to 2013, hewas with Green Charge Networks, Brooklyn, NY, USA. From 2013 to 2016,he was with the New York Independent System Operator, Albany, NY, USA.His research interests include renewable generation, energy storage, electricitymarkets, optimization, and voltage stability. He is a Registered ProfessionalEngineer (Electric Power) in New York, Massachusetts, and California.

Daniel Douglas (S’13) received the B.S. degree in electrical engineering fromTemple University, Philadelphia, PA, USA. He is currently working toward thePh.D. degree in electrical engineering from Rensselaer Polytechnic Institute,Troy, NY, USA. His research interests include power generation control, powersystem control, power engineering computing, ferroresonance, and inductivepower transmission.

Scott G. Ghiocel (M’13) received the M.S. and Ph.D. degrees from Rens-selaer Polytechnic Institute, Troy, NY, USA, in 2010 and 2013, respectively.He is a principal engineer at Mitsubishi Electric Power Products, Inc. (MEPPI),Cypress, CA, USA. His research interests include power system dynamics, volt-age stability, reactive power control, and applications of synchronized phasormeasurements.

Maximilian W. A. Liehr (S’14) received the B.S. and M.Eng. degrees in elec-trical engineering from Rensselaer Polytechnic Institute, Troy, NY, USA. He iscurrently working toward the Ph.D. degree from Rensselaer Polytechnic Insti-tute. His research interests include power system dynamics, voltage stability,and renewable energy.

Joe H. Chow (LF’16) received the B.S. degrees in electrical engineering andmathematics from the University of Minnesota, Minneapolis, MN, USA, andthe M.S. and Ph.D. degrees in electrical engineering from the University of Illi-nois, Urbana-Champaign, Champaign, IL, USA. After working in the GeneralElectric power system business in Schenectady, NY, USA, he joined Rensse-laer Polytechnic Institute in 1987, and is a professor of electrical, computer,and systems engineering. His research interests include power system dynamicsand control, voltage-source converter-based FACTS controllers, voltage stabil-ity analysis, and synchronized phasor data. He is a member of the NationalAcademy of Engineering.

Dmitry Kosterev received the B.S. and Ph.D. degrees in electrical engineeringfrom Oregon State University, Corvallis, OR, USA, in 1992 and 1996, respec-tively. He is with the Bonneville Power Administration, where his responsi-bilities include network planning, power system modeling, and power systemperformance monitoring. He chaired the WECC Modeling and Validation WorkGroup from 2007 to 2011, and the WECC Load Modeling Task Force since2002. He is actively involved with synchro-phasor technologies, and has servedas a co-chair of the Planning Implementation Task Team at North-AmericanSynchro-Phasor Initiative.

Anthony Faris received the B.S.E.E. degree from the University of Portland,Portland, OR, USA, and the M.S.E.E. degree from the University of Washing-ton, Seattle, WA, USA. He is an Engineer in the Measurements Systems group,Bonneville Power Administration (BPA), Portland, OR, USA. He began work-ing with phasor measurement systems in 2004 as a student at BPA.

Eric Heredia received the B.S. degree in electrical engineering from the Uni-versity of Portland, Portland, OR, USA, in 2002 and joined BPA in August 2002.He worked in BPA’s Transmission Planning group for 13 years, and then movedto BPA’s Operations Planning group in 2015. His work with BPA focuses ontransmission system studies determining network capabilities and limitations.

Gordon H. Matthews is a General Engineer at Bonneville Power Adminis-tration, Portland, OR, USA. He previously managed the Mechanical and CivilSections of the BPA Laboratories, including outreach to the BPA Energy Ef-ficiency Seattle Office. His other assignments included the Power ServicesWestern Area Marketing Hub, Corporate Services, and Transmission Servicesto assist Scheduling Automation, defining and refining Transmission Policy andbusiness practices, NERC Reliability, and A123 Compliance. His current as-signment is in the Technology Innovation work group in Corporate Strategy.