Stochastic Optimization of Power System Dynamics for Grid Resilience

Bryan ArguelloSandia National Laboratories

barguel@sandia.gov

Nathan StewartSandia National Laboratories

nastewa@sandia.gov

Matthew HoffmanSandia National Laboratories

mjhoffm@sandia.gov

Abstract

When faced with uncertainty regarding potentialfailure contingencies, prioritizing system resiliencethrough optimal control of exciter reference voltageand mechanical torque can be arduous due to thescope of potential failure contingencies. Optimalcontrol schemes can be generated through a two-stagestochastic optimization model by anticipating a setof contingencies with associated probabilities ofoccurrence, followed by the optimal recourse actiononce the contingency has been realized. The first stage,common across all contingency scenarios, co-optimallypositions the grid for the set of possible contingencies.The second stage dynamically assesses the impact ofeach contingency and allows for emergency controlresponse. By unifying the optimal control scheme priorand post the failure contingency, a singular policy canbe constructed to maximize system resilience.

1. Introduction

In analyzing the behavior of the electric power gridafter a major disruptive event, full dynamic models arecritical. During such an event, real or reactive power,voltages, and other system metrics may go beyond limitsenforced by steady state optimal power flow models(for general references in power system dynamics andstability, see [1, 2]). Our work focuses on the problemof positioning the grid to be resilient to a set of possibleoutage scenarios through stochastic optimization overpower system dynamics. In each scenario, we allowsecond stage recourse through generator control toimprove grid stability after the outage.

The transient stability constrained optimal powerflow (TSCOPF) problem, introduced in [3], optimizespower flow to minimize generation cost, while usingpower system dynamics to ensure stability undercontingencies. Optimizing cost subject to stabilityconstraints can leave the system close to constraintboundaries and vulnerable to additional perturbations;

we instead incentivize safety margin, to keep the systemfarther from these boundaries. We are interested instability under multiple possible contingencies, muchlike the multi-contingency transient stability constrainedoptimal power flow problem (MC-TSCOPF) introducedin [4]. Our research differs from MC-TSCOPF in that it1) assumes the system is already at some (non-optimal)steady state prior to the contingency and explicitlyaddresses the dynamic control actions necessary toimprove contingency readiness, and 2) additionallyaddresses post-contingency corrective control in thesecond stage. Our second-stage problem is similar to thetransient stability emergency control (TSEC) problem(e.g., [5]) of ensuring grid stability after a contingency(e.g., by reducing generation and shedding load).

The three main techniques for solving TSCOPF andTSEC are direct approaches, single-machine equivalent(SIME) reduction, and computational intelligence. See[6] for a recent overview of these techniques. Mostprevalent and flexible is the direct approach, whichinvolves discretizing differential equations that expresssystem dynamics. These discretized equations can besolved directly as part of the optimization (simultaneousdiscretization), in a simulation sub-problem (sequentialdiscretization), or via a hybrid of the two. For expandeddiscussion of these methods and demonstration of thehybrid method, see [7]. While our problem formulationis agnostic of discretization approach, our results areobtained via simultaneous discretization. See [8] forenhanced discretization techniques.

Direct approaches are typically solved via aninterior point method, and can benefit from reductionand parallelization techniques. For a reduced-spaceapproach and parallelized reduced-space interior pointmethod, see [9] and [10]. A GPU-based parallelizationapproach to solving TSCOPF is given in [11].Metaheuristics have occasionally been used to solvethese problems, include whale optimization [12], hybridparticle swarm, and artificial physics optimization[13].

Our optimization model prepares the grid formultiple contingencies before any one contingency

Proceedings of the 54th Hawaii International Conference on System Sciences | 2021

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occurs and optimizes stability after the contingencyis realized. Our model does not directly considereconomic generation cost in the objective function, aswe focus on system resilience to major emergencies.Our formulation maximizes safety margins for voltageand frequency, to both ensure adequate power qualitypost-contingency and reduce the chance of a cascadingfailure. Unlike aforementioned TSCOPF researchwhere power flow is optimized, we optimize directlyover the exciter reference voltage and mechanicaltorque generator controls throughout the full timehorizon. Furthermore, our research is the first tooptimize grid dynamics in preparation for a multitude ofpossible contingencies, simultaneously with optimizingthe post-contingency controls.

2. Grid Resilience and Stability Model

We use the fourth-order flux decay generator modeland turbine with no reheating model from [1]. Forrealism, loads can also be modeled dynamically [14];the importance of dynamic load modeling comparedto static load modeling is investigated in [15]. Ourpower system model uses the exponential recoverydynamic load model [16]. Our model is a nonlineartwo-stage stochastic differential algebraic equation(DAE) formulation. The differential equations ofthe generators and loads can be discretized throughany available discretization scheme including finitedifferences and collocation. We use a set ofcontingencies such as line faults and generator failuresthat occur at the same time tf to form our scenarios. Thefirst stage varies generator controls (exciter referencevoltages and mechanical torque) before tf so thatthe grid is best prepared for any scenario to occur.Meanwhile, the second stage varies generator controlsafter tf in each scenario to maximize grid stability. Thegenerator controls (indirectly) adjust generator real andreactive power output.

In the following sections, we detail our formulationby giving nomenclature, DAE equations, a descriptionof discretization, stability objectives, and finally ourtwo-stage stochastic programming model.

2.1. Sets

Set Index Symbol DescriptionB b BusesG g GeneratorsL l LoadsΨ ψ Scenarios

2.2. Indexed Sets

Set Index Symbol DescriptionGb b Generators at bus bLb b Loads at bus b

2.3. DAE Variables

Variable Index Descriptionδ g Rotor angleω g Generator frequencyE′

q g q-axis transient voltageEfd g Field voltageIq g q-axis currentId g d-axis currentTM g Shaft mechanical torqueV b Voltageθ b Phase anglePL l Active load power drawQL l Reactive load power drawxp l Load active power state variablexq l Load reactive power state variable

2.4. Control Variables

Variable Index DescriptionVref g Exciter reference voltagePref g Generator mechanical torque power

2.5. Parameters

Parameter Index DescriptionT Time horizonωs Rated synchronous speedM g Shaft inertial constantD g Damping coefficientKA g Exciter amplifier gainTA g Exciter amplifier time constantRs g Scaled resistance after dq

transformationXq g q-axis synchronous reactanceXd g d-axis synchronous reactanceX′

d g d-axis transient reactanceT′

do g Transient time constantTch g Mechanical torque damping const.bg g Bus connected to generator gPoL l Initial active powerQoL l Initial reactive powerTpL l Active power time constantTqL l Reactive power time constantαs l First active power exponent

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αt l Second active power exponentβs l First reactive power exponentβt l Second reactive power exponentbl l Bus connected to load lY b,b Admittance magnitude matrixA b,b Admittance phase angle matrixη1 V objective scaling parameterη2 ω objective scaling parameterγ1 V objective shaping parameterγ2 ω objective shaping parameter

2.6. DAE Equations

Generator model:

dδgdt

= ωg − ωs (1)

dωgdt

=TMg

Mg− E

′

qg

IqgMg− IdgIqg

Xqg −X′

dg

Mg

−Dgωg − ωsMg

(2)

dE′

qg

dt= −

EqgT′dog

−Xdg − IdgX′

dg

T′dog

+EfdgT′dog

(3)

dEfdgdt

= −EfdgTAg

+ (Vrefg − Vbg )KAg

TAg(4)

∀g ∈ G

Governor model:

dTMg

dt=Prefg − TMg

Tchg(5)

∀g ∈ G

Stator equations:

Vbg sin(δg − θbg ) +RsgIdg −XqgIqg = 0 (6)

E′

qg − Vbg cos(δg − θbg )−RsgIqg −X′

qgIdg = 0

(7)

∀g ∈ G

Exponential recovery load model:

dxpldt

=xplTpLl

+ PoLlVαslbl− PoLlV

αtlbl

(8)

dxqldt

=xqlTqLl

+QoLlVβslbl−QoLlV

βtlbl

(9)

PLl =xplTpLl

+ PoLlVαtlbl

(10)

QLl =xqlTqLl

+QoLlVβtlbl

(11)

∀l ∈ L

Balance equations:

−∑i∈B

(ViVbYi,b cos(θi − θb −Ai,b))

−∑l∈Lb

(PLl) +∑g∈Gb

(IdgVb sin(δg − θb)

+IqgVb cos(δg − θb)) = 0 (12)

−∑i∈B

(ViVbYi,k sin(θi − θb −Ai,b))

−∑l∈Lb

(QLl) +∑g∈Gb

(IdgVg cos(δg − θb)

+IqgVb sin(δg − θb)) = 0 (13)

∀b ∈ B

In addition we add two inequality constraints thatlimit the rate of change of Vref and Pref . We do thisto prevent solutions with highly oscillatory controlchoices, as this would not be replicable in reality.∣∣∣∣dVrefgdt

∣∣∣∣ ≤ k (14)∣∣∣∣dPrefgdt

∣∣∣∣ ≤ k (15)

∀g ∈ G

Note that k could be set differently for Vref and Prefor even uniquely for each generator. In practice, thisvalue would be tied to the generator ramp rate of thesystem. A rather aggressive value of k = 3 is used forthis proof of concept.

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2.7. Discretization

To approximate the differential equations withalgebraic equations, we first partition the timehorizon [0, T ] through a finite set of points P ={0, t2, . . . , tf , . . . , tn−1, T} that includes the failuretime. Common choices of P include uniform spacingand a scheme which adds more points around timeperiods where total variation is expected to be high suchas a neighborhood after tf . Next, P is used to discretizedifferential equations through a discretization schemesuch as finite difference or collocation. As an example,the forward finite difference scheme approximates everyderivative dx

dt through the forward finite difference

dx

dt≈xtk − xtk−1

tk − tk−1(16)

After every derivative in the DAE is approximated witha finite difference, the DAE is approximated through asystem of algebraic equations. Henceforth, we refer toa discretized approximation of DAE (1) - (15) as (1)’ -(15)’.

Note that numerical circumstances around dynamicsimulations are different than those around solving ourDAE model. With dynamic simulation, numericalinaccuracies are propogated forward in time as thesimulation progresses. DAE models are used to obtaina full system solution over the full time horizonsimultaneously.

2.8. Stability Metrics

Upon discretization, discrete values of bus voltageand angular velocity are available to form stabilitymetrics for various types of grid stability.

Mv(t1, t2) =∑

t∈{τ∈P|t1≤τ<t2}

∑b∈B

(1− Vb,tη1

)γ1(17)

measures how far buses deviate from a nominal voltageof 1 p.u. between t1 and t2. Similarly,

Mω(t1, t2) =∑

t∈{τ∈P|t1≤τ<t2}

∑g∈G

(ωg,t − ωsωs · η2

)γ2(18)

measures how far frequency deviates from nominalfrequency between t1 and t2. The parameters η and γcan be used to scale and shape the stability metrics.

Penalizing these deviation metrics both incentivizesimproved power quality, and increases margin againstover/under voltage and frequency limits to reducethe likelihood of protective tripping if any follow-oncontingencies occur. Prioritizing voltage and frequencydeviation should maximize grid stability in the shortterm given a failure contingency. While these aresufficient for proof of concept, additional stabilitymetrics such as transient stability can be included.

We later add these stability metrics together to formour model’s objective function.

2.9. Stochastic Model

From the discretized dynamic model of the powergrid, we construct a two stage stochastic program withrecourse. This mathematical program takes the form

minx,yψ

c(x) + E[d(yψ)] (19)

s.t. (20)f(x) ≤ b (21)gψ(yψ) ≤ fψ ∀ψ ∈ Ψ (22)h(x) + k(yψ) ≤ gψ ∀ψ ∈ Ψ (23)

This allows only choices of x that leave constraints(23) feasible. Within these constraints, x is chosen,and yψ is chosen for every scenario ψ, so that the firststage objective c(x) and expected value of second stageobjectives d(yψ) are optimal. See [17] for further detailson stochastic programming.

Unlike most stochastic programming models in theliterature, our model’s first stage and second stage areseparated by time. Our first stage decisions are made fort ∈ [0, tf ) and prepares the grid to be in a resilient statebefore any contingency occurs. Meanwhile our secondstage decisions are made for t ∈ [tf , T ] and stabilizesthe grid after any contingency is realized.

The first stage of our stochastic model optimizescontrol variables Vref and Pref over [0, tf ) to preparefor any of the scenarios in Ψ. Therefore our first stageconstraints – effectively (21) – are equations (1)’ - (15)’discretized only for t ∈ {τ ∈ P| τ < tf}.

Our model’s second stage optimizes controlvariables Vref and Pref to maximize stability in eachscenario. For every ψ ∈ Ψ, we add equations (1)’ -(15)’ discretized only for t ∈ {τ ∈ P| τ ≥ tf} as oursecond stage constraints (22).

To ensure continuity of the dynamic model between[0, tf ) and [tf , T ], we add the single discretizedconstraint for (1)’ - (5)’, (8)’ - (9)’ pertinent to tf andthe time period before it. Note that these constraints

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form (23) above; they contain second stage decisionvariables for tf and first stage decision variables for thetime before it.

Finally, using our stability metrics Mv and Mω , ourobjective (19) is

minVref ,Pref

Mv(0, tf ) +Mω(0, tf )+ (24)∑ψ∈Ψ

pψ (Mv(tf , T ) +Mω(tf , t))

Note that the first two terms use V and ω variables fromthe first stage equations while the final terms use Vand ω variables from the various second stage scenarioequations.

In effect, our model optimizes over a singular set ofcontrols with the anticipation of various potential failurecontingency scenarios – each with its own probability –occurring. The model then optimizes controls over everyscenario to maximize stability in the event that scenariooccurs.

Note that in general, the first stage could actuallyprecede the second stage by minutes or more. In thatcase, it may be prudent to incorporate a cost incentivein the first stage to prevent economically unrealisticresults. For this proof of concept, however, we arefocused on the stability problem, and present the stagesas temporally contiguous and omit a cost penalty forsimplicity.

We consider simple load, line, and generator trips asour contingencies. For ease of exposition, we define thepost-failure partition Pf = {t ∈ P|t ≥ tf}.

To model a load trip we deactivate the differentialand algebraic load constraints (8)’ - (11)’ and fix PL,QL, xpL, xqL to 0 for t ∈ Pf .

A line trip eliminates the connection between twobuses b1 and b2. This is done by setting Yb1,b2 and Yb2,b1to zero and then recalculating the diagonal entries Yb1,b1and Yb2,b2 for t ∈ Pf . Note that in our implementation,Y includes time as an additional index to allow line trips.

Finally, for generator trips, constraints (6)’ - (7)’ aredeactivated and variables Iq , Id are fixed to zero for t ∈Pf . This prevents generator current from entering thepower network.

When performing single-scenario experiments withthese failure contingencies, we are assuming that thefailure happens deterministically, and that we optimizethe stability given this future failure. If we performa multi-scenario experiment, however, we need tointroduce uncertainty in the likelihood of each scenariooccurring. We define pψ as the probability thatcontingency scenario ψ occurs.

3. Experimentation and Results

We demonstrate our model through experiments onthe WSCC 9-bus power system. Our experimentsoptimize grid resilience and stability for different failurecontingencies.

3.1. Solution Methodology

Our model is developed in the mathematicalprogramming language Pyomo [18]. Pyomo’ssublanguage Pyomo.DAE facilitates expression ofdifferential equations and has automatic discretizationcapability with various optional discretization schemes.The experiments described in this section use uniformspacing and collocation techniques implemented inPyomo.DAE.

Pyomo transforms our DAE model into a nonlinearprogram and can interface with most nonlinear solvers.In our experiments, we used both IPOPT [19] and Knitro[20].

3.2. No-Failure Deterministic Case

Let us first consider a single scenario with nofailures. Solving our model with shape parametersη = 1 and γ = 2, a time horizon T = [0, 3], and 40discretized points, we get the following results. Figures1, 2, 3, and 4 depict V , ω, Vref , and Pref respectivelyover time.

Figure 1. V (p.u) vs. t with no failure

Note that although the system is initially in a steadystate, it quickly moves away from that steady state (viagenerator controls) towards a state with more safetymargin. We can see that even without failures, thesystem can benefit from controls – here, primarily toimprove voltage quality.

Of note, the frequency separation depicted in Figure2 is both very small and brief, and not of stabilityconcern (it is causing the generator rotor angles toconverge, not diverge).

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Figure 2. ω (Hz.) vs. t with no failure

Figure 3. Vref vs. t control trajectory with no failure

Figure 4. Pref vs. t control trajectory with no failure

3.3. Deterministic Failure Cases

At tf = 1.5, we induce a line trip between buses 5and 7, a generator trip at bus 1, and a load trip at bus 5.Our model yields the following voltage responses whendealing with each scenario individually. As the firststage is effectively allowed to anticipate each upcomingfailure, these results show the (unrealistic) best possiblepreventive-corrective control actions for each scenario.

Figure 5. V vs. t with line 5-7 trip at tf = 1.5

Figure 6. V vs. t with generator trip at Bus 1

Figure 7. V vs. t with load trip at Bus 5

For this small system, even single-componentcontingencies are a major perturbation to the system.Nonetheless, coupling preventive control (before tf )and corrective control (after tf ) allows the system tostabilize the system fairly quickly. Naturally, behaviormay take significantly longer to stabilize using morerealistic ramp rates and bounds on control variables.

In the following subsection we show the results whenthe model can not anticipate which failure will occur.

3.4. Multi-Scenario Stochastic Case

Here we optimize the first stage over severalscenarios, with scenario-specific dynamics and recoursecontrol in the second stage. In this example thereare 4 scenarios of equal probability: the three failureevents in Section 3.3 plus a no-failure scenario.This multi-scenario experiment yields the followingfirst-stage results. Note that all first-stage dynamics aredue to control actions (to improve upon the non-optimalinitial condition and prepare for contingencies).

Figure 8. First stage V vs. t

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Figure 9. First stage ω vs. t

Figure 10. First stage Vref vs. t control

Figure 11. First stage Pref vs. t control

While the frequency separation in Figure 9 wouldnot be sustainable over longer time windows, in this caseit optimally pre-positions the grid for the possible setof upcoming failures (and although not shown here, iscorrected quickly after each failure).

Figures 12-15 show the voltage trajectory for eachscenario across the entire time horizon. Note that thepre-contingency behavior is the same as Figure 8 acrossall scenarios.

Figure 12. V vs. t for line 5-7 trip

Figure 13. V vs. t for load trip at Bus 5

Figure 14. V vs. t for generator trip at Bus 1

Figure 15. V vs. t with no failures

As these figures show, the optimization is able tokeep voltage within desired values (except for a briefexcursion at bus 5 in Figure 12). Frequency was alsokept within 0.2% of nominal in all cases.

Figure 14 is particularly interesting; while thevoltages diverge, they are kept centered withinreasonable bounds around 1.0 p.u. despite the suddenloss of 24% of total generation. This is significant;although not shown here, without control the systemexperiences major voltage drop across the entire system.Coupling first and second-stage controls is particularlyeffective, yielding a 65% reduction in objective valuecompared to pre-positioning alone, or 61% reductioncompared to recourse control alone.

While the results shown here are optimisticdue to relaxed rate-of-change and bound constraints,they provide proof of concept that the combinedpre-positioning and emergency control formulationcan provide significant benefit during emergencies.

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Furthermore, they demonstrate that it is conceivableto pre-position against multiple serious contingencieswhose systemic effects are considerably different.However, such pre-positioning is likely to be far fromoptimal in nominal operating conditions (economicpenalties may mitigate this somewhat, albeit at somecost to contingency readiness). In Figure 16 we see thatsignificant control actions are required for the no-failurecase to “stand down” from contingency readiness.

Figure 16. Pref vs. t > tf with no failures

4. Conclusion

We have demonstrated the ability to solve ourtwo-stage stochastic optimization model with dynamicconstraints that represent power system dynamics. Thisclass of problem is important to power system resilience,as 1) both preparation (first stage) and emergencycontrol (second stage) are needed to ensure systemoperability under severe emergencies, 2) coupling thesedecisions together in the same framework should allowbetter solutions across a wider set of contingencies,and 3) system dynamics are critical to assessing gridstability in such emergencies. Additionally, motivatedby our intended use on severe emergency cases, wedemonstrate the use of power stability and qualityobjectives, as opposed to economic objectives typicalin the literature. This framework allows preparationagainst an uncertain space of contingencies scenarios, aswell as immediate recourse control once the uncertaintyis realized. This provides the greatest margin for safetyagainst substantial voltage and frequency deviations.

4.1. Future Research

Our current work demonstrates proof of concept onthe WSCC 9-bus system. Future research will lookat scalability to larger systems and scenario sets, andeffectiveness against multi-component failure scenarios.Load shed will be added as a second-stage decisionvariable to improve feasibility in such cases. Control ofinverter-based resources may be of significant benefit aswell. Higher temporal fidelity and/or more sophisticated

temporal discretization may be needed to accuratelycapture severe post-contingency behavior.

In cases where any excursions outside boundsare unacceptable, hard constraints can be set. Forexample, to ensure transient stabiilty, rotor angleshould be no more than 100◦ from system centerof inertia. Likewise, voltage and frequency boundconstraints would ensure that the solutions do notexceed bounds that would cause protective tripping inthe real system. Additional objective penalties and/orconstraints could be considered (e.g., voltage stability,small signal stability, and transmission line powerlimits) to provide more assurance of system stability insevere contingencies.

5. Acknowledgments

We extend our warm appreciation to BethanyNicholson for her assistance with Pyomo modeling anddiscretization questions.

Sandia National Laboratories is a multimissionlaboratory managed and operated by NationalTechnology & Engineering Solutions of Sandia,LLC, a wholly owned subsidiary of HoneywellInternational Inc., for the U.S. Department of Energy’sNational Nuclear Security Administration undercontract DE-NA0003525. SAND2020-13081 C.

This paper describes objective technical results andanalysis. Any subjective views or opinions that might beexpressed in the paper do not necessarily represent theviews of the U.S. Department of Energy or the UnitedStates Government.

References

[1] P. W. Sauer, M. A. Pai, and J. H. Chow, Powersystem dynamics and stability: with synchrophasormeasurement and power system toolbox. John Wiley &Sons, 2017.

[2] P. Kundur, N. J. Balu, and M. G. Lauby, Power systemstability and control, vol. 7. McGraw-hill New York,1994.

[3] D. Gan, R. J. Thomas, and R. D. Zimmerman,“Stability-constrained optimal power flow,” IEEETransactions on Power Systems, vol. 15, no. 2,pp. 535–540, 2000.

[4] Y. Yuan, J. Kubokawa, and H. Sasaki, “A solutionof optimal power flow with multicontingency transientstability constraints,” IEEE Transactions on PowerSystems, vol. 18, no. 3, pp. 1094–1102, 2003.

[5] Z. Li, G. Yao, G. Geng, and Q. Jiang, “An efficientoptimal control method for open-loop transient stabilityemergency control,” IEEE Transactions on PowerSystems, vol. 32, no. 4, pp. 2704–2713, 2016.

[6] S. Abhyankar, G. Geng, M. Anitescu, X. Wang,and V. Dinavahi, “Solution techniques for transientstability-constrained optimal power flow–part i,” IET

Page 3366

Generation, Transmission & Distribution, vol. 11,no. 12, pp. 3177–3185, 2017.

[7] G. Geng, V. Ajjarapu, and Q. Jiang, “A hybrid dynamicoptimization approach for stability constrained optimalpower flow,” IEEE Transactions on Power Systems,vol. 29, no. 5, pp. 2138–2149, 2014.

[8] Q. Jiang and Z. Huang, “An enhanced numericaldiscretization method for transient stability constrainedoptimal power flow,” IEEE Transactions on PowerSystems, vol. 25, no. 4, pp. 1790–1797, 2010.

[9] Q. Jiang and G. Geng, “A reduced-space interior pointmethod for transient stability constrained optimal powerflow,” IEEE Transactions on Power Systems, vol. 25,no. 3, pp. 1232–1240, 2010.

[10] G. Geng and Q. Jiang, “A two-level paralleldecomposition approach for transient stabilityconstrained optimal power flow,” IEEE Transactions onPower Systems, vol. 27, no. 4, pp. 2063–2073, 2012.

[11] G. Geng, Q. Jiang, and Y. Sun, “Parallel transientstability-constrained optimal power flow using gpu ascoprocessor,” IEEE Transactions on Smart Grid, vol. 8,no. 3, pp. 1436–1445, 2016.

[12] D. Prasad, A. Mukherjee, G. Shankar, and V. Mukherjee,“Application of chaotic whale optimisation algorithm fortransient stability constrained optimal power flow,” IETScience, Measurement & Technology, vol. 11, no. 8,pp. 1002–1013, 2017.

[13] K. Teeparthi and D. V. Kumar, “Dynamic power systemsecurity analysis using a hybrid pso-apo algorithm,”Engineering, Technology & Applied Science Research,vol. 7, no. 6, pp. 2124–2131, 2017.

[14] D. Karlsson and D. J. Hill, “Modelling and identificationof nonlinear dynamic loads in power systems,” IEEETransactions on Power Systems, vol. 9, no. 1,pp. 157–166, 1994.

[15] R. Zhang, Y. Xu, W. Zhang, Z. Y. Dong, andY. Zheng, “Impact of dynamic models on transientstability-constrained optimal power flow,” in 2016IEEE PES Asia-Pacific Power and Energy EngineeringConference (APPEEC), pp. 18–23, IEEE, 2016.

[16] D. J. Hill, “Nonlinear dynamic load models withrecovery for voltage stability studies,” IEEE transactionson power systems, vol. 8, no. 1, pp. 166–176, 1993.

[17] J. R. Birge and F. Louveaux, Introduction to stochasticprogramming. Springer Science & Business Media,2011.

[18] “Pyomo documentation 5.7,” 2017. Availableat https://pyomo.readthedocs.io/en/stable/.

[19] “Ipopt documentation.” Available at https://coin-or.github.io/Ipopt/.

[20] “Knitro - artelys.” Available athttps://www.artelys.com/solvers/knitro/.

[21] C. Ledesma, Calle and Arredondo, “Multi-contingencytscopf based on full-system simulation,” IET Generation,Transmission & Distribution, vol. 11, no. 1, pp. 64–72,2017.

[22] H. Nguyen-Duc, L. Tran-Hoai, and D. V. Ngoc, “A novelapproach to solve transient stability constrained optimalpower flow problems,” Turkish Journal of ElectricalEngineering & Computer Sciences, vol. 25, no. 6,pp. 4696–4705, 2017.

[23] H. Nguyen-Duc, L. Tran-Hoai, and H. Cao-Duc, “Anapproach to solve transient stability-constrained optimalpower flow problem using support vector machines,”Electric Power Components and Systems, vol. 45, no. 6,pp. 624–632, 2017.

[24] Y. Yang, W. Liu, J. Deng, H. Liu, H. Wei, and T. Wang,“A parallel approach for multi-contingency transientstability constrained optimal power flow,” in 2017 IEEEPower & Energy Society General Meeting, pp. 1–5,IEEE, 2017.

[25] Z. Li, G. Geng, and Q. Jiang, “Transient stabilityemergency control using asynchronous parallelmixed-integer pattern search,” IEEE Transactionson Smart Grid, vol. 9, no. 4, pp. 2976–2985, 2016.

[26] M. Paramasivam, A. Salloum, V. Ajjarapu, V. Vittal,N. B. Bhatt, and S. Liu, “Dynamic optimizationbased reactive power planning to mitigate slow voltagerecovery and short term voltage instability,” IEEETransactions on Power Systems, vol. 28, no. 4,pp. 3865–3873, 2013.

[27] G. C. Zweigle and V. Venkatasubramanian, “Wide-areaoptimal control of electric power systems withapplication to transient stability for higher ordercontingencies,” IEEE Transactions on Power Systems,vol. 28, no. 3, pp. 2313–2320, 2013.

[28] E. Pavella and Ruiz-Vega, Transient Stability of PowerSystems A Unified Approach to Assessment and Control.Springer, 2010.

[29] R. Zarate-Minano, T. Van Cutsem, F. Milano, andA. J. Conejo, “Securing transient stability usingtime-domain simulations within an optimal power flow,”IEEE Transactions on Power Systems, vol. 25, no. 1,pp. 243–253, 2009.

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