1

Data-Based Distributionally RobustStochastic Optimal Power Flow – Part II:

Case StudiesYi Guo, Student Member, IEEE, Kyri Baker, Member, IEEE, Emiliano Dall’Anese, Member, IEEE,

Zechun Hu, Senior Member, IEEE, Tyler H. Summers, Member, IEEE

Abstract—This is the second part of a two-part paper on data-based distributionally robust stochastic optimal power flow (OPF).The general problem formulation and methodology have beenpresented in Part I [1]. Here, we present extensive numericalexperiments in both distribution and transmission networks toillustrate the effectiveness and flexibility of the proposed method-ology for balancing efficiency, constraint violation risk, and out-of-sample performance. On the distribution side, the methodmitigates overvoltages due to high photovoltaic penetration usinglocal energy storage devices. On the transmission side, the methodreduces N-1 security line flow constraint risks due to high windpenetration using reserve policies for controllable generators. Inboth cases, the data-based distributionally robust model predic-tive control (MPC) algorithm explicitly utilizes forecast errortraining datasets, which can be updated online. The numericalresults illustrate inherent tradeoffs between the operational costs,risks of constraints violations, and out-of-sample performance,offering systematic techniques for system operators to balancethese objectives.

Index Terms—stochastic optimal power flow, data-driven opti-mization, multi-period distributionally robust optimization, dis-tribution networks, transmission systems

I. INTRODUCTION

AS penetration levels of renewable energy sources (RESs)continue increasing to substantial fractions of total sup-

plied power and energy, system operators will require moresophisticated methods for balancing inherent tradeoffs betweennominal performance and operational risks. This is relevant forboth transmission and distribution networks, which have seenrapid recent increases in photovoltaic (PV) and wind energysources. Here, we present extensive numerical experiments inboth distribution and transmission networks to illustrate theeffectiveness and flexibility of our data-based distributionallyrobust stochastic optimal power flow (OPF) methodology forbalancing efficiency, constraint violation risk, and out-of-sampleperformance.

This material is based on work supported by the National Science Foundationunder grant CNS-1566127. (Corresponding author: Tyler H. Summers)

Y. Guo and T.H. Summers are with the Department of MechanicalEngineering, The University of Texas at Dallas, Richardson, TX, USA, email:{yi.guo2,tyler.summers}@utdallas.edu.

K. Baker is with the Department of Civil, Environmental, and Archi-tectural Engineering, The University of Colorado, Boulder, CO, USA,email:kyri.baker@colorado.edu.

E. Dall’Anese is with the Department of Electrical, Computer, and EnergyEngineering, University of Colorado Boulder, Boulder, CO, USA, email:emiliano.dallanese@colorado.edu.

Z. Hu is with the Department of Electrical Engineering, Tsinghua University,Beijing, China, email: zechhu@tsinghua.edu.cn.

A variety of stochastic OPF methods have been recentlyproposed to explicitly incorporate forecast errors of networkuncertainties for modeling risk. These methods can be cate-gorized according to assumptions made about forecast errordistributions and metrics for quantifying risk. Many makespecific assumptions about forecast error distributions anduse chance constraints, which encode value at risk (VaR),conditional value at risk (CVaR), or distributional robustnesswith specific ambiguity sets [2]–[10]. Others handle uncertaintyusing scenarios approaches, sample average approximation,or robust optimization [11]–[28]. However, in practice alldecisions about how to model forecast error distributions mustbe based on finite historical training datasets. None of theexisting methods account for sampling errors inherent in suchdatasets in high-dimensional spaces.

Our proposed data-based distributionally robust stochasticOPF methodology, developed in Part I [1], explicitly incor-porates robustness to sampling errors by considering setsof distributions centered around a finite training dataset. Amodel predictive control (MPC) approach utilizes Wasserstein-based distributionally robust optimization subproblems toobtain superior out-of-sample performance. Here in Part IIwe illustrate the effectiveness and flexibility of the method-ology in both distribution and transmission networks. On thedistribution side, the method mitigates overvoltages due to highphotovoltaic penetration using local energy storage devices. Onthe transmission side, the method reduces N-1 security lineflow constraint risks due to high wind penetration using reservepolicies for controllable generators. In both cases, the data-based distributionally robust (MPC) algorithm explicitly utilizesforecast error training datasets, which can be updated online.The numerical results illustrate inherent tradeoffs between theoperational costs, risks of constraints violations, and out-of-sample performance, offering systematic techniques for systemoperators to balance these objectives.

The rest of the paper is organized as follows. Section IIillustrates the data-based distributionally robust stochastic ACOPF for mitigating overvoltages in a modified IEEE 37-nodedistribution feeder with high PV penetration using localenergy storage devices. Section III illustrates the data-baseddistributionally robust stochastic DC OPF for reducing N-1security line flow constraint risks due to high wind penetrationusing reserve policies for controllable generators. Section IVconcludes.

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Notation: The inner product of two vectors a, b ∈ Rm isdenoted by 〈a, b〉 := aᵀb. The Ns-fold product of distributionP on a set Ξ is denoted by PNs , which represents a distributionon the Cartesian product space ΞNs = Ξ× . . .×Ξ. We use Nsto represent the number of samples inside the training datasetΞ. Superscript “ · ” is reserved for the objects that dependon a training dataset ΞNs . We use (·)ᵀ to denote vector ormatrix transpose. The operators <{·} and ={·} return the realand imaginary part of a complex number, respectively. Theoperator [ · ][a,b] selects the a-th to b-th elements of a vectoror rows of a matrix.

II. OVERVOLTAGE MITIGATION IN DISTRIBUTIONNETWORKS

In this section, we apply the data-based distributionallyrobust stochastic OPF methodology to mitigate overvoltagesin distribution networks by controlling set points in RESs andenergy storage devices. We provide further modeling details ofthe loads, inverter-based RESs, and energy storage devices. Theset points of controllable devices are repeatedly optimized overa finite planning horizon within a MPC feedback scheme. Therisk conservativeness of the voltage magnitude constraints andthe out-of-sample performance robustness to sampling errorsare explicitly adjustable by two scalar parameters.

A. System model

1) Loads: We use P tl,n and Qtl,n to denote the active andreactive power demands at bus n ∈ N . We also define twovectors ptl := [P tl,1, . . . , P

tl,N ]ᵀ and qtl := [Qtl,1, . . . , Q

tl,N ]ᵀ.

If no load is connected to bus n ∈ N , then P tl,n = 0 andQtl,n = 0. Load uncertainties are modeled based on historicaldata of forecast errors. The active and reactive loads are givenby ptl = ptl(ut)+ptl(ξt), qtl = qtl(ut)+qtl(ξt), where ptl(ut) ∈RN and qtl(ut) ∈ RN are forecasted nominal loads, which candepend on control decisions (e.g., load curtailment control).The nodal injection errors ptl(ξt) ∈ RN and qtl(ξt) ∈ RN

depend on the aggregate forecast error vector ξt.2) Renewable energy model: Let P tav,n be the maximum

availability renewable energy generation at bus n ∈ NR ⊆ N ,where the set NR denotes all buses with RESs. With highRES penetration, overvoltages can cause power quality andreliability issues. By intelligently operating set points of RESand energy storage, operators can optimally trade off risk ofconstraint violation and economic efficiency (e.g., purchase ofelectricity from the main grid, active power curtailment costs,and reactive compensation costs). The active power injectionsof RESs are controlled by adjusting an active power curtailmentfactor αtn ∈ [0, 1]. Reactive power set points of RESs can alsobe adjusted within a limit Sn on apparent power as follows√

((1− αtn)P tav,n)2 + (Qtn)2 ≤ Sn, n ∈ NR.

We define aggregate vectors: αt := [αt1, . . . , αtN ]ᵀ and ptav :=

[P tav,1, . . . , Ptav,N ]ᵀ and qtc := [Qt1, . . . , Q

tN ]ᵀ.

If bus n ∈ N\NR has no RES, by convention we setαtn = 0, P tav,n = 0 and Qtn = 0. The curtailment factor and

reactive power compensation {αtn, Qtn} together set the inverteroperating point and are also subject to a power factor constraint

|Qtn| ≤ tan(θn)[(1− αtn)P tav,n], n ∈ NR,

where cos(θn) ∈ (0, 1] is the power factor limit for RESs. Thepower factor constraint is convex, and can be discarded insettings where the inverters are not required to operate at aminimum power factor level. The premise here is that RESs canassist in the regulation of voltages by promptly adjusting thereactive power and curtailing active power as needed; RESscan provide faster voltage regulation capabilities comparedto traditional power factor correction devices (i.e., capacitorbanks). The proposed data-based distributionally robust OPFwill consider adjustments of both active and reactive powersto aid voltage regulation, which in principle can be done inboth transmission and distribution networks.

3) Energy storage model: The state-of-charge (SOC) of theenergy storage device located at bus n ∈ NB ⊆ N in kWh isrepresented as Btn. The dynamics of these devices are

Bt+1n = Btn + ηB,nP

tB,n∆, n ∈ NB , (1)

where ∆ is the duration of the time interval (t, t + 1],and P tB,n is the charging/discharging power of the storagedevice in kW. We assume the battery state is either charging(P tB,n ≥ 0) or discharging (P tB,n ≤ 0) during each time interval(t, t+ 1]. For simplicity, we suppose the round-trip efficiencyof the storage device ηB,n = 1 to avoid the nonconvexitywhen introducing binary variables. Additionally, two commonoperational constraints of energy storage devices are

Bminn ≤ Btn ≤ Bmax

n , PminB,n ≤ P tB,n ≤ Pmax

B,n,

where Bminn , Bmax

n are the rated lower and upper SOClevels, and Pmin

B,n, PmaxB.n are the minimum and maximum

charging/discharging limits. Other constraints can be addedfor electric vehicles (EVs), for example, a prescribed SOCBtn = Bmax

n at a particular time. If no energy device isconnected to a certain bus, the charging/discharging power andSOC are fixed to zero: P tB,n = 0, Btn = 0, for all n ∈ N\NB .We define the aggregate vectors ptB := [P tB,1, . . . , P

tB,N ]ᵀ, and

bt := [Bt1, . . . , BtN ]ᵀ.

B. Data-based stochastic OPF implementation

We now use the methodologies presented in Part I SectionIV, and the models of loads, RESs and energy storage devicesto develop a data-based stochastic AC OPF for solving avoltage regulation problem. This stochastic OPF aims to balancethe operational costs the total CVaR values of the voltagemagnitude constraints. We consider an operational cost thatcaptures electricity purchased by customers, excessive solar

3

energy fed back to the utility, reactive power compensationcosts and penalties for active power curtailment

J tCost(αt,qtc,p

tB , ξt) =

=∑n∈N

at1,n[P tl,n + P tB,n − (1− αtn)P tav,n

]+

+∑n∈N

at2,n[(1− αtn)P tav,n − P tl,n − P tB,n

]+

+∑n∈N

at3,n|Qtn|+∑n∈N

at4,nαtnP

tav,n.

We collect all decision variables into yt = {αt,qtc,ptB ,bt},and all RES and load forecast errors into the random vectorξt. Now the MPC subproblems take the following formData-based distributionally robust stochastic OPF

infyτ ,κ

τo ,

$1,n,$2,n

t+Ht∑τ=t

{E[JτCost] + ρ sup

Qτ∈PNsτ

N∑o=1

EQτ [Qτo ]

},

= infyτ ,κ

τo ,

$1,n,$2,n

λτo ,sτio,ς

τiko

t+Ht∑τ=t

{E[JτCost] +

N∑o=1

(λoετ +

1

Ns

Ns∑i=1

sτio

)},

(2a)subject to

ρ(bok(κτo) + 〈aok(yτ ), ξiτ 〉+ 〈ςiko,d−Hξiτ 〉) ≤ sτio, (2b)‖Hᵀςiko − ρaok(yτ )‖∞ ≤ λτo , (2c)ςiko ≥ 0, (2d)

1

Ns

Ns∑i=1

[[(1− ατn)P τ,iav,n]2 + (Qτn)2 − S2

n +$τ1,n

]+

≤ $τ1,nβ,

(2e)

1

Ns

Ns∑i=1

[tan(θn)[(1− αtn)P τ,iav,n]− |Qτn|+$τ

2,n

]+

≤ $τ2,nβ,

(2f)

Bminn ≤ Bτn ≤ Bmax

n , (2g)

PminB,n ≤ P τB,n ≤ Pmax

B,n, (2h)

Bτ+1n = Bτn + ηB,nP

τB,n∆, (2i)

0 ≤ ατn ≤ 1, (2j)∀i ≤ Ns, ∀o ≤ N`, n ∈ NR, k = 1, 2, τ = t, . . . , t+Ht,

where $τ1,n, $τ

2,n, and κτo are CVaR auxiliary valuables,and λτo , sτio, ς

τiko are auxiliary variables associated with the

distributionally robust Wasserstein ball reformulation. Forsimplicity, the power factor constraints and apparent powerlimitation constraints are not treated as distributionally robustconstraints, and instead are handled using direct sample averageapproximation.

Remark 2.1 (battery efficiency). To maintain convexity ofthe underlying problem formulation and therefore facilitate thedevelopment of computationally affordable solution methods,we utilized an approximate model for the battery dynamicswith no charging and discharging efficiency losses (2). At theexpense of significantly increasing the problem complexity,

charging and discharging efficiencies can be accommodated as[29]

Bτ+1n = Bτn + ηcP

τBc,n∆− 1

ηdP τBd,n∆,

where ηc, ηd ∈ (0, 1] denote the charging and the dischargingefficiencies, respectively; P τBc,n ≥ 0 represents the chargingrate and P τBd,n ≥ 0 the discharging rate at time τ . Additionalconstraints, however, are needed to ensure that the solutionavoids meaningless solutions where a battery is required tocharge and discharge simultaneously; in particular, one can:a) add a constraint P τBc,nP

τBd,n

= 0 [29]; or. b) introducebinary variables to indicate the charging status (e.g., charg-ing/discharging) of the batteries [17]. Either way, given thenon-convexity of the resultant problem, possibly sub-optimalsolutions can be achieved (2). In addition, exact relaxationmethods under appropriate assumptions offer an alternative wayto maintain convexity of the charging problem; see [30], [31].Extending the proposed technical approach to a setting withbinary variables or exact relaxation methods will be pursuedas a future research effort.

Remark 2.2 (battery life). The degradation of energy storagesystems may depend on the depth of discharge and the numberof charging/discharging cycles [32]. The battery aging processis usually described by partial differential equations [33];this is a practical model for industrial applications, but itintroduces significant computational challenges in optimizationtasks [34]. Additional optimization variables as well as penaltyfunctions could be included to limit the number of cyclesper day and ensure a minimum state of charge [35]–[41].Pertinent reformulations to account for battery degradation willbe pursued in future research activities.

Remark 2.3 (voltage at slack bus). Similar to the majorityof the works in the literature, the voltage at the slack bus(i.e., substation) is considered as an input of the problem (and,therefore, it is not controllable). However, it is worth notingthat discrete variables modeling changing the tap position ofthe transformer can be incorporated in (2); see e.g., [27], [28].Branch and bound techniques can then be utilized to solve theproblem.

C. Numerical results

We use a modified IEEE-37 node test feeder to demonstrateour proposed data-based stochastic AC OPF method. As shownin Fig. 1, the modified network is a single-phase equivalentand the load data is derived from real measurements fromfeeders in Anatolia, CA during the week of August 2012 [42].We place 21 photovoltaic (PV) systems in the network. Theirlocations are marked by yellow boxes in Fig. 1, and theircapacities are summarized in Table I. Based on irradiation datafrom [42], [43], we utilized a greedy gradient boosting method[44] to make multi-step ahead predictions of solar injections,and then computed a set of forecast errors from the dataset.In general forecast errors increase with the prediction horizon.Other parameters of the network, such as line impedancesand shunt admittances, are taken from [45]. The total nominalavailable solar power

∑n P

tav,n and aggregate load demand

over 24 hours is also shown in Fig. 2.

4

Fig. 1. IEEE 37-node test feeder with renewable energy resources and storagedevices.

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM0

500

1000

1500

2000

2500

3000

3500

4000

kW

Total Load DemandTotal Available Solar Energy

Fig. 2. Total available solar energy and load demand.

TABLE ICAPACITIES OF INVERTED-BASED SOLAR ENERGY GENERATIONS AND

ENERGY STORAGE DEVICES

Node Sn [kVA] Node Sn [kVA] Node Bmaxn

4 150 7 300 9 1009 300 10 600 10 10011 660 13 360 28 5016 600 17 360 29 25020 450 22 150 32 25023 750 26 300 35 12028 750 29 300 36 20030 360 31 60032 330 33 75034 450 35 45036 450

The energy storage systems are placed with PV systems atcertain nodes, as shown in Fig. 1. Their locations and capacitiesare listed in Table I. We select the capacities in the rangeof typical commercial storage systems, or aggregate of 10-12 residential-usage batteries (e.g., electric vehicles), whichare connected to the same step-down transformer. The lowerlimit of SOC, Bmin

n , is set to be zero for all batteries. The

charging/discharging rate P tB,n is also limited by 10% of theirrespective energy capacity Bmax

n . Voltage limits V max andV min are 1.05 p.u. and 0.95 p.u., respectively. The cost functionparameters are at1,n = 10, at2,n = 3, and at3,n = 3 and at4,n = 6.The decision making time period is 5 minutes.

Due to high PV penetration, overvoltage conditions canemerge during solar peak irradiation. Given the real dataavailable for the numerical tests, the numerical tests arefocused on alleviating over-voltage conditions via the proposeddistributionally-robust tools. Other constraints are approximatedvia sample average methods [6], [20], [46]; however, ingeneral it is straightforward to formulate other constraints asdistributionally robust. The power factor PF limit is 0.9 in (2f).The risk level parameter η is set to 0.01 for quantifying 1%violation probability of constraints (2e)-(2f). To emphasize theeffect of sampling errors, the number of forecast error samplesNs included in the training dataset ΞNst is limited to 30. Theforecast errors are not assumed bounded, so the parametersof the support polytope Ξt := {ξt ∈ RNξ : Hξt ≤ d} are setto zero in (2b)-(2c). We solved (2) using the MOSEK solver[47] via the MATLAB interface CVX [48] on a laptop with16 GB of memory and 2.8 GHz Intel Core i7. Solving eachtime step during solar peak hours with distibutionally robustconstraints takes 4.84 seconds. Note that our implementationis not optimized for speed and in principle could easily besped up and scaled to larger problems since the problem isultimately convex quadratic.

In our framework, there are two key parameters, ρ andε, that explicitly adjust trade offs between performance andconstraint violation risk, and robustness to sampling errors.Fig. 3 illustrates the basic tradeoffs between operational costand CVaR values of voltage constraint violations during a24-hour operation for various values of ρ and ε. It can bereadily seen that as ρ increases, operational cost increases, butCVaR decreases since the risk term is emphasized. Notice thatwith the increasing of ε, the estimated risk is higher so thatthe solution is more conservative and leads to a lower riskof constraint violation; larger Wasserstein balls lead to higherrobustness to sampling errors. These parameters offer systemoperators explicit data-based tuning knobs to systematicallyset the conservativeness of operating conditions.

Fig. 4(a)-4(c) shows the aggregated solar energy curtailmentand substation power purchases for varying risk aversion ρ andWasserstein radius ε. In order to prevent voltages over 1.05 p.u.,the available solar energy must be increasingly curtailed as therisk aversion parameter ρ increases. As a result, the networkmust import more power from the substation. The increasingcurtailment of solar energy and purchase of power drawn fromthe substation lead to significantly higher operational cost.

However, these decisions will also lead to more stable voltageprofiles, as shown in Fig. 5. When ρ is small, there is almostno curtailment, causing overvoltages at several buses. As ρincreases, more active power is curtailed, and all voltagesmove below their upper limit. Similar comments apply forvarying the Wasserstein radius ε. For example, fixing ρ andincreasing ε also results in more curtailment and lower voltagemagnitude profiles, which leads to better robustness to solarenergy forecast errors.

5

Operational

Cost in Dollars

-0.1

100 12 PM

0

9 AM

CV

aR

50 6 AM

Time

0.1

3 AM0 12 AM

increase

9 AM

Operational

Cost in Dollars

6 AM-0.1

Time

-0.05

0

100

CV

aR

0.05

3 AM

0.1

50 12 AM0

increase

-0.10 12 AM

-0.05

0

CV

aR

3 AM

0.05

0.1

Operational

Cost in Dollars Time

50 6 AM9 AM

100 12 PM

increase

12 AM3 AM

-0.1

0

-0.05

0

CV

aR

0.05

0.1

Time6 AM50 9 AM100

Operational

Cost in Dollars increase

Fig. 3. Tradeoffs between operational cost, conditional value at risk (CVaR) of voltage constraints, and robustness to sampling errors for ε =0.0000, 0.0005, 0.0010); parameters ρ and ε are varied to tests different weighting settings and radii of the Wasserstein ball, respectively. We presentfour views from different directions to avoid occlusion.

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM

0

500

1000

1500

2000

2500

3000

3500

4000

kW

= 1

= 15

= 20

= 30

= 35

= 100

= 200

(a) Total curtailed active power ε = 0.0000

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM

0

500

1000

1500

2000

2500

3000

3500

4000

kW

= 1

= 15

= 20

= 30

= 35

= 100

= 200

(b) Total curtailed active power ε = 0.0005

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM

0

500

1000

1500

2000

2500

3000

3500

4000

kW

= 1

= 15

= 20

= 30

= 35

= 100

= 200

(c) Total curtailed active power ε = 0.0010

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM

-3000

-2000

-1000

0

1000

2000

3000

kW

= 1

= 15

= 20

= 30

= 35

= 100

= 200

(d) Power drawn from substation ε = 0.0000

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM

-3000

-2000

-1000

0

1000

2000

3000

kW

= 1

= 15

= 20

= 30

= 35

= 100

= 200

(e) Power drawn from substation ε = 0.0005

12 AM 3 AM 6 AM 9 AM 12 PM 3 PM 6 PM 9 PM 12 AM

-3000

-2000

-1000

0

1000

2000

3000

kW

= 1

= 15

= 20

= 30

= 35

= 100

= 200

(f) Power drawn from substation ε = 0.0010

Fig. 4. Comparison of active power curtailment and power purchased from substation for various values of risk aversion ρ and Wasserstein radius ε. As theseparameters increase, more active power from PV is curtailed and more power is drawn from the substation, leading to a lower risk of constraint violation and ahigher operating cost.

Finally, we evaluate out-of-sample performance by im-plementing the full closed-loop distributionally robust MPCscheme over the 24 hour period with a 15 minute planninghorizon. Monte Carlo simulations with 100 realizations offorecast errors over the entire horizon are shown in in Fig.6. We subsampled new solar energy forecast errors from thetraining dataset. The closed-loop voltage profiles based onMPC decisions for all scenarios at node 28 are shown (othernodes with overvoltages show qualitatively similar results).Again, it is clearly seen that larger values of ε and ρ yieldmore conservative voltage profiles.

In summary, we conclude that the proposed data-baseddistributionally robust stochastic OPF is able to systematicallyassess and control tradeoffs between the operational costs, risks,and sampling robustness in distribution networks. The benefitsof the open-loop stochastic optimization problems are alsoobserved in the closed-loop multi-period distributionally robust

model predictive control scheme.

III. N-1 SECURITY PROBLEM IN TRANSMISSION SYSTEMS

In this section, we apply the proposed methodology fromPart I in a transmission system to handle N-1 line flow securityconstraints. The basic DC power flow approximation and devicemodeling is discussed in Part I. Here, we also incorporate N-1security constraints and associated contingency reactions dueto uncertain wind power injections.

A. System model

We consider a transmission system with NG generators(e.g., conventional thermal and wind) connected to bus subsetNG ⊆ N . There are NL loads, Nl lines, and Nb buses. Theoutages included for N-1 security consist of tripping of anysingle lines, generators or loads, yielding Nout = NG+NL+Nl

6

Fig. 5. Optimal network voltage profiles for varying ρ and ε. Overvoltages are reduced as ρ increases.

Fig. 6. Monte Carlo simulation results of the voltage profiles at node #28 resulting from the full distributionally robust closed-loop model predictive controlscheme. We validate that in closed-loop larger values of ε and ρ yield more conservative voltage profiles.

possible outages. We collect the outages corresponding to agenerator, a line and a load in different sets IG, Il, and IL.The outages in total are I = {0} ∪ IG ∪ IL ∪ Il, where {0}indicates no outage.

The formulation of a N-1 security problem is based onthe following assumptions: 1) the power flow equations areapproximated with DC power flow, as described in Part I(Section V); 2) each wind farm is connected to a single busof the network; 3) load forecasting is perfect; 4) a single lineoutage can cause multiple generator/load failures.

The objective of the data-based distributionally robuststochastic DC OPF is to determine an optimal reserve schedulefor responding to the wind energy forecast errors while takingthe network security constraints into account. We defineP jmis ∈ R for all j ∈ I as the generation-load mismatch

given by

P jmis =

P jL − P

jG, if j ∈ {0} ∪ Il

+P jL, if j ∈ IL−P jG, if j ∈ IG

,

where P jL, PjG ∈ R denote the power disconnection corre-

sponding to the outage j ∈ I. Define PG ∈ RNG , andPL ∈ RNL as nodal generation and load injection vectors.For the generator or load failures, the power disconnection P jGor P jL is corresponding to the components in the vector PG orPL. In the case of line failures j ∈ Il, the power disconnectionP jG or P jL is the sum of the power loss caused by multiplefailures. If there is no power disconnection caused by a lineoutage, or if j = 0 (no outage happens), the power mismatchis set to zero: P jmis = 0.

To respond to contingencies, we can also define anotherreserve policy response matrix Rj,d

mis,t := [Rj,d1 , . . . , Rj,dt ]ᵀ ∈

7

Rt, so that the affine reserve policy becomes

udt = Ddt ξt + Rj,d

mis,tPjmis + edt ,∀j ∈ I, d = 1, . . . , Nd. (3)

The general constraint risk function of the line flow in Part I(Section V, Equation (17c)) is then given by

f

( Nd∑d=1

Γt,jd

{rtd +Gtdξt + Ctd

[Adtx

d0 +Bdt (Dd

t ξt

+Rj,dmis,tP

jmis + edt )

]}− pt ≤ 0

), ∀j ∈ I,

(4)

where Γt,jd ∈ R2Lt×t maps the power injection of each devicein the case of j-th outage.

B. Data-based stochastic OPF implementation

We now use the modeling here and in Part I and the affinecontrol strategy (3) to formulate a data-based distributionallyrobust stochastic DC OPF for transmission systems that alsoincorporate N-1 security constraints. The goal is to balancetradeoffs between cost of thermal generation, CVaR values ofthe line flow constraints, and sampling error robustness. Thegeneration with reserve policy in the cost function is given byP d,tG =

[Ddt ξt + edt

]t. The operational cost of generators is

J tCost =∑d∈NG

c1,d[Pd,tG ]2 + c2,d[P

d,tG ] + c3,d,

which captures nominal and reserve costs of responding towind energy forecast errors. The N-1 security reserve cost isnot included to simplify presentation, but this can also easilybe included in our framework as an additional linear cost [15].

With the proposed modeling in Part I (Section V), theupdated data-based stochastic DC OPF is shown as follows.The decision variables are collected into yt = {Dt, et,Rmis,t}.The random vector ξt comprises all wind energy forecast errors.

Data-based distributionally robust stochastic DC OPF

infyτ ,στo

t+Ht∑τ=t

{E[JτCost] + ρ sup

Qτ∈PNsτ

N∑o=1

EQτ [Qτo ]

},

= infyτ ,σ

τo ,

λτo ,sτio,ς

τiko

τ+Ht∑τ=t

{E[JτCost] +

N∑o=1

(λτoετ +

1

Ns

Ns∑i=1

sτio

)},

(5a)subject to

ρ(bok(στo ) + 〈aok(yτ ), ξiτ 〉+ 〈ςτiko,d−Hξiτ 〉) ≤ sτio, (5b)‖Hᵀςτiko − ρaok(yτ )‖∞ ≤ λτo , (5c)ςτiko ≥ 0, (5d)

1

Ns

Ns∑i=1

Nd∑d=1

[Γt,jd

{rtd +Gtdξ

it + Ctd

[Adtx

d0 +Bdt (Dd

t ξit

+ Rj,d

mis,tPjmis + edt )

]}− pt

][t,t]

≤ 0, (5e)

Nd∑d=1

[(rtd + Ctd(A

dtxd0 +Bdt e

dt ))][t,t]

= 0, (5f)

Nd∑d=1

[(Gtd + CtdB

dtD

dt )][t,t]

= 0, (5g)

∀i ≤ Ns, ∀j ∈ I, ∀o ≤ N`, k = 1, 2, τ = t, . . . , t+Ht.

C. Numerical results

We consider a modified IEEE 118-bus test system [45] todemonstrate our proposed data-based distributionally robuststochastic DC OPF shown in Fig. 7. For simplicity, we onlyshow results of a single-period stochastic optimization problem.As with the distribution network, it is straightforward to extendto multi-period closed-loop stochastic control using MPC.

Fig. 7. IEEE 118-bus test network with multiple wind energy connections.

Three wind farms are connected to bus #1, bus #9, and bus#26, with the normal feed-in power 500 MW, 500 MW, and 800MW, respectively. The corresponding conventional generatorsat bus #1 and bus #26 are removed. The wind power forecasterrors are derived from the real wind sampling data fromhourly wind power measurements provided in 2012 Global

8

6 6.5 7 7.5 8 8.5

Operation Cost in Dollars 104

-5

0

5

10

15

20

CV

aR

Line 7

6 6.5 7 7.5 8 8.5

Operation Cost in Dollars 104

0

2

4

6

8

10

CV

aR

Line 37

6 6.5 7 7.5 8 8.5

Operation Cost in Dollars 104

-2

-1

0

1

2

CV

aR

Line 38

6 6.5 7 7.5 8 8.5

Operation Cost in Dollars 104

0

5

10

15

CV

aR

Line 54

6 6.5 7 7.5 8 8.5

Operation Cost in Dollars 104

0

2

4

6

8

10

CV

aR

Line 96

(a) Predicted tradeoffs between operational cost and CVaR of line constraintviolation.

0 200 400 600 800 1000

weight factor

-800

-700

-600

-500

Lin

e f

low

(M

W)

Line 7

nominal flow limit

0 200 400 600 800 1000

weight factor

350

400

450

500

550

600

Lin

e f

low

(M

W)

Line 37

nominal flow limit

0 200 400 600 800 1000

weight factor

350

400

450

500

550

Lin

e f

low

(M

W)

Line 38

nominal flow limit

0 200 400 600 800 1000

weight factor

500

600

700

800

Lin

e f

low

(M

W)

Line 54

nominal flow limit

0 200 400 600 800 1000

weight factor

200

250

300

350

400

450

Lin

e f

low

(M

W)

Line 96

nominal flow limit

(b) Out-of-sample performance is demonstrated by Monte Carlo simulation,with a controllable level of conservativeness.

Fig. 8. Five line flows are reported to illustrate the performance of the proposed data-based distributionally robust stochastic DC OPF in a transmission system.

0 200 400 600 800 1000

weight factor

0

100

200

300

po

we

r o

utp

ut

(MW

)

#4 Generator at bus 10

0 200 400 600 800 1000

weight factor

0

20

40

60

po

we

r o

utp

ut

(MW

)

#5 Generator at bus 12

0 200 400 600 800 1000

weight factor

280

300

320

340

360

po

we

r o

utp

ut

(MW

)

#35 Generator at bus 80

0 200 400 600 800 1000

weight factor

20

22

24

26

28

30

po

we

r o

utp

ut

(MW

)

#49 Generator at bus 111

0 200 400 600 800 1000

weight factor

-1

-0.8

-0.6

-0.4

-0.2

0

D1

#4 Generator at bus 10

0 200 400 600 800 1000

weight factor

-0.04

-0.03

-0.02

-0.01

0

D1

#5 Generator at bus 12

0 200 400 600 800 1000

weight factor

-0.04

-0.02

0

0.02

D1

#35 Generator at bus 80

0 200 400 600 800 1000

weight factor

-20

-15

-10

-5

0

D1

10-3 #49 Generator at bus 111

0 200 400 600 800 1000

weight factor

-0.4

-0.3

-0.2

-0.1

0D

2

#4 Generator at bus 10

0 200 400 600 800 1000

weight factor

-10

-8

-6

-4

-2

0

D2

10-3 #5 Generator at bus 12

0 200 400 600 800 1000

weight factor

-0.02

-0.01

0

0.01

0.02

D2

#35 Generator at bus 80

0 200 400 600 800 1000

weight factor

0

0.005

0.01

0.015

D2

#49 Generator at bus 111

0 200 400 600 800 1000

weight factor

-0.25

-0.2

-0.15

-0.1

-0.05

0

D3

#4 Generator at bus 10

0 200 400 600 800 1000

weight factor

-0.01

-0.008

-0.006

-0.004

-0.002

0

D3

#5 Generator at bus 12

0 200 400 600 800 1000

weight factor

-0.02

-0.015

-0.01

-0.005

0

D3

#35 Generator at bus 80

0 200 400 600 800 1000

weight factor

-8

-6

-4

-2

0

D3

10-3 #49 Generator at bus 111

Fig. 9. Comparison of the coefficients of the policies and output powers of selected generators for various values of risk aversion ρ and Wasserstein metric ε.As these parameters increase, the risk of line flow constraint decreases at the expense of higher operational costs.

Energy Forecasting competition (GEFCCom2012) [49]. Thewind energy forecast errors are evaluated based on a simplepersistence forecast, which predicts that the wind injectionin the following period remains constant. It can be seen theforecast errors are highly leptokurtic, i.e, that the errors havesignificant outliers that make the distribution tails much heavierthan Gaussian tails. We scale the forecasting errors to have zeromean and the standard deviations of 200 MW, 200 MW and300 MW for the wind farms at bus #1, #9 and #26, respectively.We consider five key lines, which deliver the wind power from

the left side of the system to the right, as marked by the redcrosses in Fig. 7. The line flow limits are shown in Table IIand marked by gray dash lines in Fig.8(b).

To simplify our presentation, only these five lines flowsare handled with distributionally robust optimization (5b)-(5d)in both directions; the remaining line flows are modeled byN-1 security constraints (5e) with nominal CVaR and sampleaverage approximation (essentially equivalent to taking ε = 0),and no other local device constraint is included. Additionally,we assume no bounds on the wind power forecast errors ξτ ,

9

TABLE IIFIVE MAIN CHANNEL LINES DATA

# of line From bus To bus Line flow limitation [MW]7 8 9 60037 8 30 50038 26 30 50054 30 38 50096 38 65 300

hence H and d in (5b)-(5c) are set to zero. It takes 10 secondsto solve (5) for each time step using the MOSEK solver [47]via the MATLAB interface with CVX [48] on a laptop with16 GB of memory and a 2.8 GHz Intel Core i7.

Fig. 8(a) illustrates the solutions of the proposed data-baseddistributionally robust stochastic DC OPF for varying riskaversion and Wasserstein radius. Once again, the numericalresults demonstrate fundamental tradeoffs between operationalcost, CVaR values of line constraints violations, and robustnessto sampling errors. The conservativeness of the generatorpolicies are controlled by adjusting ρ and ε. By explicitly usingthe forecast error training dataset and accounting for samplingerrors, risks are systematically assessed and controlled. Sincethe forecast errors are non-Gaussian, existing methods maysignificantly underestimate risk [50]. Increasing ε providesbetter robustness to sampling errors and guarantees out-of-sample performance. Note that CVaR values can be negativewhen the worst-case expected line flow is below the constraintboundary.

Fig. 8(b) illustrates the out-of-sample performance of thedecisions via Monte Carlo simulation. For every value of ρutilized to obtain the results of Fig. 8(a), we i) sampled (new)values from the training dataset, ii) implemented the decisionsbased on the solution of the problem, and iii) calculated theempirical line flows. The dashed gray line indicates the line flowlimit. The number of scenarios for the Monte Carlo simulationsis 1000. From the simulation results, it is seen that larger εensures smaller line constraint violation for all lines exceptline 38. This happens because the risk objective is the sumof all five CVaRs, and a lower overall risk is achieved forcertain values of ρ and ε by allowing higher risk of violatingthe flow limit of line 38. In general, it is possible to prioritizecertain important constraints by weighting their associated riskhigher compared to lower priority constraints. Again, MonteCarlo simulations demonstrate that conservativeness can becontrolled explicitly by changing the Wasserstein radius ε andthe risk aversion parameter ρ.

Fig. 9 illustrates the output powers and the coefficients ofthe reserve polices for selected generators for different valuesof the risk aversion ρ and the Wasserstein radius ε. In orderto satisfy the limit on the line flows, the scheduled poweroutput of some generators (mostly located on the left side ofthe feeder or with cheaper cost profiles) are reduced as therisk aversion parameter ρ increase. As a result, some of thegenerators (mostly located on the right side of the feedersor with high cost profiles) increase the power injection tosupply the demand. With these settings, the risk of line flowconstraint decreases as shown in Fig. 8(b), at the expense ofhigher operational costs.

IV. CONCLUSION

We have illustrated the effectiveness and flexibility of our pro-posed multi-period data-based distributionally robust stochasticOPF methodology. We performed numerical experiments tobalance overvoltages in distribution networks and N−1 securityline flow risks in transmission networks. The flexibility ofcontrollable devices was exploited to balance efficiency andrisk due to high penetration renewable energy sources. Incontrast to existing work, our method directly incorporatesforecast error training datasets rather than making strongassumptions on the forecast error distribution, which allows usto leverage distributionally robust optimization techniques toachieve superior out-of-sample performance. Parameters in theoptimization problems allow system operators to systematicallyselect operating strategies that optimally trade off performanceand risk.

SUPPLEMENTARY MATERIALS

Implementation codes for 1) data-based distributionallyrobust stochastic OPF and 2) data-based distributionally robuststochastic DC OPF can be download from [Link]. The generalproblem formulation and methodologies are presented in [1].

ACKNOWLEDGEMENTS

The authors would like to thank Prof. Jie Zhang and Mr.Mucun Sun, at the Department of Mechanical Engineering,The University of Texas at Dallas, for their support on thesolar forecasting.

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