Data-based distributionally robust stochastic optimal ...tyler.summers/... · distributionally robust optimization, data-driven optimization, reliability and stability, power systems

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2018.2878385, IEEETransactions on Power Systems

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

MethodologiesYi Guo, Student Member, IEEE, Kyri Baker, Member, IEEE, Emiliano Dall’Anese, Member, IEEE,

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

Abstract—We propose a data-based method to solve a multi-stage stochastic optimal power flow (OPF) problem basedon limited information about forecast error distributions. Theframework explicitly combines multi-stage feedback policies withany forecasting method and historical forecast error data. Theobjective is to determine power scheduling policies for control-lable devices in a power network to balance operational costand conditional value-at-risk (CVaR) of device and networkconstraint violations. These decisions include both nominal powerschedules and reserve policies, which specify planned reactionsto forecast errors in order to accommodate fluctuating renewableenergy sources. Instead of assuming the uncertainties acrossthe networks follow prescribed probability distributions, weconsider ambiguity sets of distributions centered around a finitetraining dataset. By utilizing the Wasserstein metric to quantifydifferences between the empirical data-based distribution andthe real unknown data-generating distribution, we formulatea multi-stage distributionally robust OPF problem to computecontrol policies that are robust to both forecast errors andsampling errors inherent in the dataset. Two specific data-baseddistributionally robust stochastic OPF problems are proposed fordistribution networks and transmission systems.

Index Terms—stochastic optimal power flow, multi-perioddistributionally robust optimization, data-driven optimization,reliability and stability, power systems.

I. INTRODUCTION

THE continued integration of renewable energy sources(RESs) in power systems is making it more complicated

for system operators to balance economic efficiency and systemreliability and security. As penetration levels of RESs reachsubstantial fractions of total supplied power, networks will facehigh operational risks under current operational paradigms. Asit becomes more difficult to predict the net load, large forecasterrors can lead to power security and reliability issues causingsignificant damage and costly outages. Future power networks

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,[email protected].

K. Baker is with the Department of Civil, Environmental, and Archi-tectural Engineering, University of Colorado Boulder, Boulder, CO, USA,email:[email protected].

E. Dall’Anese is with the Department of Electrical, Computer, and EnergyEngineering, University of Colorado Boulder, Boulder, CO, USA, email:[email protected].

Z. Hu is with the Department of Electrical Engineering, Tsinghua University,Beijing, China, email: [email protected].

will require more sophisticated methods for managing theserisks, at both transmission and distribution levels.

The flexibility of controllable devices, including power-electronics-interfaced RESs, can be utilized to balance effi-ciency and risk with optimal power flow methods [1]–[9],which aim to determine power schedules for controllabledevices in a power network to optimize an objective function.However, most OPF methods in the research literature andthose widely used in practice are deterministic, assuming pointforecasts of exogenous power injections and ignoring forecasterrors. Increasing forecast errors push the underlying distributedfeedback controllers that must handle the transients caused bythese errors closer to stability limits [10].

More recently, research focus has turned to stochastic androbust optimal power flow methods that explicitly incorporateforecast errors, in order to more systematically trade offeconomic efficiency and risk and to ease the burden onfeedback controllers [11]–[34] Many formulations assumethat uncertain forecast errors follow a prescribed probabilitydistribution (commonly, Gaussian [14], [20], [27], [33]) andutilize analytically tractable reformulations of probabilisticconstraints. However, such assumptions are unjustifiable dueto increasingly complex, nonlinear phenomena in emergingpower networks, and can significantly underestimate risk.

In practice, forecast error probability distributions are neverknown; they are only observed indirectly through finite datasets.Sampling-based methods have been applied with a focus onquantifying the probability of constraint violation [15], [16] andfor constraining or optimizing conditional value at risk (CVaR)[18], [21], [29]. The prediction-realization approach [35], [36]solves an online stochastic optimal power flow problem bya reconciliation algorithm, which ensures feasibility for anyforecast error. Distributionally robust approaches use data toestimate distribution parameters (e.g., mean and variance) andaim to be robust to any data-generating distribution consistentwith these parameters [21], [24], [25], [28], [29], [34]. Otherstake a robust approach, assuming only knowledge of boundson forecast errors and enforcing constraints for any possiblerealization, e.g., [17], [22]. Overall, this line of recent researchhas explored tractable approximations and reformulations ofdifficult stochastic optimal power flow problems. However,none of the existing work explicitly accounts for samplingerrors arising from limited data, which in operation can cause



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poor out-of-sample performance1. Even with sophisticatedrecent stochastic programming techniques, decisions can beoverly dependent on small amounts of relevant data from ahigh-dimensional space, a phenomenon akin to overfitting instatistical models.

We propose a multi-period data-based method to solvea stochastic optimal power flow problem based on limitedinformation about forecast error distributions available throughfinite historical training datasets. A preliminary version of thiswork appeared in [37], and here we significantly expand thework in several directions into the present two part paper. Themain contributions are as follows

1) We formulate a multi-stage distributionally robust optimalcontrol problem for optimal power flow. A distribution-ally robust model predictive control algorithm is thenproposed, which utilizes computationally tractable data-driven distributionally robust optimization techniques[38] to solve the subproblems at each stage. Whereasdistributionally robust optimization approaches focus onsingle-stage problems, here we extend these approachesto multi-stage settings to obtain closed-loop feedbackcontrol policies. This allow us to update forecast errordatasets, and in turn re-compute decisions with thelatest knowledge. In principle, the framework allowsany forecasting methodology and a variety of ambiguityset parameterizations. We focus on Wasserstein balls [39]around an empirical data-based distribution [38], [39],which allows controllable conservativeness by adjustingthe Wasserstein radius. In contrast to previous work, weobtain policies that are explicitly robust to sampling errorsinherent in the dataset. This approach achieves superiorout-of-sample performance guarantees in comparisonto other stochastic optimization approaches, effectivelyregularizing against overfitting the decisions to limitedavailable data.

2) We leverage pertinent linear approximations of the ACpower-flow equations (see, e.g., [40]–[44]) to facilitatethe development of computationally-affordable chance-constrained AC OPF solutions that are robust to dis-tribution mismatches, and provide a unified frameworkthat is applicable to both transmission and distributionsystems. Formulations for distribution networks incor-porate inverted-based RESs and energy storage systems,and focus on addressing the voltage regulation problemunder uncertainty. The transmission system formulationincorporates synchronous generators and power injectionsfrom RESs, and it focuses on probabilistic N − 1security constraints on active transmission line flows. Theframework yields set points and feedback control policiesfor controllable devices that are robust to variations insolar and wind injections and sampling errors inherentto the finite training datasets.

3) The effectiveness and flexibility of the proposed method-ologies are demonstrated with extensive numerical exper-

1Out-of-sample performance is an evaluation of the optimal decisions usinga dataset that is different from the one used to obtain the decision, which canbe tested with Monte Carlo simulation

iments in a 37-node distribution network and in a 118-bus transmission system. These extensive case studiesare presented in Part II [45]. We demonstrate inherenttradeoffs between economic efficiency and robustnessto constraint violations and sampling errors due toforecasting. By explicitly incorporating forecast errorand sampling uncertainties, the methodology can helpnetwork operators to better understand these risks and in-herent tradeoffs, and to design effective optimization andcontrol strategies for appropriately balancing efficiencyobjectives with security requirements.

The rest of the paper is organized as follows: Section IIpresents a general formulation of the proposed data-baseddistributionally robust stochastic OPF problem. Section IIIdescribes the modeling of network, grid-connected components,and network constraints. Section IV specializes the data-basedstochastic OPF for distribution networks with an approximateAC power flow. Section V adapts the proposed methods fortransmission systems with DC power flow. Section VI con-cludes the paper. Extensive simulation results from numericalexperiments and discussions are presented in Part II.

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. STOCHASTIC OPF AS STOCHASTIC OPTIMAL CONTROL

In this section, we formulate a stochastic OPF problem asa distributionally robust stochastic optimal control problem.We first pose the problem generically to highlight the overallapproach, and in subsequent sections we detail the modeland objective and constraint functions more explicitly for bothdistribution networks and transmission systems. This frameworkis more general than most stochastic OPF and distributionallyrobust optimization approaches in the literature, which typicallyfocus only on individual or single-stage optimization problems.

Let xt ∈ Rn denote a state vector at time t that includesthe internal states of all devices in the network. Let ut ∈Rm denote a control input vector that includes inputs for allcontrollable devices in the network. Let ξt ∈ RNξ denote arandom vector in a probability space (Ω,F ,Pt) that includesforecast errors of all uncertainties in the network. Forecast errordistributions are never known in practice, so Pt is assumed to beunknown but belonging to an ambiguity set Pt of distributionswith a known parameterization, which will be discussed indetail shortly. We define the concatenated forecast error over anoperating horizon T as ξ0:T := [ξᵀ0 , . . . , ξ

ᵀT ]ᵀ ∈ RNξT , which

has joint distribution P and corresponding ambiguity set P .Since forecast errors are explicitly included, we seek closed-

loop feedback policies of the form ut = π(x0, . . . , xt, ξ0:t,Dt),



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where the term Dt collects all network and device modelinformation and the parameterization of the ambiguity set ofthe forecast error distribution. The control decisions at time tare allowed to be functions of the entire state and forecast errorhistory up to time t; this is called a history-dependent stateand disturbance feedback information pattern. This generalformulation allows for the design of not only nominal plansfor controllable devices inputs, but also for planned reactionsto forecast errors as they are realized.2 The policy functionπ maps all available information to control actions and is anelement of a set Π of measurable functions.

This leads us to the following multi-stage distributionallyrobust stochastic optimal control problem

infπ∈Π

supP∈P

EPT∑t=0

ht(xt, ut, ξt), (1a)

subject to xt+1 = ft(xt, ut, ξt), (1b)ut = π(x0, . . . , xt, ξt,Dt), (1c)(xt, ut) ∈ Xt. (1d)

The goal is to compute a feedback policy that minimizes theexpected value of an objective function ht : Rn × Rm ×RNξ → R under the worst-case distribution in the forecasterror ambiguity set P . The objective function ht will includeboth operating costs and risks of violating various networkand device constraints and is assumed to be continuous andconvex for every fixed ξt. The system dynamics function ft :Rn ×Rm ×RNξt → Rn models internal dynamics and othertemporal interdependencies of devices, such as state of chargefor batteries and ramp limits of generators. The constraint set Xtincludes network and device constraints, such as power balanceand generator and storage device bounds (some constraintsmay be modeled deterministically and others may be includedas risk terms in the objective function).

The main challenges to solving (1) are the multi-stagefeedback nature of the problem, the infinite dimensionalityof the control policies, the possible nonlinearity of devicedynamics, and how to appropriately parameterize and utilizeour available knowledge about forecast error distributions. Wewill tackle these using a distributionally robust model predictivecontrol scheme with affine feedback policies and linear modelsfor device dynamics, where stage-wise distributionally robuststochastic optimization problems are repeatedly solved overa planning horizon. To incorporate forecast error knowledge,we will use tractable reformulations of ambiguity sets basedexplicitly on empirical training datasets of forecast errors.

A. Ambiguity sets and Wasserstein balls

There is a variety of ways to reformulate the generalstochastic OPF problem (1) to obtain tractable subproblems thatcan be solved by standard convex optimization solvers. Theseinclude assuming specific functional forms for the forecasterror distribution (e.g., Gaussian) and using specific constraintrisk functions, such as those encoding value at risk (i.e., chance

2The reactions can be interpreted as pre-planned secondary frequency controlallocations [17] or contingency reactions in response to forecast errors basedon device dynamics and parameters describing forecast error distributions.

constraints), conditional value at risk (CVaR), distributionalrobustness, and support robustness. In all cases, the out-of-sample performance of the resulting decisions in operationalpractice ultimately relies on 1) the quality of data describingthe forecast errors and 2) the validity of assumptions madeabout probability distributions. Many existing approaches makeeither too strong or too weak assumptions that possible leadto underestimation or overestimation of risk. In this paper, weextend a recently proposed tractable method [38] to a multi-period data-based stochastic OPF, in which the ambiguity setis based on a finite forecast error training dataset ΞNs .

Within the area of distributionally-robust optimization,moment-based ambiguity sets are utilized to model distributionsfeaturing specified moment constraints such as unimodality[25], [46], directional derivatives [47], symmetry [24], and log-concavity [48]. The ambiguity sets can be defined as confidenceregions based on goodness-of-fit tests [49]. Another line ofworks considers ambiguity sets as balls in the probability space,with radii computed based on the Wasserstein metric [50], theKullback-Leilber divergence [51], and the Prohorov metric [52].This paper formulates ambiguity sets by leveraging Wassersteinballs. Relative to other approaches, Wasserstein balls providean upper confidence bound, quantified by Wasserstein radiusε [38], to achieve the superior out-of-sample performance;they also enable power system operators to “control” theconservativeness of the solution, thus ensuring he flexibilityin the power system operation. Additionally, the approach inthis paper seeks the worst-case expectation subjected to alldistributions contained in the ambiguity set. The worst-caseexpectation of the stochastic OPF problem over Wassersteinambiguity set can be reformulated as a finite-dimensionalconvex problem, and can be solved using existing convexoptimization solvers.

The Wasserstein metric defines a distance in the spaceM(Ξ)of all probability distributions Q supported on a set Ξ withEQ[‖ξ‖] =

∫Ξ‖ξ‖Q(dξ) < ∞. In this paper, we assume the

support set is polytopic of the form the uncertainty set is apolytope Ξ := ξ ∈ RNξ : Hξ ≤ d.

Definition [Wasserstein Metric]. Let L be the space of allLipschitz continuous functions f : Ξ → R with Lipschitzconstant less than or equal to 1. The Wasserstein metric dW :M(Ξ)×M(Ξ)→ R is defined ∀Q1,Q2 ∈M(Ξ) as

dW (Q1,Q2) = supf∈L

(∫Ξf(ξ)Q1(dξ)−

∫Ξf(ξ)Q2(dξ)

).

Intuitively, the Wasserstein metric quantifies the minimum“transportation” cost to move mass from one distribution toanother. We can now use the Wasserstein metric to define anambiguity set

PNs :=

Q ∈M(Ξ) : dw(PNs ,Q) ≤ ε

, (2)

which contains all distributions within a Wasserstein ball ofradius ε centered at a uniform empirical distribution PNs onthe training dataset ΞNs . The radius ε can be chosen so thatthe ball contains the true distribution P with a prescribedconfidence level and leads to performance guarantees [38].The radius ε also explicitly controls the conservativeness of



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the resulting decision. Large ε will produce decisions thatrely less on the specific features of the dataset ΞNs and givebetter robustness to sampling errors. This parameterization willbe used in the next subsection to formulate a distributionallyrobust optimization.

B. Data-based distributionally robust model predictive control

The goal of our data-based distributionally robust stochasticOPF framework is to understand and to illustrate inherenttradeoffs between efficiency and risk of constraint violations.Accordingly, the objective function comprises a weighted sumof an operational cost function and a constraint violation riskfunction: ht = J tCost + ρJ tRisk, where ρ ∈ R+ is a weight thatquantifies the network operator’s risk aversion. The operationalcost function is assumed to be linear or convex quadratic

J tCost(xt, ut) := fᵀxxt +1

2xt

ᵀHxxt + fᵀuut +1

2ut

ᵀHuut,

where Hx and Hu are positive semidefinite matrices. Thisfunction can capture several objectives including thermalgeneration costs, ramping costs, and active power losses.

The risk function JRisk associated with the constraint viola-tion comprises a sum of the conditional value-at-risk (CVaR)[53] of a set of N` network and device constraint functions;specifically, it is defined as

J tRisk :=

N∑i=1

CVaRβP[`i(xt, ut, ξt)],

where β ∈ (0, 1] refers to the confidence level of the CVaRunder the distribution P of the random variable ξt. Intuitively,the constraint violation risk function JRisk could be understoodas the sum of networks and devices constraint violationmagnitude at a “risk level” β, which penalizes both frequencyand expected severity of constraint violations [53]. Furtherdetails will be provided in subsequent sections.

Data-based distributionally robust model predictive con-trol for stochastic OPF. The general problem (1) will beapproached with a distributionally robust model predictivecontrol (MPC) algorithm. MPC is a feedback control techniquethat solves a sequence of open-loop optimization problemsover a planning horizon Ht (which in general may be smallerthan the overall horizon T ). At each time t, we solve thedistributionally robust optimization problem over a set Πaffineof affine feedback policies using the Wasserstein ambiguity set(2)

infπ∈Πaffine

supP∈PNs

EPt+Ht∑τ=t

JτCost(xτ , uτ , ξτ ) + ρJτRisk(xτ , uτ , ξτ ),

(3a)subject to xτ+1 = fτ (xτ , uτ , ξτ ), (3b)

uτ = π(x0, . . . , xτ , ξτ ,Dτ ), (3c)(xτ , uτ ) ∈ Xτ . (3d)

Only the immediate control decisions for time t are imple-mented on the controllable device inputs. Then time shiftsforward one step, new forecast errors and states are realized,the optimization problem (3) is re-solved at time t + 1, and

the process repeats. This approach allows any forecastingmethodology to be utilized to predict uncertainties over theplanning horizon. Furthermore, the forecast error dataset PNs ,which defines the center of the ambiguity set PNs , can beupdated online as more forecast error data is obtained. It isalso possible to remove outdated data online to account fortime-varying distributions.

In the rest of the paper, we will derive specific modelsfor both distribution and transmission networks and griddevices where the subproblems (3) have exact tractable convexreformulations as quadratic programs [38] and can be solvedto global optimality with standard solvers.

III. SYSTEM MODEL

We now consider a symmetric and balanced electric powernetwork model in steady state, where all currents and voltagesare assumed to be sinusoidal signals at the same frequency.

A. Network model

Consider a power network (either transmission or distribu-tion), denoted by a graph (N , E), with a set N = 1, 2, ..., Nof buses, and a set E ⊂ N ×N of the power lines connectingbuses. Let V ti ∈ C and Iti ∈ C denote the phasors for theline-to-ground voltage and the current injection at node i ∈ N .Define the complex vectors vt := [V t1 , V

t2 , ..., V

tN ]ᵀ ∈ CN and

it := [It1, ..., ItN ] ∈ CN . Let zij denote the complex impedance

of the line between bus i and bus j, then the line admittance isyij = 1/zij = gij + jbij . We model the lines using a standardPi Model. The admittance matrix Y ∈ CN×N has elements

Yij =

∑l∼i yil + yii if i = j

−yij (i, j) ∈ E0 (i, j) /∈ E

, (4)

where l ∼ i indicates that bus i and bus j are connected. ViaKirchoff’s and Ohm’s laws, we have it = Yvt. Net complexpower bus injections are given by

st = vt (it)∗ = diag(vt) (

Yvt)∗. (5)

The components of st = [St1, St2, . . . , S

tN ]ᵀ ∈ CN can be

expressed in rectangular coordinates as Sti = P ti + jQti, whereP ti is active power and Qti is reactive power. Positive Pi and Qimeans that bus i generates active/reactive power, and negativePi and Qi mean that bus i absorbs the active/reactive power.Vectors of active and reactive power pt = [P t1 , P

t2 , . . . , P

tN ]ᵀ

and qt = [Qt1, Qt2, . . . , Q

tN ]ᵀ are further divided into nominal

and error terms: pt = pt(ut) + pt(ξt) and qt = qt(ut) +qt(ξt). The nominal active and reactive power injection vectorspt(ut) ∈ RN and qt(ut) ∈ RN depend on control decisions,and the forecast errors pt(ξt) ∈ RN and qt(ξt) ∈ RN dependon the random vector ξt.

To handle nonconvexity of the power flow equations (5), weutilize two different linearization methods that are effective inboth distribution and transmission networks.



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B. Dynamic of grid-connected devices

We consider Nd grid-connected devices, which may include1) traditional generators and inverter-based RESs; 2) fixed,deferrable, and curtailable loads; 3) storage devices likebatteries and plug-in electric vehicles, which are able to actas both generators and loads. There are two types of devices:devices with controllable power flow affected by decisionvariables (e.g., conventional thermal and RESs generators,deferrable/curtailable loads and storage devices); and deviceswith fixed or uncertain power flow which will not be affectedby decision variables (e.g., fixed loads). The power flow ofeach controllable device is modeled by a discrete-time lineardynamical system

xdt+1 = Adxdt + Bdudt ,

where device d at time t has internal state xdt ∈ Rnd , dynamicsmatrix Ad ∈ Rnd×nd , input matrix Bd ∈ Rnd×md , and controlinput udt ∈ Rmd . The first element of xdt corresponds to thepower injection of device d at time t into the network at its bus,and other elements describe internal dynamics, such as state-of-charge (SOC) of energy storage devices. At time t, state andinput histories are expressed by xdt := [(xd1)ᵀ, . . . , (xdt )

ᵀ]ᵀ ∈Rndt and udt := [(ud0)ᵀ, . . . , (udt−1)ᵀ]ᵀ ∈ Rmdt with

xdt = Adtxd0 +Bdt u

dt ,

where

Adt :=

Ad

(Ad)2

...(Ad)t

, Bdt :=

Bd 0 . . . 0

AdBd Bd. . . 0

.... . . . . .

...(Ad)t−1Bd . . . AdBd Bd

.

C. Network constraints

The AC power flow equations render prototypical ACOPF formulation nonconvex and NP-hard; what is more,in the present context, they hinder the development ofcomputationally-affordable chance-constrained AC OPF formu-lations where CVaR arguments are leveraged as risk measures.

For distribution systems, we refer the reader to the linearapproximation methods proposed in e.g., [40]–[44], [54], withthe latter suitable for multi-phase systems with both wyeand delta connections; these approximate models have beenshown to provide high levels of accuracy in many existingtest systems. For transmission systems, one can consider thetradition DC power-flow model to approximate the voltageangles and active power flows in the system; see, e.g., [17],[21], [55], [56]. Alternatively, one can consider alternativelinearizations; see e.g., [54]. As long as an accurate linearmodel exists, the proposed technical approach can be utilized toformulate and solve a distributionally-robust chance-constrainedAC OPF problem. In particular, linear models can be utilizedto formulate convex (in fact, linear) constraints on line flowsand voltage magnitudes, e.g.

V min ≤ |Vi| ≤ V max, ∀i ∈ N . (6)

Grid-connected devices have various local constraints includ-ing, e.g., state of charge limitations for energy storage devices,

allowable power injection ranges, generator ramping limits, andother device limits. These can be modeled (or approximated)as linear inequalities of the form

Ttdx

dt + Ut

dudt + Ztdξt ≤ wd, d = 1, . . . , Nd, (7)

where Ttd ∈ Rld×ndt, Ut

d ∈ Rld×mdt, and Ztd ∈ Rld×Nξt,and wd ∈ Rld is a local constraint parameter vector.

IV. DATA-BASED DISTRIBUTIONALLY ROBUSTSTOCHASTIC OPF FOR DISTRIBUTION NETWORKS

A. Distribution network model

In this section, we specialize the model to symmetric andbalanced power distribution networks, connected to the gridat a point of common coupling (PCC). Loads and distributedgenerators (e.g., thermal generators, inverter-based RESs, andenergy storage devices) may be connected to each bus. Weaugment the bus set with node 0 as the PCC.

The voltage and injected current at each bus are definedas V tn = |V tn|ej∠V

tn , and Itn = |Itn|ej∠I

tn . The absolute values

|V tn| and |Itn| correspond to the signal root-mean-square values,and the phase ∠V tn and phase ∠Itn correspond to the phase ofthe signal with respect to a global reference.

Node 0 is modeled as a slack bus and the others are PQbuses, in which the injected complex power are specified. Theadmittance matrix can be partitioned as[

It0it

]=

[y00 yᵀ

y Y

] [V0

vt

].

The net complex power injection is then

st = diag(vt)(Y∗(vt)∗ + y∗(vt0)∗

). (8)

The nonconvexity of this equation in the space of power injec-tions and bus voltages is a source of significant computationaldifficulty in optimal power flow problems. In the rest of thissection, we formulate a convex and computationally efficientdata-based stochastic OPF problem based on a particularlinear approximation of (8) that is appropriate for distributionsnetworks. This approximation occurs on a specific point of apower flow manifold that preserves network structure for bothreal and reactive power flows and allows direct application ofstochastic optimization techniques for incorporating forecasterrors.

B. Leveraging approximate power flow

Collect the voltage magnitudes |V tn|n∈N into the vector|vt| := [|V t1 |, . . . , |V tN |]ᵀ ∈ RN . To develop computationally-feasible approaches, the technical approach in this paper lever-ages linear approximations of the AC power-flow equations;in particular, linear approximations for voltages, as a functionof the injected power st, are given by

vt ≈(

Hpt + Jqt + c

), |vt| ≈

(Mpt + Nqt + a

). (9)

Using these approximations, the voltage constraints V min ≤|V tn| ≤ V max can be approximated as V min1N Mpt +Nqt+a V max1N . The coefficient matrices of the linearized



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voltages, and the normalized vectors a and c can be obtainedas shown in [42]–[44], [54]. For completeness, in the remainderof this subsection we briefly outline the approach taken in [42],[54] to derive a linear model for the voltages.

Suppose that vt = v + ∆vt, where v = |v|∠θ ∈ CN is apre-determined nominal voltage vector and ∆vt ∈ CN denotesa deviation from nominal. Then we have

st = diag(v + ∆vt)

(Y∗(v + ∆vt)∗ + y∗V ∗0

). (10)

Neglecting second-order terms diag(∆vt)Y∗(∆vt)∗, the powerbalance (8) becomes Λ∆vt + Φ(∆vt)∗ = st + Ψ, whereΛ := diag(Y∗v∗ + y∗V ∗0 ), Φ := diag(v)Y∗, Ψ :=−diag(v)(Y∗v∗ + y∗V ∗0 ). We consider a choice of the nom-inal voltage v = Y−1yV0, which gives Λ = 0N×N andΨ = 0N . Therefore the linearized power flow expression isst = diag(v)Y∗(∆vt)∗, the deviation ∆vt becomes ∆vt =Y−1diag−1(v∗)(st)∗.

Let us denote Y−1 = (G + jB)−1 = ZR + jZI . Thenexpanding ∆vt in rectangular form gives

M =

(ZR diag

(cos(θ)|v|

)− ZI diag

(sin(θ)

|v|

)),

N =

(ZI diag

(cos(θ)|v|

)+ ZR diag

(sin(θ)

|v|

)),

which define the rectangular matrices H := M + jN, J :=N− jM, and the coefficient c is v. If v dominates ∆vt, thenthe voltage magnitudes are approximated by |v|+<∆vt, andlinearized coefficients of voltage magnitudes become M := M,N := N, and a := |v|. It is worth noting that the approachproposed in [44] accounts for multiphase systems with both wyeand delta connections. Accordingly, the proposed frameworkis applicable to generic multiphase feeders with both wye anddelta connections.

C. Data-based stochastic OPF formulation for distributionnetworks

Using the introduced linearized relationship between voltageand power injection vectors pt and qt, we express the voltagemagnitude in the following form

gt[pt(ut, ξt),q

t(ut, ξt)]

:= M(I−diagαt)ptav+Nqt+|v|,

where αt ∈ RN is a control policy defined as the fraction ofthe active power curtailment by the renewable energy powerinjection. A system state vector ptav ∈ RN collects the activepower injection including loads and the available RES power.We aim to optimize the set points αt,qt of nodal powerinjections, which can be achieved by adjusting controllableloads and generators. More details of system modeling andcomponent dynamics will be introduced in Part II.

Broadly speaking, we quantify a violation risk of voltagemagnitude constraints (6) and local device constraints (7) foreach node and each time as follows

E Rgtn[pt(ut, ξt),q

t(ut, ξt)]− V max

≤ 0, (11a)

E RV min − gtn

[pt(ut, ξt),q

t(ut, ξt)]≤ 0, (11b)

E R

Ttdx

dt + Ut

dudt + Ztdξt − wd

≤ 0, d = 1, . . . , Nd,

(11c)

where gtn( · ) is the n-th element of the function value gt( · ),and R denotes a generic transformation of the inequalityconstraints into stochastic versions. Using a prior on theuncertainty and possibly introducing auxiliary variables, thegeneral risk functions (11a)-(11c) can be reformulated usinge.g., scenario-based approaches, [30], [31] or moment-baseddistributionally robust optimization [21], [25]. This paper seeksa reformulation by leveraging the CVaR [53]. A set of con-straints will be approximated using the proposed distributionallyrobust approach, while other constraints will be evaluated usingsample average methods.

We define a set Vt that contains a subset of N` affineconstraints (11a)-(11c) that will be treated with distributionallyrobust optimization techniques. This allows some or all ofthe constraints to be included. We express them in terms ofa decision variable vector yt and uncertain parameters ξt,where yt consists of all the decision variables including theRES curtailment ratio vector ατ and other controllable deviceset-points, and ξt contains the uncertain parameters acrossthe network including the active and reactive power injectionforecast errors. For simplicity, we consider the risk of eachconstraint individually; it is possible to consider risk of jointconstraint violations, but this is more difficult and we leave itfor future work. Each individual affine constraint in the set Vtcan be written in a compact form as follows

Cto(yt, ξt) = [A(yt)]oξt + [B(yt)]o, o = 1, ..., N`,

where Cto(·) is the o-th affine constraint in the set Vt. We use[ · ]o to denote the o-th element of a vector or o-th row of amatrix. The CVaR with risk level β of the each individualconstraint in the set Vt is

infκtoEξt

[Cto(yt, ξt) + κto]+ − κtoβ

≤ 0, (12)

where κto ∈ R is an auxiliary variable [53]. The expressioninside the expectation in (12) can be expressed in the form

Qto = maxk=1,2

[〈aok(yt), ξt〉+ bok(κto)

]. (13)

This expression is convex in yt for each fixed ξt since it is themaximum of two affine functions. Our risk objective function isexpressed by the distributionally robust optimization of CVaR

J tRisk =

t+Ht∑τ=t

N∑o=1

supQτ∈PNsτ

EQτ maxk=1,2

[〈aok(yτ ), ξτ 〉+ bok(κτo)

].

The above multi-period distributionally robust optimization canbe equivalently reformulated the following quadratic program,the details of which are described in [38]. The objective is tominimize a weighted sum of an operational cost function andthe total worst-case CVaR of the affine constraints in set Vt(e.g., voltage magnitude and local device constraints).Data-based distributionally robust stochastic OPF

infyτ ,κτo

t+Ht∑τ=t

E[JτCost] + ρ sup

Qτ∈PNsτ

N∑o=1

EQτ [Qτo ]

,



7

= infyτ ,κ

τo ,

λτo ,sτio,ς

τiko

t+Ht∑τ=t

E[J tCost] +

N∑o=1

(λoετ +

1

Ns

Ns∑i=1

sτio

),

(14a)subject to

ρ(bok(κτo) + 〈aok(yτ ), ξiτ 〉+ 〈ςiko,d−Hξiτ 〉) ≤ sτio, (14b)‖Hᵀςiko − ρaok(yτ )‖∞ ≤ λτo , (14c)ςiko ≥ 0, (14d)∀i ≤ Ns,∀o ≤ N`, k = 1, 2, τ = t, ..., t+Ht,

where ρ ∈ R+ quantifies the power system operators’ riskaversion. This is a quadratic program that explicitly uses thetraining dataset ΞNsτ = ξiτi≤Ns . The risk aversion parameterρ and the Wasserstein radius ετ allow us to explicitly balancetradeoffs between efficiency, risk and sampling errors inherentin ΞNsτ . The uncertainty set is modeled as a polytope Ξ :=ξ ∈ RNξ : Hξ ≤ d. The constraint ςiko > 0 holds since theuncertainty set is not-empty; on the other hand, in a case withno uncertainty (i.e, ςiko = 0), the variable λ does not play anyrole and sτio = ρ(bok(κτo) + 〈aok(yτ ), ξiτ 〉).

V. DATA-BASED DISTRIBUTIONALLY ROBUST STOCHASTICOPTIMAL POWER FLOW FOR TRANSMISSION SYSTEMS

A. Illustrative explanation for the DC approximation

In transmission networks, we employ a widely used “DC”linearization of the nonlinear AC power flow equations thatmakes the following assumptions [17], [21]:• The lines (i, j) ∈ E are lossless and characterized by their

reactance =1/yik;• The voltage magnitudes |Vn|n∈N are constant and

close to one per unit;• Reactive power is ignored.

Under a DC model, line flows are linearly related to the nodalpower injections; therefore, flow constraints in the transmissionlines can be expressed as

Nd∑d=1

Γtd(rtd +Gtdξt + Ctdx

dt ) ≤ pt, (15)

where Γtd ∈ R2Lt×t maps the power injection (or consump-tions) et each node to line flows and can be constructed fromthe network line impedances [55], [57]; on the other hand,pt ∈ R2Lt denotes a limit on the line flows. The powerinjections in (15) features controllable and non-controllablecomponents; the non-controllable power for a device/node dis modeled as rtd + Gtdξt (with positive values denoting thenet power injection in the network), where the vector rtd ∈ Rt

denotes the nominal power injection and Gtdξt models uncertaininjections. Finally, xdt is the vector of controllable powers, andthe matrix Ctd ∈ Rt×ndt selects the first element of xdt at eachtime, i.e., Ctd := It ⊗ Ctd, where Ctd = [1,01×(nd−1)], It is at-dimensional identity matrix and ⊗ denotes the Kroneckerproduct operator. The power balance constraint is

Nd∑d=1

(rtd +Gtdξt + Ctdxdt ) = 0. (16)

B. Reserve policies

Deterministic OPF formulations ignore the uncertainties ξtand compute an open-loop input sequence for each device. Ina stochastic setting, one must optimize over causal policies,which specifies how each device should respond to the currentsystem states and forecast errors as they are discovered. We cannow formulate a finite horizon stochastic optimization problem

infπ∈Πaffine

t+Ht∑τ=t

E[JτCost(xτ , πτ , ξτ )

], (17a)

subject toNd∑d=1

[rtd +Gtdξt + C tdx

dt

][t,t ]

= 0, (17b)

E R Nd∑d=1

[Γtd(r

td +Gtdξt + C tdx

dt )− pt

][2Lt,2Lt ]

≤ 0,

(17c)

E R[Tτdx

dτ + Uτ

dudτ + Zτdξτ − wd

]≤ 0, (17d)

d = 1, . . . , Nd, τ = t, . . . , t+Ht,

where R denotes a general constraint risk function. The defini-tion and discussion of the general stochastic transformation of(17c)-(17d) is same as (11a)-(11c), and can be found in SectionIV.C. Here, [t, t] denotes the finite time interval [t, t + Ht]for brevity. The cost function is proportional to the first andsecond moments of the uncertainties ξt, since we assume theoperational cost function is convex quadratic. Optimizing overgeneral policies makes the problem (17) infinite dimensional,so we restrict attention to affine policies

udt = Ddt ξt + edt ,

where each participant device d (e.g., traditional generators,flexible loads or energy storage devices) power schedule udt ∈Rt is parameterized by a nominal schedule edt ∈ Rt plus alinear function Dd

t ∈ Rt×Nξt of prediction error realizations.To obtain causal policies, Dd

t must be lower-triangular. TheDdt matrices can be interpreted as planned reserve mechanisms

involving secondary frequency feedback controllers, whichadjust conventional generator power outputs using frequencydeviations to follow power mismatches [17]. Under affinepolicies, the power balance constraints are linear functions ofthe distribution of ξt, which are equivalent toNd∑d=1

(rtd + Ctd(Adtxd0 +Bdt e

dt )) = 0,

Nd∑d=1

(Gtd + CtdBdtD

dt ) = 0.

C. Data-based stochastic OPF formulation for transmissionsystems

We now use the above developments to formulate a data-based distributionally robust OPF problem to balance anoperating efficiency metric with CVaR values of line flow andlocal device constraint violations. We collect the line flow andlocal device constraints into a set Vt, which will be evaluatedwith distributionally robust optimization techniques in a totalamount of N`. This allows some or all of the constraints to



8

be included. For simplicity, we again consider the risk of eachconstraint individually. We express the constraints in terms ofdecision variables Dd

t , edt and uncertain parameters ξt. Each

individual affine constraint in the set Vt can be written in acompact form as follows

Cto(Dt, et, ξt) = [A(Dt)]oξt + [B(et)]o, o = 1, ..., N`,

where Cto(·) is the o-th affine constraint in the set Vt. Thedecision variables Dt and et both appear linearly in Cto. TheCVaR with risk level β of the each individual constraint in theset Vt is

infσtoEξt

[Cto(Dt, et, ξt) + σto]+ − σtoβ

≤ 0, (18)

where σto ∈ R is an auxiliary variable [53]. The expressioninside the expectation (18) is expressed in the form

Qto = maxk=1,2

[〈aok(yt), ξt〉+ bok(σto)

], (19)

where the decision variable vector y consists of optimizationvariables D, e, σ. The convex expression (19) is maximumof two affine functions, and we consider the following distri-butionally robust total CVaR objective

J tRisk =

t+Ht∑τ=t

N∑o=1

supQτ∈PNsτ

EQτ maxk=1,2

[〈aok(yτ ), ξτ 〉+ bokσ

τo )

].

The above multi-period distributionally robust optimization canbe equivalently reformulated as a quadratic program. The detailsof the linear reformulation are shown in [38]. The objective isto minimize a weighted sum of an operational cost functionand the total worst-case CVaR of the affine constraints in theset Vt (e.g., line flow constraints and local device constraints).Data-based distributionally robust stochastic DC OPF

infyτ ,στo

t+Ht∑τ=t

E[JτCost] + ρ sup

Qτ∈PN`τ

N∑o=1

EQτ [Qτo ]

,

= infyτ ,σ

τo ,

λτo ,sτio,ς

τiko

t+Ht∑τ=t

E[JτCost] +

N∑o=1

(λτoετ +

1

Ns

Ns∑i=1

sτio

),

(20a)subject toNd∑d=1

[(rtd + C td(A

dtxd0 +Bdt e

dt ))]

[t,t]= 0, (20b)

Nd∑d=1

[(Gtd + C tdB

dtD

dt )]

[t,t]= 0, (20c)

ρ(bok(στo ) + 〈aok(yτ ), ξiτ 〉+ 〈ςτiko,d−Hξiτ 〉) ≤ sτio, (20d)‖Hᵀςτiko − ρaok(yτ )‖∞ ≤ λτo , (20e)ςτiko ≥ 0, (20f)∀i ≤ Ns,∀o ≤ N`, k = 1, 2, τ = t, . . . ,Ht,

where ρ ∈ R+ quantifies the power system operators’ riskaversion. Once again, this is a quadratic program that explicitlyuses the training dataset ΞNsτ := ξiτi≤Ns , and the riskaversion parameter ρ and the Wasserstein radius ετ allow

us to explicitly balance tradeoffs between efficiency, risk,and sampling errors inherent in ΞNsτ . The uncertainty setis modeled as a polytope Ξ := ξ ∈ RNξ : Hξ ≤ d.The constraint ςiko > 0 holds since the uncertainty set isnot-empty; on the other hand, in a case with no uncertainty(i.e, ςiko = 0), the variable λ does not play any role andsτio = ρ(bok(στo ) + 〈aok(yτ ), ξiτ 〉).

VI. CONCLUSIONS

In this paper, we have proposed a framework for the data-based distributionally robust stochastic OPF based on finitedataset descriptions of forecast error distributions across thepower network. The method allows efficient computation ofmulti-stage feedback control policies that react to forecasterrors and provide robustness to inherent sampling errors inthe finite datasets. Tractability is obtained by exploiting convexreformulations of ambiguity sets based on Wasserstein ballscentered on empirical distributions. The general framework isadapted to both distribution networks and transmission systemsby allowing general device models and utilizing differentlinearizations of the AC power flow equations. The effectivenessand flexibility of our proposed method is demonstrated ina companion paper, Part II [45], which uses the method tohandle a over-voltages in a distribution network with high solarpenetration and to address line flow risks in a transmissionsystem with high wind penetration. These numerical results alsoshow that the method has superior out-of-sample performanceand allows a more principled and systematic tradeoff ofefficiency and risk.

SUPPLEMENTARY MATERIALS

Implementation codes for 1) data-based distributionallyrobust stochastic OPF and 2) data-based distributionally robuststochastic DC OPF can be download from https://github.com/TSummersLab/Distributionally-robust-stochastic-OPF. Numer-ical case studies (e.g., overvoltage in distribution networksand N -1 security in transmission systems), and discussions onsimulation results are presented in [45].

APPENDIX

Pertinent reformulations of the probabilistic line flow con-straints in transmission systems are described in this section.Consider, for simplicity, the time period between [0, t] andrewrite the line flow constraint as

E R Nd∑d=1

Γtd(rtd +Gtdξt + Ctdx

dt )− pt

≤ 0,

where xdt = Adtxd0 +Bdt u

dt and the affine reserve policies are

udt = Ddt ξt + edt . The CVaR counterpart of each individual

constraints is given by [cf. (18)]

infσtoEξtmax

[ Nd∑d=1

Γtd(rtd +Gtdξt + Ctd(A

dtxd0 +BdtD

dt ξt

+Bdt edt ))− pt

]o

+ σto

− σtoβ ≤ 0.

(21)

https://github.com/TSummersLab/Distributionally-robust-stochastic-OPF

https://github.com/TSummersLab/Distributionally-robust-stochastic-OPF



9

Since the decision variables Dt, et, σto enter as linear terms

in (21), a compact expression similar to (19) can be obtained;i.e.,

Qto = maxk=1,2

[〈aok(yt), ξt〉+ bok(σto)

],

where

k = 1,

ao1(yt) =[[∑Nd

d=1 ΓtdCtdB

dtD

dt

]o,[∑Nd

d=1 ΓtdCtdB

tdedt

]o

],

bo1(σto) =[− p+

Nd∑d=1

Γtd(rtd+Gtdξ+CtdA

dtxd0)]o

+σto−σtoβ,

k = 2,

ao2(yt) =[0ᵀNξt

, 0],bo2(σto) = −σtoβ,

for all o. Similar steps can be followed to obtain the constraintsfor each device and the voltage magnitude.

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Yi Guo (S’13) received the B.S. degree in electricalengineering from Xi’an JiaoTong University, Xi’an,China, in 2013. He earned his M.S. degree in elec-trical engineering from the University of Michigan,Dearborn, MI, USA. He is pursuing his Ph.D. degreein Mechanical Engineering at The University ofTexas at Dallas, Richardson, TX, USA. His researchinterests are in control and optimization in networks,with applications to electric power networks.

Kyri Baker (S’08-M’15) received her B.S., M.S,and Ph.D. in Electrical and Computer Engineeringat Carnegie Mellon University in 2009, 2010, and2014, respectively. Since Fall 2017, she has beenan Assistant Professor at the University of Colorado,Boulder, in the Department of Civil, Environmental,and Architectural Engineering, with a courtesy ap-pointment in the Department of Electrical, Computer,and Energy Engineering. Previously, she was aResearch Engineer at the National Renewable EnergyLaboratory in Golden, CO. Her research interests

include power system optimization and planning, building-to-grid integration,smart grid technologies, and renewable energy.

Emiliano Dall’Anese (S’08-M’11) received theLaurea Triennale (B.Sc Degree) and the LaureaSpecialistica (M.Sc Degree) in TelecommunicationsEngineering from the University of Padova, Italy,in 2005 and 2007, respectively, and the Ph.D. inInformation Engineering from the Department ofInformation Engineering, University of Padova, Italy,in 2011. From January 2009 to September 2010, hewas a visiting scholar at the Department of Electricaland Computer Engineering, University of Minnesota,USA. From January 2011 to November 2014 he

was a Postdoctoral Associate at the Department of Electrical and ComputerEngineering and Digital Technology Center of the University of Minnesota,USA. From December 2014 to August 2018 he was a Senior Researcher atthe National Renewable Energy Laboratory, Golden, CO, USA. Since August2018 he has been an Assistant Professor within the Department of Electrical,Computer, and Energy Engineering at the University of Colorado Boulder.

His research interests lie in the areas of optimization, signal processing, andcontrol. Application domains include cyber-physical systems and networkedsystems, with specific current efforts on power systems, wind farms, andcollaborative computing.

Zechun Hu (M’09-SM’17) received the B.S. andPh.D. degrees in electrical engineering from Xi’anJiaoTong University, Xi’an, China, in 2000 and 2006,respectively. He worked in Shanghai JiaoTong Univer-sity after graduation and also worked in Universityof Bath as a research officer from 2009 to 2010.He joined the Department of Electrical Engineeringat Tsinghua University in 2010 where he is nowan associate professor. His major research interestsinclude optimal planning and operation of powersystems, smart grid, electric vehicles and energy

storage systems.

Tyler Holt Summers is an Assistant Professor ofMechanical Engineering with an affiliate appointmentin Electrical Engineering at the University of Texasat Dallas. Prior to joining UT Dallas, he was anETH Postdoctoral Fellow at the Automatic ControlLaboratory at ETH Zurich from 2011 to 2015. Hereceived a B.S. degree in Mechanical Engineeringfrom Texas Christian University in 2004 and anM.S. and PhD degree in Aerospace Engineering withemphasis on feedback control theory at the Universityof Texas at Austin in 2007 and 2010, respectively.

He was a Fulbright Postgraduate Scholar at the Australian National Universityin Canberra, Australia in 2007-2008. His research interests are in feedbackcontrol and optimization in complex dynamical networks, with applications toelectric power networks and distributed robotics.

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Data-based distributionally robust stochastic optimal ...tyler.summers/... · distributionally robust optimization, data-driven optimization, reliability and stability, power systems