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Security Constrained Optimal Power Flowwith Distributionally Robust Chance Constraints

Line Roald,Student Member, IEEE,Frauke Oldewurtel,Member, IEEE,Bart Van Parys,Student Member, IEEE,and Göran Andersson,Fellow, IEEE

Abstract—The growing amount of fluctuating renewable in-feeds and market liberalization increases uncertainty in powersystem operation. To capture the influence of fluctuations inoperational planning, we model the forecast errors of the un-certain in-feeds as random variables and formulate a securityconstrained optimal power flow using chance constraints. Thechance constraints limit the probability of violations of technicalconstraints, such as generation and transmission limits, butrequire a tractable reformulation. In this paper, we discussdifferent analytical reformulations of the chance constraints,based on a given set of assumptions concerning the forecasterror distributions. In particular, we discuss reformulat ions thatdo not assume a normal distribution, and admit an analyticalreformulation given only a mean vector and covariance matrix.We illustrate our method with a case study of the IEEE 118bus system, based on real data from the European system. Thedifferent reformulations are compared in terms of both achievedempirical violation probability and operational cost, which allowsus to provide a suggestion for the most appropriate reformulationin an optimal power flow setting. For a large number ofuncertainty sources, it is observed that the distributionsof theline flows and generator outputs can be close to normal, eventhough the power injections are not normally distributed.

Index Terms—Renewable integration, Chance ConstrainedOptimal Power Flow, N-1 security

I. I NTRODUCTION

A fundamental tool in power system analysis is theoptimalpower flow(OPF) [1]. Several tasks central to power systemoperation, such as unit commitment, reserve procurement,market clearing and security assessment rely on the solution ofan OPF. The main goal of the OPF is to minimize operationalcost, while ensuring secure operation that respects technicallimits of the power system. In current operational schemes,the system is considered secure if it remains within theoperational limits during normal operation and duringoutageof any single component. This principle is referred to as theN−1 criterion, and is reflected in the OPF through additionalconstraints, leading to asecurity constrained optimal powerflow (SCOPF). While theN − 1 criterion secures the systemagainst individual outages,forecast uncertaintyis another kindof disturbance affecting the system. Forecast uncertaintyarisesfrom unforeseen fluctuations in the power injections, such asinaccurate predictions of load or renewable in-feeds, as wellas from short-term electricity trading. While load profilesare

L. Roald, F. Oldewurtel and G. Andersson are with the Power Systems Lab-oratory at the Department of Electrical Engineering, ETH Zurich, Switzerland.Email: {roald | oldewurtel| andersson}@eeh.ee.ethz.ch. B. Van Parys is withthe Automatic Control Laboratory at the Department of Electrical Engineering,ETH Zurich, Switzerland. Email: bartvan@control.ee.ethz.ch

relatively predictable, higher shares of electricity productionfrom renewable sources and liberalization of energy markets(particularly in Europe) have increased the forecast uncertaintyby orders of magnitude [2]. In current operational planning,uncertainty is usually ignored and uncertain quantities aretypically replaced by a forecast value. While this approachhasprovided good solutions in the past, the increased levels ofun-certainty lead to frequent N-1 violations in real-time operation.To mitigate these problems, it is proposed to explicitly accountfor uncertainty during operational planning, in particular whilesolving the OPF.

There are different approaches to account for uncertaintywithin the OPF. Robust and worst-case methods, e.g. [3], en-sure secure operations for all possible forecast errors, but oftenprovide very conservative and thus costly solutions. Stochasticprogramming methods give the operator more freedom totrade-off cost and security. One example is two stage stochasticprogramming for unit commitment and reserve scheduling,e.g. [4], which minimizes the expected cost of operationsbased on a set of scenarios. Another example is chance con-strained programming, which explicitly limits the probabilityof constraint violations [5], [6], [7]. Since the main goal inshort-term operational planning is to ensure secure operations,we consider the latter method, and formulate the OPF as aprobabilistic SCOPF(pSCOPF) with chance constraints. Theacceptable violation probability, which is treated as a designparameter in the optimization problem, allows the operatortochoose an appropriate trade-off between cost and security ofoperations.

Although the pSCOPF allow us to account for uncertaintyin a comprehensive way, it is generally hard to reformu-late chance constraints as tractable constraints. Two mainapproaches for reformulation have been applied to the OPFproblem, based either on sampling or analytical reformulation.In [5], the SCOPF is formulated as ajoint chance constrainedproblem (limiting the probability that any of the constraints areviolated), which is reformulated using the scenario approachbased on [8]. The formulation was extended to include marketclearing with co-optimization of energy and reserves in [9],where a different sampling based reformulation based on[10] was used. Both sampling based reformulations requireno knowledge about the underlying distribution, except foravailability of a given number of samples (which increaseswith the problem size).

In contrast, the SCOPF formulated in [7] usesseparatechance constraints (limiting the probability for each constraintseparately) to formulate the pSCOPF. Assuming that the

http://arxiv.org/abs/1508.06061v1

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random variables follow a Gaussian distribution, an exactanalytical reformulation is obtained. The same type of Gaus-sian reformulation is performed for an OPF without securityconstraints in [6].

While the assumption of a Gaussian distribution limits theapplicability of the analytical reformulation from [7], [6], theanalytical reformulation has some attractive properties.First,it is scalable to a large number of random variables, as thenumber of random variables does not influence the problemsize or complexity of the OPF itself. Second, the solutionis more transparent than a sample based solution since itis possible to trace the influence of each random variablethrough the analytical relations. Finally, the solution based onthe analytical reformulation is deterministic, i.e., the OPF willalways find the same optimal solution with the same optimalcost. While this might seem trivial, the OPF solution basedon the scenario approach is actually random, since it dependson the choice of the samples. The same problem might thuslead to different solutions with different costs, depending onwhich samples were chosen.

This paper investigates how the good qualities of the an-alytical reformulation can be preserved, while moving awayfrom the limiting assumption of a Gaussian distribution. Usingoptimal probability inequalities, we obtaindistributionallyrobustreformulations of the chance constraints. This approachis well-known in operations research and control theory andhas been investigated in, e.g., [11], [12], [13]. In [14], theapplication of some distributionally robust reformulations tothe optimal power flow problem were also discussed. Here,we introduce reformulations based on assumptions like uni-modality and symmetry of the forecast errors, and explain whythose are relevant in the optimal power flow context. We aim toprovide recommendations for the most suitable reformulations,depending on the sources of uncertainty (e.g., RES fluctua-tions, load variations, short term trading) and the time frame(e.g., day-ahead planning, real-time operation). To compare thedifferent analytical reformulations, we introduce the concept ofan uncertainty margin. The uncertainty margin has a physicalinterpretation as a security margin against forecast errors, andrepresents a reduction of available transmission and generationcapacity. A larger uncertainty margin thus increases security,but also the operational cost. The empirical performance ofthe proposed reformulations is assessed through a case studyfor the IEEE 118 bus system, with uncertainties representedthrough historical forecast errors from the Austrian PowerGrid. We investigate which reformulation is the most appro-priate for a chance constrained SCOPF problem, consideringempirical violation probability, nominal operational cost andthe accuracy of the distributional assumptions.

The remainder of this paper is organized as follows: SectionII and III present the uncertainty modeling and the formulationof the chance constrained SCOPF. Section IV discusses differ-ent analytical reformulations for these chance constraints, andSection V applies them to the power flow equations. Section Vdemonstrates the proposed formulation in a case study for theIEEE 118 bus system. Section VI summarizes and concludesthe paper.

II. M ODELING FORECAST UNCERTAINTY

Forecast errors arise from uncontrolled power in-feeds thatdeviate from their forecasted values, caused by, e.g., fluctua-tions in load or renewable production. The characteristicsofforecast errorsδR differ between systems, depending on theirgeneration mix, load characteristics and market structure, andalso depend on parameters such as the time of the day, or theforecast horizon. For example, the minute-to-minute variationin wind in-feeds in the central European system follows aStudent t-distribution [15], while the day-ahead distributionof wind forecast depends on the forecasted wind power,and is typically non-symmetric. The uncertain in-feeds canbe modeled as random variables with continuous probabilitydistributions. We define the vector ofn uncertain in-feeds as

P̃R = PR + δR . (1)

Here, the uncertain in-feed̃PR is the sum of the forecastedvalue PR and a random deviationδR. Since the sources ofuncertainty differ both within and between systems, the fulldistribution of δR is generally not known. However, we willassume some partial information about the distribution. Inparticular, we assume that the meanµR ∈ Rn and covarianceΣR ∈ R

n×n of the forecast errors exist, and can be estimatedeither based on historical data or through forecasting methods.We allow for non-zero mean, since forecasts are not necessar-ily based on the expectation ofPR, but rather on the mostprobable realization (which are not the same, e.g., for skeweddistributions).

III. O PTIMAL POWER FLOW FORMULATION

We now introduce the mathematical formulation of thepSCOPF for a system withn buses, based on the formulationin [7]. The setsG, D and R represent the conventionalgenerators, the fixed loads and the uncertain in-feeds (con-sisting of, e.g., in-feeds from wind and solar power plants),respectively. To simplify notation, we assume that there isone generator, load and generator connected at each bus, suchthat |G| = |D| = |R| = n. This assumption is however notnecessary for the method itself. The set of transmission linesis denoted byL, and there are|L| = nL lines in the system.The contingencies considered for the N-1 security criterioninclude outage of any line or generator, in totalnC = nL +noutages.

A. Generator modeling

The nominal generation output of the generators,PG ∈ Rn

are the optimization variables of the problem. In addition tokeeping the system balanced in nominal operation conditions,any power deviation arising from either forecast errors orgeneration outages must be balanced by the generators. Thecontribution of balancing energy from each generator can bechosen in different ways. Here, we assume that each generatorcontributes according to its maximum nominal output, similarto [16]. When all generators operate, the balancing contribu-tion of each generatorg is given by

di(g) =PmaxG,g∑nj=1 P

maxG,j

, (2)

3

where the superscript0 refers to normal operating condition(i = 0), or situation with line outages (i ∈ L). Duringthe outage of generatori, the compensation vector of thegenerators is given by

di(g) =PmaxG,g∑n

j=1,j 6=i PmaxG,j

∀G\i, di(i) = 0 . (3)

The vectorsdi ∈ Rn thus describe the compensation of anypower mismatch in the system, for any outage situationi. Wenote that by definition ofd, the system remains balanced afterany fluctuations or generator outages.

B. Line flow modeling

Similar to the setup in [5], the line flows are expressed aslinear functions of the active power injections in both normaland outage conditions.

P il = AiP iinj , for all i = 0, ..., nC . (4)

Here,Ai ∈ RnL×n describes the relation between the activepower injectionsP iinj ∈ R

n and the line flowsP il after outagei, with i = 0 being the normal operation condition.Ai is givenby

Ai = Bif

[(B̃ibus)

−10

0 0

](5)

where Bif ∈ RnL×n is the line susceptance matrix and

B̃ibus ∈ Rn−1×n−1 the bus susceptance matrix (without the

last column and row) after outagei [5]. The power injectionsare given by

P iinj = PG + di(PG(i) − 1δδR) + PR + δR − PD . (6)

Here,PD ∈ Rn is the vector of loads, and the power mismatchdue to generation outagePG(i) is non-zero only fori ∈ G.The vector1δ ∈ R1×n is a vector of ones, such that1δδRrepresents the sum of the forecast errors.

C. Chance constrained optimal power flow

Using the modeling assumptions presented above, we canformulate the pSCOPF as

minPG

cTPG (7)

subject to

11×n(PG + PR − PL) = 0 (8)

P[PG(g) + di(g)(PG(i) − 1δδR) ≤ P

maxG(g) ] ≥ 1− ε , (9)

P[PG(g) + di(g)(PG(i) − 1δδR) ≥ P

minG(g)] ≥ 1− ε , (10)

P[Ai(l,·)Piinj ≤ P

maxL(l) ] ≥ 1− ε , (11)

P[Ai(l,·)Piinj ≥ −P

maxL(l) ] ≥ 1− ε , (12)

for g = 1, ..., n, l = 1, ..., nL, i = 1, ..., nC .

The objective (7) is to minimize generation cost, withcrepresenting the bids of the generators. Constraint (8) ensurespower balance in the system. The constraints (9)-(12) are thegeneration and transmission constraints, withPminG andP

maxG

being the minimum and maximum generation levels andPmaxLbeing the transmission capacity of the lines. Those constraints

depend on the realization of the random variableδR, and areformulated as single chance constraints. The chance constraintensures that probability of a constraint violation (e.g., alineflow exceeding the limit) remains smaller thanε. We will referto ε as the violation probability and to1 − ε as the securitylevel. The value ofε is an input parameter to the optimization.

IV. CHANCE CONSTRAINT REFORMULATION

To obtain a tractable optimization problem, the chanceconstraints (9)-(12) must admit a deterministic and tractablereformulation. These constraints (9)-(12) are all univariate orsingle chance constraints of the general form

P[a(PG) + b(PG)δR ≤ c] ≥ 1− ε . (13)

wherea(PG) ∈ R and b(PG) ∈ R1×n are affine functions ofthe decision variablesPG andc is a constant. The terma(PG)represents the nominal generation output or the nominal lineflows (without forecast errors) andc represents the generationor line flow limit. The vectorb(PG) expresses the influence ofthe forecast errorsδR on the respective constraint. Regardlessof the exact expressions forb(PG), and for any dimension ordistribution of the random vectorδR, the left hand side of theconstraint is a scalar random variableδ = a(PG) + b(PG)δRwith meanµ(PG) and varianceσ(PG) given by

µ(PG) = a(PG) + b(PG)µR , σ(PG) =‖ b(PG)Σ1/2R ‖2 .

What is of interest when reformulating the constraint (13) isnot the distribution of the forecast uncertaintyδR, but thedistribution ofδ, which represents the variations in line flowsor generation outputs. Depending on the system,δ mightfollow different distributions. We will now present differentdistributional assumptions forδ which are relevant in thecontext of the SCOPF. The applicability of each assumptiondepends mainly on the source of uncertainty (e.g., load, renew-ables or short-term trading), the time frame of the forecast(e.g., day-ahead planning or close to real-time operation)and the availability of data (e.g., historical forecast errors orprobabilistic forecasts).

1) Normal distribution (Φ): The normal distribution is agood distribution model in two different cases. First, whenδR follows a multivariate normal distribution (which mightbe the case, e.g., for load uncertainty), which meansδ willbe normally distributed as well. Second, when the numberof uncertainty sources is large and not highly correlated,arguments similar to the central limit theorem (e.g., [17])imply that the distribution ofδ (which is a weighted sum ofδR) is expected to be close to a normal distribution.

2) Student’st-distribution (t): When the forecast fluctua-tions are heavy tailed (e.g., as for the minute-to-minute vari-ability in the European grid [15]), the Student’st-distributioncan be a more appropriate representation. Particularly whenconsidering small violation probabilities (ε < 0.03), Student’st distribution provides additional robustness compared tothenormal distribution.

In many cases, only limited knowledge about the distri-bution of δ is available. It might therefore be desirable toonly assume some general properties of the distribution ofδ,

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rather than a specific distribution. This leads to the followingdistributionally robust reformulations, that are valid for allprobability distributions that share the general properties:

3) Symmetric, unimodal distributions (S): If the distribu-tion is likely to be close to normal, but we do not know howclose, we can resort to the general assumption of unimodal,symmetric distribution with known mean and covariance.

4) Unimodal distributions (U ): In systems where the fore-cast uncertainty is related mainly to load, wind and PVproduction, the distribution ofδR is likely to be unimodal,with fluctuations centered around the forecasted value. Undersuch conditions, it is highly probable that the distribution ofδ is also unimodal.

5) Known mean and covariance (C): In systems whereintra-day electricity trading is not controlled by the trans-mission system operator, for example in Europe, intra-daytransactions introduce uncertainty in the power injections fromconventional power plants. The transactions might followalmost any probability distribution, and can even be discrete.In this case, we reformulate the chance constraint based onlyon a known (and finite) mean and covariance.

For all distributional assumptions 1) - 5), the chance con-straint (13) can be reformulated to the following analyticexpression

a(PG) ≤ c− b(PG)µR − f−1(1− ε) ‖ b(PG)Σ

1/2R ‖2 . (14)

Analyzing (14), we see that the left part,a(PG) ≤ c, representsthe “nominal” constraint, i.e., the constraint we would obtainif we neglect the forecast uncertainty. The second and thirdterm represents a reduction of the nominally available capacityc, which is necessary to secure the system against forecastdeviations. This reduction can thus be interpreted as a securitymargin against uncertainty, i.e., anuncertainty margin. Noticethat the largerf−1(1− ε), the larger the uncertainty margin.

Depending on which assumption 1) - 5) is deemed appro-priate, we definef−1(1 − ε) according to either an inversecumulative distribution function (for known distributions 1),2)) or a probability inequality (when only partial informationis available 3) - 5)). The exact expressions forf−1(1− ε) areshown in Table I, and their derivations as well as the derivationof (14) are given in the Appendix. We note that for 1) and 2),the reformulation is tight (the chance constraint holds withequality). The distributionally robust reformulations 3)-5) aretypically not tight, and will usually lead to empirical violationprobabilities lower thanε.

Since the reformulations 1) - 5) differ only in the definitionof f−1(1− ε), we can compare them by comparing the valueof f−1(1− ε) for differentε. In Fig. 1,f−1(1− ε) is plottedagainst the security level1−ε. We observe that allf−1(1−ε)increase asε decreases, indicating that a larger uncertaintymargin is necessary to achieve a lower violation probability.With more information, tighter probabilistic bounds can bedefined and thus a lower value off−1(1 − ε) is necessaryto ensure the desired security level (i.e.,f−1C > f

−1U > f

−1S ).

The lowest values are obtained when we assume knowledge ofthe actual distribution, i.e., for the normal and the Student’s tdistribution. Note that all reformulations assuming symmetryhave fS(0.5) = fΦ(0.5) = ft(0.5) = 0. Finally, Student’s

TABLE IEXPRESSIONS FORf−1(1 − ε).

Φ: Cumulative distribution function of the standard normal distribution.tν,σT : Cumulative distribution function of the Student t distribution withzero mean,ν degrees of freedom and scale parameterσT = (ν − 2)/ν.1) Normal f−1Φ (1 − ε) = Φ−1(1 − ε)

2) Student’st f−1t (1 − ε) = t−1ν,σT (1− ε)

3) Symmetric, unimodal fS(1 − ε) =

√

29ε

for 0 ≤ ε ≤ 16

√3(1− 2ε) for 1

6< ε < 1

2

0 for 12≤ ε ≤ 1

4) Unimodal f−1U

(1 − ε) =

√

49ε

− 1 for 0 ≤ ε ≤ 16

√

3(1−ε)1+3ε

for 16< ε ≤ 1

5) Mean, covariance f−1C

(1 − ε) =√

1−εε

for 0 ≤ ε ≤ 1

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1

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1 (1−

ε)

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1 (1−

ε)

fΦ−1 Normal

fq

ν

−1 t with ν=4

fC−1 Chebyshev

fU−1 Unimodal

fS−1 Symm+Uni

Fig. 1. Values off−1(1 − ε) for the normal distribution, the Student tdistribution with 4 degrees of freedom, the Chebyshev inequality, unimodaldistributions and symmetric, unimodal distributions. Theleft part shows allsecurity levels, while the right part is a zoom in on high security levels.

t distribution has a more pronounced peak and heavier tailsthan the normal distribution. This is reflected in that for lowersecurity levels,f−1t (1−ε) < f

−1Φ (1−ε), while at high security

levels,f−1t (1− ε) > f−1Φ (1 − ε).

V. REFORMULATED CONSTRAINTS

With the reformulation presented above, we can reformulatethe chance constraints (9)-(12) as

PG(g) + di(g)PG(i) ≤ P

maxG(g)− (15)

di(g)1δµR − f−1(1− ε) ‖ di(g)1δΣ

1/2R ‖2,

PG(g) + di(g)PG(i) ≥ P

minG(g)− (16)

di(g)1δµR + f−1(1− ε) ‖ di(g)1δΣ

1/2R ‖2,

Ai(l,·)(PG + diPG(i) + PR − PD) ≤ P

maxL(l) − (17)

Ai(l,·)(I−di1δ)µR − f

−1(1−ε)‖Ai(l,·)(I−di1δ)Σ

1/2R ‖2,

Ai(l,·)(PG + diPG(i) + PR − PD) ≥ −P

maxL(l) − (18)

Ai(l,·)(I−di1δ)µR + f

−1(1−ε)‖Ai(l,·)(I−di1δ)Σ

1/2R ‖2,

for g = 1, ..., n, l = 1, ..., n, i = 1, ..., nC ,

whereI ∈ Rn×n is the identity matrix. When comparing (15)- (18) to (14), we recognize the same structure. The first partrepresent the constraint we would obtain by neglecting the

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forecast uncertainty, where as the second and third term onthe right hand side represent the uncertainty margin.

Since a higher uncertainty margin leads to a reduction inthe available transmission and generation capacity, a higheruncertainty margin will not only reduce the probability ofviolation, but also increase the nominal cost of operation (i.e.,the cost of the pSCOPF). The acceptable violation probabilityε and the distributional assumption (which defines the functionf−1(1 − ε)) should therefore be chosen carefully to obtain agood trade-off between security against forecast errors andcost of operation.

Note that the reformulated chance constraints (15) - (18)are linear, since the uncertainty margin is not dependent onany decision variables and can be pre-computed. The pSCOPFproblem (7), (8), (15) - (18) is thus a linear program with thesame computational complexity as a traditional DC SCOPF.

VI. CASE STUDIES

The purpose of this case study is to demonstrate the chance-constrained SCOPF, and investigate which distributional as-sumptions are most appropriate for power systems operation.We base our study on the IEEE 118-bus system [18], with afew modifications as follows. The generation cost is assumedto be linear, and is based on the linear cost coefficientsof the data provided with Matpower 4.1 [19]. Although theformulation could be extended to include unit commitment, itis not considered here. Therefore, the minimum generationoutput of the conventional generators is set to zero. Theforecast uncertaintyδR is modeled based on historical datafor 1 year from the Austrian Power Grid (APG). We definethe forecast error as the difference between the the so-calledDACF (Day-Ahead Congestion Forecast) and the snapshot(the real-time power injections) for all hours and buses withavailable data (8492 data points for 28 buses). Since the systemis constantly evolving and might exhibit seasonal patterns, weassume that the power system operator only uses data from thepast three months. We use two three-month periods to definethe forecast uncertainty for this case study, such that we obtain2207 data samples for a total of 54 buses.

The historical data was assigned to different load busesthroughout the system, and modified such that the standarddeviation corresponds to 20 % of the forecasted load. Themean µR and covarianceΣR used in the pSCOPF werecalculated based on this modified data (i.e, assuming perfectknowledge ofµR, ΣR). Fig. 2 shows the forecast errors fromsome representative nodes, including the histograms and pair-wise scatter plots of the forecast errors. By inspection, itisclear that the forecast errors are not normally distributed.

In the following, we assess how the different distributionalassumptions impact the solution of the pSCOPF. We solve thepSCOPF for all five reformulations assuming an acceptableviolation probability of ǫ = 0.1. The results are comparedwith each other and to the solution of the correspondingdeterministic SCOPF. To assess the quality of the solution,wecompare the number of empirical constraint violations (basedon the historical samples) and the relative cost of the solutions.Further, we run statistical tests to check if the data is normally

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Fig. 2. Forecast errors for 4 selected nodes of case study. The diagonal plotsshow the histograms of the forecast errors (x-axis: deviation in MW, y-axis:number of occurences), while the off-diagonal plots show the scatter plotsbetween two corresponding forecast errors (x- and y-axis: deviation in MW).

0) Det 1) Norm 2) Stud t 3) Symm 4) Uni 5) M&C0

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f f−

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ε) [−

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c)

b)

a)

Fig. 3. Results derived from the different SCOPF solutions.From left to right:0) the deterministic SCOPF, and the pSCOPF based on the assumption of 1)a normal distribution, 2) a Student t distribution, 3) a symmetric, unimodaldistribution, 4) a unimodal distribution and 5) known mean and covariance.From top to bottom, the figure shows a) empirical violation probabilities forthe all constraints, b) nominal dispatch cost, and c) value of the functionf−1(1− ε) for ε = 0.1.

or unimodally distributed, and investigate the accuracy oftheestimated uncertainty margins.

1) Number of empirical violations:The empirical violationprobabilitiesε̂ are evaluated for all constraints based on the2207 data samples. The results are shown in Fig. 3 a), startingwith 0) the deterministic solution, then the solution basedon 1) a normal and 2) a Student t distribution, and thenthe distributionally robust solutions 3)-5). From left to right,we thus assume lessened knowledge about the distribution.The violation probabilities of the non-active constraintsε̂n−aand activeε̂a constraints are plotted in yellow and orange,respectively. The average violation probabilitŷεavg of theactive constraints is plotted in black.

As seen on the left, the empirical violation probability of

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the deterministic solution is very high, with some constraintsviolating 80% of the cases. This highlights the need for prob-abilistic methods to avoid frequent violations of operationallimits. The pSCOPF solutions have much lower violationprobabilities. The active constraints (where the distancefromthe nominal flow to the flow limit is given by the uncertaintymargin) have a higher empirical violation probability thanthenon-active constraints (which have some additional margin).The solution based on a normal distribution violates theaccepted violation probabilityε < 0.1 for some constraints,but the violation is small,̂ε < 0.11, and the average vi-olation probability is acceptablêεavg < 0.1. The solutionbased on a Student t distribution, which assumes a morepeaked distribution, has larger empirical violation probabilitiesε̂ > 0.15. The distributionally robust solutions oversatisfythe accepted violation probabilityε, with ε̂avg = 0.07 forthe symmetric, unimodal solution,0.03 for the unimodalsolution and0.0025 for the solution based only on mean andcovariance. Although the chance constraints are satisfied,itdoes not necessarily imply that the underlying assumption(e.g., symmetry and unimodality) is accurate. Since the re-formulations are distributionally robust, we might get a lowempirical violation probability, even if we assumed the wrongfamily of distributions.

2) Operational cost:Fig 3 b) shows the generation costobtained with the pSCOPF solutions, normalized by the cost ofthe deterministic problem (shown to the left). All probabilisticsolutions have higher cost than the deterministic solution,showing that the consideration of uncertainties increase thenominal cost of operation. The reformulations which assumemore knowledge about the distribution 1), 2) lead to lowercost than the more general reformulations 3)-5). The mostexpensive solution is obtained for reformulation 5), whichonlyassumes knowledge of mean and covariance.

The cost differences are explained by the different valuesf−1(1−ε), which defines the uncertainty margin and thus theconstraint tightening. The value off−1(1−ε) is plotted in Fig.3 c). Comparing Fig. 3 a), b) and c), we observe how a largerf−1(1−ε) leads to an increase in nominal operation cost, butat the same time reduces the empirical violation probability.This highlights two important aspects of the pSCOPF. First,we need to defineε such that it reflects a reasonable trade-off between cost and security. Second, we want to achievean empirical violation probabilitŷε as close as possible tothe accepted violation levelε. A reformulation with too manyviolations (̂ε >> ε) leads to unsecure operations, but at thesame time a too conservative solution (ε̂ 0.95, andrejected for p-valuesp < 0.05. In between, we can neitherreject nor confirm the hypothesis.

In Fig. 4, the p-values from both tests are plotted as ahistogram. The bars show the percentage of constraints withp-values in the indicated p-value interval. We observe thatthep-values from the Shapiro-Wilk test are below the thresholdp < 0.05 for most constraints, while the p-values fromHartigans Dip Test are abovep > 0.95 for the majority of theconstraints. We thus conclude that unimodality is a reasonableassumption, while the original data is probably not normallydistributed.

Although the statistical test rejects normality, the normaldistribution might still be a good assumptions for the partsof the distribution which we are interested in. In Fig. 5,the empirical distribution of the line flow deviations for oneactive transmission constraint is shown. This constraint hadthe lowest p-value among the active constraints in the Shapiro-Wilk test. Fig. 5 also show the empirical uncertainty margin(corresponding to an empirical violation probabilityε̂ = 0.1),as well as the uncertainty margins obtained with the fivedistributional assumptions. We observe that the uncertaintymargins based on the normal distribution (plotted in red)match very closely to the empirical margins (plotted in green).The Student t distribution underestimates the margin and thedistributionally robust reformulations lead to too high margins.

4) Discussion: Based on the above results, we concludethat the reformulation based on a normal distribution canprovide a good trade-off between cost and security, partic-ularly in systems with a large number of uncertainty sources.Although assuming a normal distribution does not guaranteeempirical violation probabilitieŝε < ε, the assumption mightbe useful whenε can be interpreted as a guideline ratherthan a hard limit. Statistical tests and assessments as in Fig.5 can be used to assess whether the normal distribution isa reasonable approximation. If the system operator wants ahigher confidence in enforcing the actual violation levelεand is willing to tolerate a larger increase in operational cost,

7

−60 −40 −20 0 20 40 600

20

40

60

80

100

Line Flow Deviations

Num

ber

of o

ccur

ence

s

EmpiricalNormalStudentSymmetricUnimodalChebyshev

Fig. 5. Histogram of post-contingency line flow deviations.The uncertaintymargins are computed empirically (green), and for 1) a normal distribution(red), 2) a Student t distribution (yellow), 3) a symmatric,unimodal distribu-tion (magenta), 4) a unimodal distribution (light blue) and5) a distributionwhere only the mean and covariance are known (dark blue).

assuming a unimodal distribution would be reasonable.

VII. SUMMARY AND CONCLUSION

This paper discusses different analytic reformulations forchance constraints and their applicability in the pSCOPFcontext. The chance constraints are reformulated either byassuming a known probability distribution (such as normalor Student t distribution) or by using distributionally robustreformulations assuming general properties of the distribution(i.e., known mean and variance, symmetry, unimodality).

The reformulated chance constraints all have a similarform, and are easily comparable. They are similar to thenominal constraints of the deterministic problem, except forthe uncertainty margin (a security margin against forecastdeviations), which represents a reduction of the transmissionor generation capacity. With a larger uncertainty margin, theprobability of violations decreases, but the nominal operationalcost increases. Therefore, it is desirable to find a reformulationwhich leads to an uncertainty margin which is sufficientlylarge, yet as small as possible.

In the case study based on the IEEE 118 bus system andforecast errors from Austria, the trade-off between securityand cost is highlighted. Although the choice of reformulationdiffers between systems with different uncertainty charac-teristics, we show that the normal distribution might be agood approximation in cases where the acceptable violationprobability can be interpreted as a guideline, rather than ahard constraint. If the transmission system operator wantstoenforce the violation probability as a strict limit, choosing amore conservative, distributionally robust reformulations basedon, e.g., unimodality will provide more confidence.

In general, we believe that the pSCOPF with analyticallyreformulated chance constraints provides a transparent andscalable approach to assess the effect of uncertainty in powersystem operational planning. Future work will investigatehow the approach can be extended towards corrective controlactions for uncertainties (e.g., HVDC and PSTs), to furtherreduce the cost of handling uncertainty. Further, we plan toinvestigate extensions towards AC power flow.

ACKNOWLEDGMENT

This research work described in this paper has been partiallycarried out within the scope of the project ”Innovative tools for

future coordinated and stable operation of the pan-Europeanelectricity transmission system (UMBRELLA)”, supportedunder the 7th Framework Programme of the European Union,grant agreement 282775. We thank our partners, especiallyAustrian Power Grid, for providing historical data.

APPENDIX

A. Exact tractable reformulations of single chance constraints

We discuss here how the chance constraint (13) can bereformulated as the deterministic constraint (14). Using theequalityδ = a(PG) + b(PG)δR it is clear that constraint (13)is equivalent to

P[δ < c] ≥ 1− ǫ (19)

where the constraint (19) should be satisfied for all distribu-tionsP consistent with the distributional assumptions made onδ. The constraint (19) can equivalently be represented as

P

[δ − µ(PG)

σ(PG)<

c− µ(PG)

σ(PG)

]≥ 1− ǫ (20)

where it can be remarked that the scaled random variableδn := (δ − µ(PG)) /σ(PG) has zero mean and unit varianceby construction.

In order to unify the analysis for all distributional assump-tions made in Section IV, we consider the general situation inwhich the distributionP of δn is merely known to belong to aset of distributionsP . Specifically, we have thatP correspondto {N(0, 1)} in the Gaussian case 1) and to{tν

(0,√(ν−2ν )

)}

in the Student’st case 2) withν degrees of freedom. In theformer two cases, the setP is a singleton as the distributionof δn is known. In the most general case 5), the setPconsists of all distributions of zero mean and unit variance.The setP is additionally required to contain only unimodalor unimodal symmetric distributions in the unimodal case 4)and the unimodal symmetric case 3), as both notions are scaleinvariant. That is, ifδ is unimodal or symmetric unimodal thanαδ+ β is unimodal or symmetric unimodal as well for anyαandβ real numbers.

Given a set of distributionsP representing the distributionalassumptions made, we will first show that the chance con-straint (13) is equivalent to the deterministic constraint(14)for fP(k) := infP∈P P [δn < k].

We can trivially rewrite constraint (20) equivalently as theconstraintP[δn < (c− µ(PG)) /σ(PG)] ≥ 1−ǫ for all P ∈ P .Using the definition offP we can demand alternatively thatfP((c− µ(PG)) /σ(PG)) ≥ 1−ǫ. Finally as the functionfP isincreasing it has a well defined generalized inversef−1P (λ) =inf {k | fP(k) ≥ λ}. From the definition off it follows thatwe must have that(c− µ(PG)) /σ(PG) ≥ f

−1P (1 − ǫ). After

reordering the terms we obtain

a(PG) + b(PG)µR ≤ c− f−1P (1− ǫ)‖b(PG)Σ

1/2R ‖2.

Finally in the remainder of this section, we show forthe distributional assumptions discussed above we obtain theresults mentioned in Table I.1–2) For non-atomic distributionsP it follows by continuity

that f{P} = P[δn < k] = P[δn ≤ k] explaining the firsttwo entries in Table I.

8

3) Let the setP corresponds to the set of all symmetric uni-modal distributionsS with zero mean and unit variancethen we can leverage the classical Gauss bound [23],[24]. Indeed, in this situation we have that the mode ofthe distributions is known as it coincides with the meanbecause of symmetry. For anyk > 0, we have again bysymmetry that the equalityP[δn ≥ k] = 12P[|δn| ≥ k]holds. As the mode ofP is known, the classical Gaussbound [23], [24] can be used to establish

fU (k) = 1− supP∈S

P [δn ≥ k]

=

{1− 12 supP∈U P [|δn| ≥ k] if k > 0

0 otherwise

=

9k2−29k2 if k ≥

√43

12 +

k2√3

k > 0

0 otherwise

(21)

4) The unimodal case, in whichP = U consists of zeromean and unit variance unimodal measures, can bedealt with using the one-sided Vysochanskij–Petunininequality [25], i.e.

fU (k) = 1− supP∈U

P [δn ≥ k]

= 1−

49(1+k2) if k ≥

√53

1− 43k2

1+k2 if k ≥ 0

1 otherwise

=

1− 49(1+k2) if k ≥√

53

43

k2

1+k2 k ≥ 0

0 otherwise

(22)

5) Lastly, when the setP corresponds to the setC con-taining all distributions of zero mean and variance, theclassical Cantelli inequality [26] establishes that

fC(k) = 1− supP∈C

P [δn ≥ k]

= 1−

{1

1+k2 if k ≥ 0

1 otherwise

=

{k2

1+k2 if k ≥ 0

0 otherwise(23)

which after taking the inverse gives us the correspondingresult in Table I.

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https://www.entsoe.eu/fileadmin/user_upload/_library/resources/LCFR/2012-06-14_SOC-AhT-OR_Report_final_V9-3.pdfhttp://motor.ece.iit.edu/data/JEAS_IEEE118.dochttp://www.R-project.orghttp://dx.doi.org/10.1007/s10107-015-0878-1

I IntroductionII Modeling forecast uncertaintyIII Optimal Power Flow FormulationIII-A Generator modelingIII-B Line flow modelingIII-C Chance constrained optimal power flow

IV Chance constraint reformulationIV-1 Normal distribution ()IV-2 Student's t-distribution (t)IV-3 Symmetric, unimodal distributions (S)IV-4 Unimodal distributions (U)IV-5 Known mean and covariance (C)

V Reformulated constraintsVI Case StudiesVI-1 Number of empirical violationsVI-2 Operational costVI-3 Testing the distributional assumptions and the accuracy of estimated uncertainty marginVI-4 Discussion

VII Summary and ConclusionAppendixA Exact tractable reformulations of single chance constraints

References