Chance-Constrained AC Optimal Power Flow Integrating HVDC Lines and Controllability
Andreas Venzke, Lejla Halilbasic, Adelie Barre, Line Roald, Spyros Chatzivasileiadis
aa r X i v : . [ c s . S Y ] J un Chance-Constrained AC Optimal Power FlowIntegrating HVDC Lines and Controllability
Andreas Venzke ∗ , Lejla Halilbaˇsi´c ∗ , Ad´elie Barr´e ∗ , Line Roald † and Spyros Chatzivasileiadis ∗∗ Center for Electric Power and Energy, Technical University of Denmark, Kgs. Lyngby, Denmark † Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, USAEmail: { andven, lhal } @elektro.dtu.dk, [email protected], [email protected], [email protected] Abstract —The integration of large-scale renewable generationhas major implications on the operation of power systems, twoof which we address in this work. First, system operators haveto deal with higher degrees of uncertainty due to forecast errorsand variability in renewable energy production. Second, withabundant potential of renewable generation in remote locations,there is an increasing interest in the use of High Voltage DirectCurrent lines (HVDC) to increase transmission capacity. TheseHVDC transmission lines and the flexibility and controllabilitythey offer must be incorporated effectively and safely into thesystem. In this work, we introduce an optimization tool thataddresses both challenges by incorporating the full AC powerflow equations, chance constraints to address the uncertaintyof renewable infeed, modelling of point-to-point HVDC lines,and optimized corrective control policies to model the generatorand HVDC response to uncertainty. The main contributions aretwofold. First, we introduce a HVDC line model and the cor-responding HVDC participation factors in a chance-constrainedAC-OPF framework. Second, we modify an existing algorithm forsolving the chance-constrained AC-OPF to allow for optimizationof the generation and HVDC participation factors. Using realisticwind forecast data, for 10 and IEEE 39 bus systems withHVDC lines and wind farms, we show that our proposed OPFformulation achieves good in- and out-of-sample performancewhereas not considering uncertainty leads to high constraintviolation probabilities. In addition, we find that optimizing theparticipation factors reduces the cost of uncertainty significantly.
Index Terms —AC optimal power flow, chance constraints,HVDC transmission, uncertainty.
I. I
NTRODUCTION
A. Motivation
Power system operators have to deal with higher degrees ofuncertainty. Increasing shares of unpredictable renewable gen-eration, and stochastic loads, can lead to additional costs andjeopardize system security if uncertainty is not explicitly con-sidered and addressed. In addition, with abundant renewablepotential being available further away from load centers, e.g.off-shore, High-Voltage Direct Current lines (HVDC) becomethe preferred technology for transmitting large amounts ofrenewable energy over longer distances. In order to deal withuncertainty, operators carry out both preventive and correctivecontrol actions in their system [1]. HVDC lines and grids canoffer corrective control actions in the form of real-time controlof active and reactive power flows. The AC optimal powerflow (AC-OPF) problem is a key tool for addressing thesechallenges [2]. The AC-OPF problem minimizes an objective function (e.g., generation cost) subject to the power systemoperational constraints (e.g. limits on the transmission lineflows and bus voltages). The goal of this paper is to proposean AC optimal power flow (AC-OPF) formulation that a)considers uncertainty in wind power infeed, b) incorporatesan HVDC line model and c) allows for an optimization ofthe generator and HVDC control response to fluctuations inrenewable generation.
B. Literature Review
Existing literature considers uncertainty within the OPFproblem using methods such as scenario-based or chance-constrained stochastic programming (e.g. [3]–[5]), robust op-timization methods (e.g. [6]–[11]), or distributionally robustoptimization (e.g. [12]–[15]). Stochastic formulations can in-clude a set of scenarios describing possible realizations ofuncertainty, or chance constraints which define a maximumallowable probability of constraint violation. Robust optimiza-tion methods on the other hand often assume a pre-defineduncertainty set and secure the system against the worst-caserealization inside this set. To deal with the higher complexityarising from the uncertain parameters, existing approacheseither assume a DC-OPF or use different techniques to achievea tractable formulation of the AC-OPF under uncertainty.Examples of approaches utilizing robust optimization for theAC-OPF under uncertainty include [11], which uses explicitmaximization to approximate the AC-OPF under uncertaintyas a mixed-integer program, and [10], which develops a convexinner approximation by assuming controllable loads at allbuses.In this paper we focus on the chance-constrained OPF.Chance-constrained DC-OPF results to a faster and morescalable algorithm, but the DC-OPF is an approximation thatneglects losses, reactive power, and voltage constraints. Refs.[3] and [4] formulate a chance constrained DC-OPF assuminga Gaussian distribution of the forecast errors. In [16], acombination of randomized and robust optimization is usedto achieve a tractable formulation of the chance constrainedDC-OPF including N-1 security constraints. The same authorsextended their work to consider a convex relaxation of the non-linear AC OPF problem in [17]. Related to convex relaxationideas in [17], several works [18]–[20] have investigated usingconvex relaxations of the non-convex AC-OPF to achieve atractable formulation of the chance-constrained AC-OPF. In18], the semidefinite relaxation is applied and both sample-based and analytical solution approaches are discussed. Thelatter approach is extended to include interconnected AC andHVDC grids in [19]. Using the semidefinite relaxation, thework in [20] proposes a distributed solving approach for asuitable approximation of the chance constrained AC-OPFusing the alternating direction method of multipliers (ADMM).A variety of other approaches have been proposed, typicallybased either on full or partial linearization [5], [15], [21]–[23]. The work in [21] uses linearization and back-mappingto achieve a tractable formulation, while [22], [23] obtainsanalytical reformulations based on linearized AC power flowequations. The work in [15] uses the Wasserstein metric as dis-tance measure between probability distributions and proposesa tractable formulation of the chance constrained AC-OPF as-suming that the true probability distribution is within a definedWasserstein distance to the empirical distribution based on datasamples. The work presented in this paper is most closelyrelated to the approach in [5], where a linearization used tomodel the impact of uncertainty is combined with the full ACpower flow equations for the forecasted operating point. Thepapers devises a scalable, iterative solution algorithm, whichis observed to produce close to optimal solutions in [24].
C. Contributions
Previous work included simultaneous consideration ofpower injection uncertainty and operation of HVDC in a singleoptimization problem. For example, Refs. [25]–[27] considerstochastic OPF formulations which also incorporate HVDClines and HVDC grids. However, they all assume a DC-OPF formulation. The focus of this paper is to avoid mostof these simplifications to the extent that it is possible, andinstead use the full non-linear AC power flow equations asthe DC-OPF can lead to substantial errors [28]. The AC-OPF formulation further allows to fully utilize the controlcapabilities of the HVDC converters, including voltage andreactive power control. Our work differs in two importantaspects from the work in [19]. First, by relying on a non-convex AC-OPF formulation, and a linearization around theforecasted operating point, our approach is more scalable thanthe semidefinite relaxation used in [19]. Second, the work in[19] uses a sample-based approach and robust optimizationto approximate the chance constraints which can result toconservative results and very low empirical chance constraintviolation probabilities. Here, we assume a normal distributionof the forecast errors which can lead to less conservativesolutions compliant with the maximum allowable violationprobabilities. In this paper, we propose an iterative chance-constrained AC-OPF for AC grids with HVDC lines, develop-ing further the work described in [5] and elaborated in [24].The main contributions of our work are:1) We integrate an HVDC line model and HVDC correctivecontrol policies in a non-convex chance-constrained AC-OPF framework considering uncertainty in wind power. 2) We enable optimization of both generator and HVDCparticipation factors to react to forecast errors within acomputationally efficient iterative solution algorithm.3) To improve computational tractability, we propose toutilize a constraint generation method.4) Using realistic wind forecast data and a Monte CarloAnalysis, for 10 and 39 bus systems with HVDC linesand wind farms, we show that (i) not considering un-certainty leads to high constraint violation probabilitieswhereas our proposed approach achieves compliancewith the target chance constraint violation probabilitiesand (ii) optimizing both generator and HVDC participa-tion factors reduces the cost of uncertainty significantly.The structure of this paper is as follows. Section II statesthe chance-constrained AC-OPF formulation. In Section III,the HVDC line model and HVDC corrective control policy isexplained. Section IV introduces the iterative solution algo-rithm. Section V evaluates the performance of the proposedapproach on 10 and 39 bus test cases. Section VI concludes.II. O
PTIMAL P OWER F LOW F ORMULATION
This section states the chance-constrained AC-OPF andpresents a tractable reformulation of the chance constraints,which is based on the work from [5] and [24]. For ease ofreference, we follow the notation of [24] wherever possible.
A. Chance-Constrained AC Optimal Power Flow
A power network consists of the set N of buses, a subsetof those denoted by G have a generator connected. The busesare connected by a set ( i, j ) ∈ L of transmission lines frombus i to j . The AC-OPF problem minimizes an objectivefunction (e.g., generation cost) subject to the power systemoperational constraints (e.g. limits on the transmission lineflows and bus voltages). For a comprehensive review of theAC-OPF problem, the reader is referred to [29].The chance-constrained AC-OPF aims at determining theleast-cost operating point, which reduces the probability ofviolating the limits of system components to an acceptablelevel ǫ for a range of uncertainty realizations (e.g. ǫ = 1% ).Consequently, the AC-OPF variables, commonly defined inthe space of x := { P , Q , V , θ } variables, are not only subjectto one possible set of realizations of the uncertain parametersbut to a range of uncertain realizations depending on theirforecast errors ω . P , Q , V and θ denote vectors of nodalactive and reactive power injections as well as nodal voltagemagnitudes and angles, respectively. We assume wind powerforecast errors ω to be the the only source of uncertainty andto follow a multivariate Gaussian distribution with zero meanand known covariance, as the authors in [24] have shown isreasonably accurate, even when ω is not normally distributed.The actual wind power realization ˜P W is modelled as the sumof its expected value P W and the forecast error ω , ˜ P W,i = P W,i + ω i , ∀ i ∈ W , (1)where W denotes the subset of network nodes with windgenerators connected to them. Note that our framework readilyxtends to consider other sources of uncertainty in powerinjections, e.g. of loads. We assume that wind power plantsare operated with a constant power factor, which meansthat their reactive power output follows their active poweroutput, i.e., ˜ Q W,i = γ ( P W,i + ω i ) , where the power ratio γ = q − cos φ cos φ depends on the power factor cos φ and can be aparameter or an optimization variable. The actual realizationsof the OPF decision variables are modelled as the sum oftheir optimal set-points at the forecasted wind infeed x andtheir reactions to a change in wind power injection ∆x ( ω ) ,i.e., ˜x ( ω ) = x + ∆x ( ω ) . This gives rise to the followingformulation of the chance-constrained AC-OPF: min x c T2 P + c T1 P G + c (2a)s.t. f i ( x ) = 0 , ∀ i ∈ N (2b) P ( P G,k + ∆ P G,k ( ω ) ≤ P max G,k ) ≥ − ǫ, ∀ k ∈ G (2c) P ( P min G,k ≤ P G,k + ∆ P G,k ( ω )) ≥ − ǫ, ∀ k ∈ G (2d) P ( Q G,k + ∆ Q G,k ( ω ) ≤ Q max G,k ) ≥ − ǫ, ∀ k ∈ G (2e) P ( Q min G,k ≤ Q G,k + ∆ Q G,k ( ω )) ≥ − ǫ, ∀ k ∈ G (2f) P ( V i + ∆ V i ( ω ) ≤ V max i ) ≥ − ǫ, ∀ i ∈ N (2g) P ( V min i ≤ V i + ∆ V i ( ω )) ≥ − ǫ, ∀ i ∈ N (2h) P ( P L,ij + ∆ P L,ij ( ω ) ≤ P max L,ij ) ≥ − ǫ, ∀ ( i, j ) ∈ L (2i) P ( P min L,ij ≤ P L,ij + ∆ P L,ij ( ω )) ≥ − ǫ, ∀ ( i, j ) ∈ L (2j)The chance-constrained AC-OPF (2a) – (2j) minimizes thetotal generation cost for the forecasted operating point. Theterms P G , Q G denotes the active and reactive power dis-patch of the generators, and c , c , c denote the quadratic,linear and constant cost factors, respectively. The term P L denotes the active power line flow. Constraint (2b) enforces then = 2 |N | nodal active and reactive power balance equationsfor the forecasted operating point where N represents theset of network nodes. Note that we do not explicitly enforcethe power balance for ω = 0 . Instead, as will be outlinedin the following, our formulation ensures satisfaction of thelinearized AC equations around the operating point, which incombination with the chosen control policies has been shownto perform well on the non-linear system for reasonable levelsof uncertainty [24]. The inequality constraints in (2c) – (2j)include upper and lower limits on active and reactive powergeneration, voltage magnitudes, as well as active power flows P L . They are formulated as individual chance constraintsand enforced with a confidence level of (1 − ǫ ) . The chanceconstraints account for the entire range of ω , as they can beanalytically reformulated to tractable deterministic constraintsusing a first order Taylor expansion, which will be discussedin detail in Section II-A2.
1) Affine Policies:
We model the control policies as affinefunctions of the uncertainty ω . Conventional generators areassumed to balance fluctuations in active power generationaccording to their generator participation factors α for eachgenerator k ∈ G according to ˜ P G,k ( ω ) = P G,k + ∆ P G,k ( ω ) = P G,k − α k ω + δ Pk , (3) where the term δ P denotes the contribution to the compen-sation of the unknown changes in active power losses, represents an all-ones row vector of size |W| . This gener-ator response mimics Automatic Generation Control (AGC)commonly used in power system operation. The generatorparticipation factors α are thus defined w.r.t. to the total winddeviation Ω = P i ∈W ω i and can be either pre-determined(e.g., as a result of a reserve procurement) or optimized withinthe OPF. The condition P i ∈G α i = 1 ensures balance of thetotal power mismatch, i.e., P i ∈G α i P i ∈W ω i = Ω . Activepower losses vary non-linearly with the wind power deviationand are usually compensated by the generator at the referencebus; this results in the loss term δ P being equal to zero forgenerators at PV and PQ buses. All other variables of interest ∆x ( ω ) := { ∆Q G , ∆V , ∆ θ, ∆P line } are modeled similarly, ˜ x i ( ω ) = x i + Γ x i ω, (4)where Γ x i is a (1 × |W| ) vector defining the response ofvariable x i to each wind power deviation. In general, the re-sponse is modeled as follows: ∆x ( ω ) = ∂ x ∂ω ω = Γ x ω , where Γ x represents a matrix of linear sensitivities w.r.t. ω . Theterm Γ x also includes expressions for the unknown changesin active power losses δ P and is derived from the first orderTaylor expansion of the AC power flow equations around theforecasted operating point, (cid:20) ∆P∆Q (cid:21) = J (cid:12)(cid:12)(cid:12) x ∗ (cid:20) ∆ θ ∆V (cid:21) . (5)The term J denotes the Jacobian matrix. The left-hand sideof (5) can also be expressed in terms of the wind deviation ω , the power ratio γ , the generator participation factors α aswell as the unknown nonlinear changes in active and reactivepower (i.e., δ P , ∆Q ), (cid:20) Idiag ( γ ) (cid:21) ω + (cid:20) − α H0 (cid:21) ω + (cid:20) δ P ∆Q (cid:21) = Ψ ω + (cid:20) δ P ∆Q (cid:21) . (6)The terms I , H and denote ( |N | × |W| ) identity, all-onesand zero matrices, respectively. The matrix of GenerationDistribution Factors (GDF) Ψ depends linearly on α and γ (for a detailed derivation refer to [30]). Replacing the left-hand side in (5) with (6) yields: Ψ ω + (cid:20) δ P ∆Q (cid:21) = J (cid:12)(cid:12)(cid:12) x ∗ (cid:20) ∆ θ ∆V (cid:21) (7)In accordance with common practices in power systemoperations, some variables are assumed not to change underdifferent wind power realizations, such as the voltage mag-nitude at PV and reference buses, the voltage angle at thereference bus and the reactive power injection at PQ buses.We summarize the nonzero changes of unknown active andreactive power injections in δ := [ δ Pref ∆ Q ref ∆Q T PV ] T .Analogously, ∆ˆx denotes the nonzero changes in voltagemagnitudes and angles, i.e., ∆ˆx := [ ∆ θ TPV ∆ θ TPQ ∆V TPQ ] T . C,i Q C,i i P
C,j Q C,j j DC system P loss Fig. 1. HVDC line model connecting AC bus i to AC bus j with activeHVDC converter injections P C , reactive HVDC converter injections Q C and an active loss term P loss . Rearranging the resulting system of equations in (7) accordingto the groups of zero and nonzero elements (cid:20) δ (cid:21) = (cid:20) J Imod J IImod J IIImod J IVmod (cid:21) (cid:20) (cid:21) − (cid:20) Ψ Imod Ψ IImod (cid:21) ω, (8)allows us to derive expressions (9a) and (9b) for the changein variables as a function of the uncertainty ω . ∆ˆx = (cid:16) J IVmod (cid:17) − Ψ IImod ω = Γ ˆx ω (9a) δ = (cid:16) J IImod ( J IVmod ) − Ψ IImod − Ψ Imod (cid:17) ω = Γ δ ω (9b) J mod and Ψ mod denote the modified Jacobian and GDFmatrices, where the rows and columns have been rearrangedaccording to δ and ∆ˆx . Thus, the linear sensitivities Γ x depend on the GDF matrix Ψ , which is a linear function ofthe generator participation factors α and the power ratio γ .
2) Reformulating the Chance Constraints:
Given the lineardependency of the OPF variables on ω in the region around theoperating point and the assumption of a multivariate normaldistribution for ω , we are able to reformulate the individualchance constraints (2c)–(2j) to tractable deterministic con-straints. The linear chance constraint P ( x i + Γ x i ( Ψ ) ω ≤ x max i ) ≥ − ǫ is reformulated to x i ≤ x max i − Φ − (1 − ǫ ) q Γ x i Σ ( Γ x i ) T , (10)where Φ − denotes the inverse cumulative distribution func-tion of the Gaussian distribution. It can be observed that theoriginal constraint x i ≤ x i is tightened by an uncertaintymargin λ x i := Φ − (1 − ǫ ) p Γ x i Σ ( Γ x i ) T , which securesthe system against variations in wind infeed [5]. Given thedependency of Γ x on Ψ , optimizing over the generationresponse α explicitly represents its impact on the uncertaintymargins of the remaining variables within the optimization.III. HVDC L INE M ODELING
In this section, we present a model to include HVDC linesin the chance-constrained AC-OPF and we introduce HVDCparticipation factors to allow for corrective control. We assumethat the HVDC lines are modeled as presented in Fig. 1 withindividual active and reactive power injections P C , Q C atthe two AC buses the HVDC line is connected to and alumped loss term P loss for the DC system losses. The set c ∈ N C denotes the HVDC converter and for each two HVDCconverter comprising an HVDC line the set ( i, j ) ∈ L C denotes the AC buses the HVDC converters are connected to,respectively. We approximate the active and reactive power P C,c Q C,c m max q,c S nom C,c m min q,c S nom C,c − m p,c S nom C,c m p,c S nom C,c
Feasibleoperating region
Fig. 2. Active and reactive power capability curve of HVDC converter c [31]. capability of the converter as a rectangular box with thefollowing constraints: P min C,c ≤ P C,c ≤ P max C,c ∀ c ∈ C (11a) Q min C,c ≤ Q C,c ≤ Q max C,c ∀ c ∈ C (11b)Expressing the lower and upper active and reactive HVDCconverter limits P min C , P max C , Q min C , Q max C as a function ofthe nominal converter rated power S nom C and assuming thatthe lower and upper bounds on active power are symmetric(i.e. P min C = − P max C ) yields: − m p,c S nom C,c ≤ P C,c ≤ m p,c S nom C,c ∀ c ∈ C (12a) m min q,c S nom C,c ≤ Q C,c ≤ m max q,c S nom C,c ∀ c ∈ C (12b)The resulting feasible operating region is visualized in Fig. 2.For a more detailed modeling of the active and reactive powercapability of HVDC converter the interested reader is referredto [32]. In order to link the active power injections betweenthe two AC buses that the HVDC line is connected to, anactive power balance constraint has to be included. To modelthe DC system losses P loss , we use a constant loss term a defined as a share of the nominal apparent power rating forbuses ( i, j ) ∈ L C and converters c ∈ C : P HVDC ,i + P HVDC ,j + P loss ,c = 0 with P loss ,c = 2 aS nom C,c (13)This term gives an estimate of the HVDC converter losses.Note that we neglect the DC line losses. The reactive powerinjections at both AC buses ( i, j ) ∈ L C can be chosenindependently from each other within the HVDC converterlimits. To allow for corrective control, we assign a participationfactor β c for each HVDC converter c ∈ C similarly to thecase of generators. As the HVDC line itself cannot generateactive power, the participation factor is positive at one end ofthe HVDC line and negative at the other end, i.e. β i = − β j for buses ( i, j ) ∈ L C . This controllability can be used to e.g.reroute power to reduce congestion in case of different forecasterror realizations. The GDF matrix Ψ is modified as follows: Ψ = (cid:20) I − ( α + β ) Hdiag ( γ ) (cid:21) (14)The HVDC participation factors β are nonzero only for theconverter connected AC buses and its sign depends on whichend of the HVDC line the AC bus is connected to. Similarto the engineering constraints of the AC grid, the converterlimits need to be considered as chance constraints in order tonsure secure operation with sufficient probability throughoutthe uncertainty range, e.g., P ( − m p,c S nom C,c ≤ P C,c + β c ω ) ≥ − ǫ ∀ c ∈ C , (15a) P ( m p,c S nom C,c ≥ P C,c + β c ω ) ≥ − ǫ ∀ c ∈ C . (15b)These can be reformulated for each converter c ∈ C : − m p,c S nom C,c + Φ − (1 − ǫ ) p β c β c T ≤ P C,c , (16a) m p,c S nom C,c − Φ − (1 − ǫ ) p β c β c T ≥ P C,c . (16b)Note that the uncertainty margins λ P C introduced in (16a)and (16b) depend linearly on the HVDC participation factor β . The degree of controllability is determined by α and β , bothof which can be either pre-determined or optimized within thechance-constrained AC-OPF.IV. I TERATIVE C HANCE - CONSTRAINED
AC-OPF
OPTIMIZING G ENERATOR AND
HVDC
CONTROL POLICIES
The reformulated chance-constrained AC-OPF (17a) – (17e)considering HVDC lines extends the variable set x to includethe active and reactive power set-points of the HVDC convert-ers [ P C , Q C ] : min x c T2 P + c T1 P G + c (17a)s.t. f ac ( x ) = 0 (17b) f dc ( P C ) = 0 (17c) x ≤ x max − λ x ( α, β ) (17d) x ≥ x min + λ x ( α, β ) (17e)If the corrective control actions provided by conventionalgenerators and HVDC lines are optimized within the sameframework, the participation factors α and β extend the vari-able set x and the following additional equations are included: β i = − β j ∀ ( i, j ) ∈ L C (18a) α k ≥ ∀ k ∈ G (18b) X k ∈G α k = 1 (18c)The problem (17a) – (17e) introduces for each HVDC linean additional power balance equation (17c) according to(13) considering the losses in the DC system. All inequalityconstraints are tightened with their corresponding uncertaintymargins λ x ( α, β ) = [ λ P G , λ Q G , λ V , λ P L , λ P C ]( α, β ) . Theuncertainty margins do not only depend on the generatorand HVDC participation factors but also on the Jacobianmatrix of the AC power flow equations as can be observedin (9a) and (9b). Including the Jacobian terms as optimizationvariables would introduce even more non-linearities in the AC-OPF and thus, substantially increase the complexity of theproblem. To this end, the authors in [24] have introduced acomputationally efficient iterative solution algorithm, whichdecouples the uncertainty assessment (i.e., the derivation ofthe uncertainty margins) from the optimization. Algorithm 1
Iterative Chance-Constrained AC-OPF Optimiz-ing Generator and HVDC Corrective Control Policies Set iteration count: k ← initialize λ x , = while || λ x ,k − λ x ,k − || ∞ > ρ do if k = 0 then solve (17a) – (17e) for x \ { α, β } and obtain x opt evaluate Jacobian at x opt else include λ x ,k ( α k , β k ) according to (17d) and (17e) solve (17a) – (17e), (18a) – (18c) to obtain x k opt evaluate Jacobian, Γ k opt and λ k opt at x k opt , α k opt and β k opt end if derive expressions for Γ x ,k +1 and λ x ,k +1 as functionsof optimization variables α k +1 and β k +1 k ← k + 1 end while .To maintain computational efficiency, we extend the iter-ative framework of [24] and evaluate the Jacobian at eachiteration for the current operating point. In [24], the uncertaintymargins were constants and were computed in an outer itera-tion. In the current paper, the sensitivity factors are constants,while α and β are kept as optimization variables, which allowsus to optimize these at the expense of adding non-linear(but convex) second order cone (SOC) terms. We define thesteps in Algorithm 1, where subscript opt denotes the optimalsolution of an OPF. The algorithm converges as the change inuncertainty margins between two consecutive iterations fallsbelow a defined tolerance value ρ .If we include the participation factors as optimization vari-ables in the iterative solution algorithm, the right hand sidesof the inequalities (17d)-(17e) are a non-linear function ofthe participation factors in the OPF problem. As a result, thecomputational complexity is increased. To maintain scalability,we propose to use a constraint generation method to solve theAC-OPF in each step of Algorithm 1 based on [25]: First, wesolve the AC-OPF excluding all uncertainty margins (i.e. theyare set to zero), except the uncertainty margins for the gener-ators (2c) – (2d) and the HVDC active power (16a) – (16b).Note that for these constraints, we can simplify the uncertaintymargin to a linear function in the participation factors andincluding these is computationally cheap. Then, based on theOPF results, we iteratively evaluate all uncertainty margins forthe optimized values of the participation factors. Only thoseinequality constraints in (17d)–(17e), which are violated forthe current optimized state variables and participation factorsare included in the next OPF problem. The OPF problem isresolved until the solution complies with all constraints (17d)–(17e). As we will show in Section V-D, this allows us to reducethe number of considered uncertainty margins significantly andmaintain scalability of our approach.In case the actual true uncertainty distribution cannot bewell captured by a Gaussian distribution or only limited fore-cast data is available, it is possible to formulate distribution-lly robust versions of the chance constraints. An increasingnumber of papers consider distributional robustness, includinge.g. [12]–[15]. Distributional robustness can be understoodin terms of the ambiguity regarding the parameters of thedistribution [12], [14] or regarding the type of distribution [13].Different types of ambiguity and associated uncertainty setswill result in different problem reformulations, which may bemore or less tractable. The work in [15] uses the Wassersteinmetric as distance measure between probability distributionsand proposes a tractable formulation of the chance constrainedAC-OPF assuming that the true probability distribution iswithin a defined Wasserstein distance to the empirical distri-bution based on data samples. Note that some distributionallyrobust approaches, such as the one presented in [13], allowsfor a similar reformulation of the individual chance constraints(2c)–(2j). Essentially, it is possible to obtain valid chance-constraint reformulation for any random variables with finitemean and covariance by replacing Φ − (1 − ǫ ) by a different(constant) function. This will lead to a more conservative, butsafe solution. V. S IMULATIONS AND R ESULTS
We specify the simulation setup. In the first part, we showthe benefit of optimizing the generator participation factorsfor the proposed iterative chance-constrained AC-OPF for a10 bus system. In the second part, we include an HVDC linein this system to relieve congestion in the AC system andinvestigate optimizing both the generator and HVDC controlpolicies and, in addition, the convergence behaviour of theiterative solution algorithm. In the third part, we consider anIEEE 39 bus system and evaluate the benefit of controllability.
A. Simulation Setup
To evaluate the performance of the proposed approacheswe use two metrics. First, we compute the cost of uncertaintywhich is the increase in generation cost by including chanceconstraints. Let f denote the objective value of the AC-OPFwithout considering uncertainty, i.e. all uncertainty marginsare set to zero: λ x = 0 . Note that we will refer to thisOPF problem as AC-OPF ( λ x = 0 , w/o uncertainty) in thefollowing. Let f U denote the objective value of the chance-constrained AC-OPF, i.e. all uncertainty margins are computedaccording to the presented iterative solution algorithm, and theOPF formulation takes into account the uncertainty in the windpower injections. Then, we can compute the cost of uncertaintyas follows: Cost of Uncertainty = f U − f f × (19)The cost of uncertainty is expressed in percent and is alwayslarger or equal to zero as the resulting tightening of theright hand side of (17d) and (17e) shrinks the OPF feasiblespace. Second, we perform an in- and out-of-sample analysisto compute the empirical individual chance constraint viola-tion probability. To determine the mean and variance of theGaussian distribution, we use a limited amount of samplesfrom realistic wind forecast data. For the in-sample analysis ˜ = ˜ =C1G5G1G4 W1 Fig. 3. 10 bus system with two wind farms located at buses 4 and 10. AnHVDC line (marked in blue) replaces the congested AC line (marked in red)between buses 2 and 10. we draw 10’000 samples from this Gaussian distributionand evaluate the performance (i.e. the occurring constraintviolations) of our proposed OPF formulation. For the out-of-sample analysis we use 10’000 samples from same databaseof realistic wind forecast data. This allows a first assessmentof how our proposed OPF formulation performs if the windrealizations do not exactly match a Gaussian distribution. Forthe in- and out-of-sample Monte Carlo Analysis we assume aminimum violation limit of 0.1% to exclude numerical errors.Note that for each type of individual chance constraint, wereport the maximum observed empirical violation probability.To compute the constraint violations, we use AC power flowsin MATPOWER [33]. The maximum allowable constraintviolation limit is set to ǫ = 5% . We consider a convergencecriterion of ρ = 10 − . All simulations are carried out ona laptop with processor Intel(R) Core(TM) i7-7820HQ CPU2.90 Ghz and 32GB RAM. The optimization problems areimplemented with YALMIP [34] in MATLAB and are solvedwith IPOPT [ ? ]. The wind farm power factor γ is set to 1.The 10 bus system which is considered in the followingfirst two subsections is shown in Fig. 3. The grid parametersare provided in [35]. The generator at bus 3 is selected tobe the slack bus. Upper and lower voltage limits of . p.u.and . p.u. are assumed. As we consider the active branchflow limit we set the maximum active branch limit to ofthe apparent branch flow limit. In this system configuration,the flow of power is from the upper left to the main loadunits at buses 7 to 10 and the transmission line from bus2 to bus 10 is congested. Two wind farms are located atbuses 10 and 4 with a maximum power of 1.0 GW and of2.5 GW, respectively. To compute the covariance matrix Σ ofthe forecast errors, we use realistic day-ahead wind forecastscenarios from [36]. The forecasts are based on wind powermeasurements in the Western Denmark area from 15 differentcontrol zones collected by the Danish transmission systemoperator Energinet. We select control zone 7 and 9 at time step4 to correspond to the wind farms at bus 2 and 10, respectively. ABLE IE
MPIRICAL CONSTRAINT VIOLATION PROBABILITY FOR BUS TESTCASE WITHOUT
HVDC
LINE
Constraint limits on P G Q G V P L In-sample analysis with 10’000 samples (%)AC-OPF ( λ x = 0 , w/o uncertainty) 49.0 0.0 6.7 49.7CC-AC-OPF (fixed α ) 5.3 0.0 2.8 5.3CC-AC-OPF (opt. α ) 4.9 0.0 2.9 4.9Out-of-sample analysis with 10’000 samples (%)AC-OPF ( λ x = 0 , w/o uncertainty) 43.2 0.0 4.6 49.2CC-AC-OPF (fixed α ) 5.8 0.0 3.4 6.1CC-AC-OPF (opt. α ) 5.8 0.0 3.4 5.6 In order to construct the covariance matrix we draw 100random samples from this data. The forecasted wind infeed iscomputed as the mean of these 100 samples. Note, for the 10bus system, due to the small system size, we do not employ theconstraint generation method proposed in Section IV but wedirectly solve the OPF problem with all uncertainty marginsincluded. For the 39 bus system we employ the constraintgeneration method to maintain scalability.
B. Optimization of Generator Participation Factors
In this section, for the 10 bus test case, we show the benefitin terms of generation cost of optimizing the generator par-ticipation factors α instead of assigning uniform participationfactors. The fixed participation factors are chosen to be α =[0 . . . . . , i.e. each generator equally compensatesthe deviation in wind power. We compare the performanceof an AC-OPF without considering uncertainty, the iterativechance-constrained AC-OPF (CC-AC-OPF) with fixed gener-ator participation factors and the latter (CC-AC-OPF) withoptimizing the generator participation factors. For the 10bus test case, the overall dispatch cost without consideringuncertainty (uncertainty margins set to zero) is . × h ,with considering uncertainty and fixed participation factors is . × h and with optimizing the participation factors is . × h . As a result, the cost of uncertainty evaluates to . for fixed participation factors. This can be reduced to . by optimizing the participation factors. The number ofiterations for fixed α is 5 and for variable α is 6. The averagesolving time for the AC-OPF iteration is 0.4 seconds for fixed α and 0.9 seconds for optimizing α as the computationalcomplexity is increased by the including α as optimizationvariable in the uncertainty margins (17d)–(17e).The results for the Monte Carlo Analysis for in- and out-of-sample testing are shown in Table I. Both in the in-and out-of-sample analyses the AC-OPF without consideringuncertainty leads to large empirical violation probabilities forthe active generator limits and the active branch flow limitsas the response of generators to the wind power deviationsis not considered. Voltage violations are observed as well. Incase we use the proposed iterative chance-constrained AC-OPF with fixed and optimized generator participation factorswe reduce the empirical violation probability both in- andout-of-sample very close to the desired . The remaining G1 G2 G3 G4 G5 . . . . (a) – Normalized generation dispatch P G / P m a x G ( − ) G1 G2 G3 G4 G5 (b) – Uncertainty margins for active power λ P G ( M W ) AC-OPF CC-AC-OPF (fixed α ) CC-AC-OPF (opt. α )Fig. 4. A comparison of (a) normalized generation dispatch and (b) uncer-tainty margins for active power for AC-OPF without considering uncertaintyand the chance-constrained AC-OPF with fixed and optimized generatorparticipation factors. Note that lower active limits of all generators is zero.TABLE IIE MPIRICAL CONSTRAINT VIOLATION PROBABILITY FOR BUS TESTCASE WITH
HVDC
LINE
Constraint limits on P G Q G V P L P C In-sample analysis with 10’000 samples (%)AC-OPF ( λ x = 0 , w/o uncertainty) 50.5 0.0 45.3 12.4 0.0CC-AC-OPF (fixed α and β ) 5.1 0.0 3.8 3.8 0.0CC-AC-OPF (opt. α and β ) 0.9 0.0 3.9 3.5 0.0CC-AC-OPF (mod.) 4.8 0.0 2.0 3.8 4.6Out-of-sample analysis with 10’000 samples (%)AC-OPF ( λ x = 0 , w/o uncertainty) 43.2 0.0 47.8 11.5 0.0CC-AC-OPF (fixed α and β ) 5.8 0.0 3.4 3.9 0.0CC-AC-OPF (opt. α and β ) 0.4 0.0 3.2 3.8 0.0CC-AC-OPF (mod.) 5.7 0.0 1.0 4.6 4.0 minor mismatch can be either attributed to a wrong estimationof the mean and covariance in the out-of-sample analysisor to the approximation we make by using the first-orderTaylor expansion to linearize the system behaviour around theforecasted operating point. Note that the forecast errors drawnfrom the realistic forecast data are not Gaussian distributed andthe observed violations out-of-sample can therefore be larger.However, we observe that they are still close to the desired indicating good performance of the proposed algorithm.If we optimize the generator participation factors, we obtain α opt = [0 . .
30 0 .
57 0 . . . In Fig. 4 we compare thegeneration dispatch and the uncertainty margins for the threeformulations. We can observe that by optimizing the participa-tion factors the generator response is shifted to the generatorsG2, G3 and G5 with mainly generator G3 compensating thewind power mismatch. The cheap generators G1 and G4operate at their maximum power output for the forecastedsystem operating state. This significantly reduces the cost ofuncertainty from 2.03% to 0.39% while maintaining systemreliability. . Including HVDC Line and HVDC Control Policies We replace the AC line between buses 2 and 10 in Fig.3with an HVDC line of S nom C = 4 GVA, and investigate therelief of congestion and decrease of the cost of uncertainty.We assume the converters are of the multi-modular converter(MMC) technology and that the total losses per converterstation are approximately c = 1% per HVDC converteraccording to [37], and for the active and reactive powercapability of the converter the limits are chosen as m P = 0 . , m min q = 0 . , m max q = 0 . [31]. The generation cost for the AC-OPF without considering uncertainty is decreased by 4.3% to . × h due to upgrading the AC to the HVDC lineand thereby reducing the congestion level of the system. Incase we again assume fixed generator participation factors α = [0 . . . . . and HVDC participation factor β = 0 , the cost of uncertainty amounts to 2.2%. By optimizingboth the generator and HVDC participation factors, the cost ofuncertainty can be reduced to 0.0%, i.e. the available HVDCand generator controls are sufficient to absorb the uncertaintyassociated with the two wind farms without any cost increase.The number of iterations for both fixed and variable α and β is 6. The average solving time for the AC-OPF iteration is 0.4seconds for fixed α and β and is 1.6 seconds for optimizing α and β , indicating that the computational complexity is furtherincreased by considering β as an optimization variable.In Table II, the empirical constraint violation probabilityfor an AC-OPF without considering uncertainty, an iterativeCC-AC-OPF with fixed α and β and an iterative CC-AC-OPF with optimized α and β is shown. We observe againthat without considering uncertainty, large violations of thegenerator active, voltage, and active branch flow limits occur.Both the CC-AC-OPF with fixed and optimized α and β achieve a satisfactory performance in- and out-of-sample. Forthe considered test case, the optimized generator participationfactors evaluate to α = [0 . . . . . and the optimizedHVDC participation factor β evaluates to 0.1032.In Fig. 5 the uncertainty margins and participation factorsfor each iteration of the chance constrained AC-OPF frame-work are shown for the 10 bus test system with one HVDCline. The participation factors are optimization variables. Notethat in the first iteration, the Jacobians are not available. Wecan observe that after the second iteration the uncertaintymargins do not vary significantly showcasing the robustness ofthe iterative solution framework. The convergence behaviourof the iterative solution algorithm without considering theparticipation factors as optimization variables is investigatedin detail in [38].To investigate the ability of the introduced framework tocomply with the chance constraints on the active HVDCconverter set-points (16a) – (16b), we consider a modifiedsetup, where the HVDC line capability S nom C is reduced to2 GVA, resulting in congestion on the HVDC line. We assigna fixed participation factor of β = 0 . to this HVDC line, andallow for an optimization of the generator participation factors α . The resulting empirical violation probability is shown in λ P G ( M W ) G1 G2G3 G4G5 λ Q G ( M V a r) G1 G2G3 G4G5 · − λ V ( p e r un it ) λ P L ( M W ) λ P C ( M W ) C1/C2 − . . Iterations ( α , β ) α α α α α β Fig. 5. Uncertainty margins λ and participation factors α , β for each iterationof the chance constrained AC-OPF framework for the 10 bus test system withone HVDC line. The participation factors as optimization variables.1.2 1.4 1.6 1.8 P C, (GW) O cc u rr e n ce s P C, (GW)Fig. 6. Histograms of the in-sample (left) and out-of-sample (right) MonteCarlo analysis for the active HVDC converter injection P C, at bus 2 for10’000 samples. Note, that both in- and out-of-sample the empirical violationprobability (4.6%, 4.0%) complies with the target value of 5%. The red dashedline indicates the maximum active power limit of the HVDC converter. Table II with the entry CC-AC-OPF (mod.) and achievessatisfactory performance as well. Note, that both in- and out-of-sample the empirical violation probability of the HVDCchance constraints (4.6%, 4.0%) complies with the target valueof 5%. This is confirmed in Fig. 6 which shows a histogramof the in- and out-of-sample analysis for the HVDC converteractive power injection P C, at bus . D. IEEE 39 bus New England system
In the following, we investigate the performance of ourproposed iterative chance constrained AC-OPF algorithm onan IEEE 39 bus New England system with 2 HVDC lines and2 wind farms. We obtained the system data from the IEEEPES PGLib-OPF v19.01 benchmark library [39]. We placetwo farms at buses 4 and 16 with a maximum power of 0.5
ABLE IIIE
MPIRICAL CONSTRAINT VIOLATION PROBABILITY FOR
IEEE 39
BUSTEST CASE WITH
LINES AND WIND FARMS
Constraint limits on P G Q G V P L P C In-sample analysis with 10’000 samples (%)AC-OPF ( λ x = 0 , w/o uncertainty) 49.5 49.3 5.3 51.3 0.0CC-AC-OPF (fixed α , β ) 4.9 4.2 0.0 5.5 0.0CC-AC-OPF (opt. α , β ) 4.2 3.3 0.0 5.1 4.8Out-of-sample analysis with 10’000 samples (%)AC-OPF ( λ x = 0 , w/o uncertainty) 41.6 58.0 1.3 43.7 0.0CC-AC-OPF (fixed α , β ) 4.1 0.0 1.7 4.4 0.0CC-AC-OPF (opt. α , β ) 4.1 0.0 1.2 4.3 4.2 GW and of 1.0 GW, respectively. The maximum wind powerinjection corresponds to 24.0% of the total system load. Tocompute the covariance matrix, and forecast errors, we followthe same procedure as for the 10 bus system. We select controlzone 7 and 9 at time step 4 to correspond to the wind farms atbus 4 and 16, respectively. We place two HVDC lines from bus4 to bus 30 and from bus 16 to bus 38 with S nom C = 500 MVA,respectively. We assume that only the generators at buses 30,32, and 36 have a non-zero participation factor α . We reducethe line limits to 80% to obtain a more congested system.For the remaining parameters not specified in [39] we keepprevious assumptions.First, we fix the participation factors to be equal inthe chance constrained AC-OPF, i.e. α = [
13 13 13 ] , and setthe HVDC participation factors to be zero, i.e. β = [0 0] .The overall dispatch cost without considering uncertainty is . × h . The cost of uncertainty for fixed participa-tion factors evaluates to 1.7%. If both the generator andHVDC participation factors are optimization variables, forthe considered test case, the optimized generator participationfactors evaluate to α = [0 . . . and the optimized HVDCparticipation factors evaluate to β = [0 . . . The utilizedcontrollability allows us to reduce the cost of uncertainty to0.7%. The average solving time for the iterative AC-OPFis 0.8 seconds with 4 iterations for fixed α and β and is1.6 seconds for optimizing α and β with 13 iterations. Notethat for this test case, we employ the constraint generationmethod explained in Section IV. We observe that only 8 outof the 146 possible uncertainty margins need to be included,thereby reducing the computational effort significantly. InTable III, the empirical constraint violation probability foran AC-OPF without considering uncertainty, the iterative CC-AC-OPF with fixed α and β and the iterative CC-AC-OPFwith optimized α and β is shown. We observe that withoutconsidering uncertainty, in this test case, large violations ofthe generator active and reactive power limits occur. Boththe CC-AC-OPF with fixed and optimized α and β achievea satisfactory performance in- and out-of-sample.VI. C ONCLUSIONS
In this work, we propose an AC optimal power flowformulation that a) considers uncertainty in wind power infeed,b) incorporates an HVDC line model and c) allows for an optimization of the generator and HVDC control response tofluctuations in renewable generation. For this purpose, we pro-pose a computationally efficient iterative chance-constrainedAC-OPF formulation extending [5], [24]. Using realistic windforecast data and a Monte Carlo Analysis, for 10 and IEEE39 bus systems with HVDC lines and wind farms, weshow that our proposed chance constrained OPF formulationachieves good in- and out-of-sample performance whereasnot considering uncertainty leads to high empirical constraintviolation probabilities. In addition, we find that optimizingthe participation factors reduces the cost of uncertainty signif-icantly. Our directions for future work are twofold: First, thepresented framework could be extended to take into accountinterconnected AC and HVDC grids, in particular DC buseswith multiple HVDC line connections. Second, data-drivenapproaches such as [40], [41] could be incorporated to includestability criteria (e.g. small-signal stability) in the chance-constrained OPF by encoding the feasible space using mixedinteger programming.A
CKNOWLEDGMENT
This work is supported by the multiDC project, fundedby Innovation Fund Denmark, Grant Agreement No. 6154-00020B. The authors would like to thank Pierre Pinson forsharing the forecast data.R
EFERENCES[1] E. Karangelos and L. Wehenkel, “Probabilistic reliability managementapproach and criteria for power system real-time operation,” in , June 2016, pp. 1–9.[2] P. Panciatici, M. C. Campi, S. Garatti, S. H. Low, D. K. Molzahn,A. X. Sun, and L. Wehenkel, “Advanced optimization methods for powersystems,” in , 2014.[3] D. Bienstock, M. Chertkov, and S. Harnett, “Chance-constrained optimalpower flow: Risk-aware network control under uncertainty,”
SIAMReview , vol. 56, no. 3, pp. 461–495, 2014.[4] L. Roald, F. Oldewurtel, T. Krause, and G. Andersson, “Analytical refor-mulation of security constrained optimal power flow with probabilisticconstraints,” in
IEEE PowerTech , Grenoble, France, 2012.[5] J. Schmidli, L. Roald, S. Chatzivasileiadis, and G. Andersson, “Stochas-tic AC optimal power flow with approximate chance-constraints,” in
IEEE Power and Energy Society General Meeting , Boston, US, 2016.[6] J. Warrington, P. J. Goulart, S. Mariethoz, and M. Morari, “Policy-basedreserves for power systems,”
IEEE Transactions on Power Systems ,vol. 28, no. 4, pp. 4427–4437, 2013.[7] R. A. Jabr, S. Karaki, and J. A. Korbane, “Robust multi-period opf withstorage and renewables,”
IEEE Transactions on Power Systems , vol. 30,no. 5, pp. 2790–2799, Sep. 2015.[8] M. Lubin, Y. Dvorkin, and S. Backhaus, “A robust approach to chanceconstrained optimal power flow with renewable generation,”
IEEETransactions on Power Systems , vol. 31, no. 5, pp. 3840–3849, Sep.2016.[9] ´A. Lorca, X. A. Sun, E. Litvinov, and T. Zheng, “Multistage adaptiverobust optimization for the unit commitment problem,”
OperationsResearch , vol. 64, no. 1, pp. 32–51, 2016.[10] R. Louca and E. Bitar, “Robust ac optimal power flow,”
IEEE Transac-tions on Power Systems , 2018.[11] X. Bai, L. Qu, and W. Qiao, “Robust ac optimal power flow for powernetworks with wind power generation,”
IEEE Transactions on PowerSystems , vol. 31, no. 5, pp. 4163–4164, 2016.[12] M. Lubin, Y. Dvorkin, and S. Backhaus, “A robust approach to chanceconstrained optimal power flow with renewable generation,”
IEEETransactions on Power Systems , vol. 31, no. 5, pp. 3840 – 3849, 2016.[13] L. Roald, F. Oldewurtel, B. Van Parys, and G. Andersson, “SecurityConstrained Optimal Power Flow with Distributionally Robust ChanceConstraints,”
ArXiv e-prints , Aug. 2015.14] Y. Zhang, S. Shen, and J. L. Mathieu, “Distributionally robust chance-constrained optimal power flow with uncertain renewables and uncertainreserves provided by loads,”
IEEE Transactions on Power Systems ,vol. 32, no. 2, pp. 1378–1388, 2017.[15] C. Duan, W. Fang, L. Jiang, L. Yao, and J. Liu, “Distributionally robustchance-constrained approximate ac-opf with wasserstein metric,”
IEEETransactions on Power Systems , vol. 33, no. 5, pp. 4924–4936, 2018.[16] M. Vrakopoulou, K. Margellos, J. Lygeros, and G. Andersson, “A prob-abilistic framework for reserve scheduling and N-1 security assessmentof systems with high wind power penetration,”
IEEE Transactions onPower Systems , vol. 28, no. 4, pp. 3885–3896, 2013.[17] M. Vrakopoulou, M. Katsampani, K. Margellos, J. Lygeros, and G. An-dersson, “Probabilistic security-constrained ac optimal power flow,” in . IEEE, 2013, pp. 1–6.[18] A. Venzke, L. Halilbasic, U. Markovic, G. Hug, and S. Chatzivasileiadis,“Convex relaxations of chance constrained AC optimal power flow,”
IEEE Transactions on Power Systems , vol. 33, no. 3, pp. 2829–2841,May 2018.[19] A. Venzke and S. Chatzivasileiadis, “Convex relaxations of probabilisticac optimal power flow for interconnected ac and hvdc grids,”
IEEETransactions on Power Systems , pp. 1–1, 2019.[20] V. Rostampour, O. ter Haar, and T. Keviczky, “Distributed stochasticreserve scheduling in ac power systems with uncertain generation,”
IEEETransactions on Power Systems , vol. 34, no. 2, pp. 1005–1020, 2019.[21] H. Zhang and P. Li, “Chance constrained programming for optimalpower flow under uncertainty,”
IEEE Transactions on Power Systems ,vol. 26, no. 4, pp. 2417–2424, 2011.[22] K. Baker, E. Dall’Anese, and T. Summers, “Distribution-agnosticstochastic optimal power flow for distribution grids,” in
North AmericanPower Symposium (NAPS) , Denver, US, 2016.[23] M. Lubin, Y. Dvorkin, and L. Roald, “Chance constraints for improvingthe security of ac optimal power flow,”
IEEE Transactions on PowerSystems , pp. 1–1, 2019.[24] L. Roald and G. Andersson, “Chance-constrained AC optimal powerflow: Reformulations and efficient algorithms,”
IEEE Transactions onPower Systems , vol. PP, no. 99, pp. 1–1, 2017.[25] L. Roald, S. Misra, T. Krause, and G. Andersson, “Corrective control tohandle forecast uncertainty: A chance constrained optimal power flow,”
IEEE Transactions on Power Systems , vol. 32, no. 2, pp. 1626–1637,2017.[26] M. Vrakopoulou, S. Chatzivasileiadis, and G. Andersson, “Probabilisticsecurity-constrained optimal power flow including the controllability ofhvdc lines,” in
IEEE PES ISGT Europe 2013 , Oct 2013, pp. 1–5.[27] R. Wiget, M. Vrakopoulou, and G. Andersson, “Probabilistic securityconstrained optimal power flow for a mixed HVAC and HVDC grid withstochastic infeed,” in
Power Systems Computation Conference , 2014.[28] K. Dvijotham and D. K. Molzahn, “Error bounds on the DC powerflow approximation: A convex relaxation approach,” in , Dec 2016, pp. 2411–2418.[29] D. K. Molzahn and I. A. Hiskens, “A Survey of Relaxations andApproximations of the Power Flow Equations,”
Foundations and Trendsin Electric Energy Systems , vol. 4, no. 1-2, pp. 1–221, February 2019.[30] L. Roald, “Optimization Methods to Manage Uncertainty and Risk inPower System Operation,” Ph.D. dissertation, ETH Zurich, 2016.[31] M. C. Imhof, “Voltage Source Converter based HVDC–modelling andcoordinated control to enhance power system stability,” Ph.D. disserta-tion, ETH Zurich, 2015.[32] J. Beerten, S. Cole, and R. Belmans, “Generalized steady-state VSCMTDC model for sequential AC/DC power flow algorithms,”
IEEETransactions on Power Systems , vol. 27, no. 2, pp. 821–829, May 2012.[33] R. D. Zimmerman, C. E. Murillo-S´anchez, R. J. Thomas et al. , “Mat-power: Steady-state operations, planning, and analysis tools for powersystems research and education,”
IEEE Transactions on power systems ,vol. 26, no. 1, pp. 12–19, 2011.[34] J. L¨ofberg, “Yalmip : A toolbox for modeling and optimization inmatlab,” in
In Proceedings of the CACSD Conference , Taipei, Taiwan,2004.[35] S. Chatzivasileiadis, T. Krause, and G. Andersson, “Flexible AC trans-mission systems (FACTS) and power system security - A valuationframework,” in
IEEE Power and Energy Society General Meeting ,Detroit Michigan, US, 2011.[36] P. Pinson, “Wind energy: Forecasting challenges for its operationalmanagement,”
Statistical Science , vol. 28, no. 4, pp. 564–585, 2013. [37] P. S. Jones and C. C. Davidson, “Calculation of power losses for MMC-based VSC HVDC stations,” in
Power Electronics and Applications(EPE), 2013 15th European Conference on . IEEE, 2013, pp. 1–10.[38] L. A. Roald, D. K. Molzahn, and A. F. Tobler, “Power system opti-mization with uncertainty and AC power flow: Analysis of an iterativealgorithm,” in ,2017.[39] The IEEE PES Task Force on Benchmarks for Validation of EmergingPower System Algorithms, “PGLib Optimal Power Flow Benchmarks,”Published online at https://github.com/power-grid-lib/pglib-opf, 2019.[40] L. Halilbaˇsic, F. Thams, A. Venzke, S. Chatzivasileiadis, and P. Pinson,“Data-driven security-constrained AC-OPF for operations and markets,”in
XX Power Systems Computation Conference, Dublin , 2018.[41] I. Konstantelos, G. Jamgotchian, S. H. Tindemans, P. Duchesne, S. Cole,C. Merckx, G. Strbac, and P. Panciatici, “Implementation of a massivelyparallel dynamic security assessment platform for large-scale grids,”