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Contents lists available at ScienceDirect

Electric Power Systems Research

journal homepage: www.elsevier.com

Multi-period stochastic security-constrained OPF considering the uncertainty sourcesof wind power, load demand and equipment unavailability

Hossein Sharifzadeh a, Nima Amjady a, Hamidreza Zareipour b,

a University of Semnan, Iranb University of Calgary, Canada

A R T I C L E I N F O

Article history:Received 2 September 2016Received in revised form 2 January2017Accepted 5 January 2017Available online xxx

Keywords:Optimal power flowSecurity constraintsUncertainty sourcesWind generationLoad demand

A B S T R A C T

The uncertainty sources of the intermittent generation and load demand as well as transmission line unavailability threatenthe security of power systems. In this paper, to treat these uncertainties, a new stochastic optimal power flow consid-ering the security constraints is proposed. A scenario generation method is also presented to model the uncertainties ofwind generations and load demands considering their correlations. In the proposed model, the uncertainties are copedwith through combination of optimal here-and-now and wait-and-see decisions. The effectiveness of the proposed modelis shown on the well-known IEEE 24-bus test system. Higher effectiveness of the proposed model compared with fourdeterministic methods and one other stochastic method to determine procured reserve and ‘after-the-fact’ conditions isnumerically illustrated. Additionally, the impact of the number of scenarios on the performance of the proposed model isevaluated by means of a sensitivity analysis. It has also been shown that the scenarios generated considering correlationshave more smooth variations and can more effectively capture the uncertain behavior of load.

© 2016 Published by Elsevier Ltd.

Nomenclature

Sets and indices

Set of network busesBus-generator incidence matrixBus-branch incidence matrixSet of non-islanding branch contingenciesBranch contingency indexSet of conventional generating unitsIndex of conventional generating unitsNetwork bus indicesThe set of scenarios including the same realizations ofthe uncertain variables from hour 1 to hour

S Set of wind generation and load demand (W&L) sce-narios

S′ Set of trial scenarioss, s1, s2 W&L scenario indexess′ Trial scenario indexT Number of time steps in the scheduling horizon

Index of time steps in the scheduling horizonIndex of wind generating units

Parameters

Corresponding author. Fax: +1 4032826855.

Email address: hzareipo@ucalgary.ca (H. Zareipour)

Offer cost for up and down spinning reserve capacityby conventional unit g for hour t, respectivelyMaximum flow limit of the branch connecting bus iand bus jMinimum and maximum generation limits of conven-tional unit gSpecified load demand of bus i in hour t and W&Lscenario s, obtained through the scenario generationmethodSpecified generation for wind unit w in hour t andW&L scenario s, obtained through the scenario gener-ation method

Maximum available down/up spinning reserve ca-pacity of conventional unit g in hour t

Up and down ramp rate limit, respectivelyLoad shedding cost for load i in hour tBranch reactance between buses i and jOffer cost for energy and deployed up and down spin-ning reserve by conventional unit g for hour t, respec-tivelyOffer cost of conventional unit g for up and down gen-eration shift in hour t, respectivelyProbability of contingency c, W&L scenario s, andtrial scenario s', respectively

Variables

http://dx.doi.org/10.1016/j.epsr.2017.01.0110378-7796/© 2016 Published by Elsevier Ltd.

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Power flow of the branch connecting bus i and busj in hour t and W&L scenario s, in normal state andpost-contingent state c, respectivelyScheduled generation of conventional unit g for hour tin normal stateGeneration of conventional unit g in hour t, W&L sce-nario s, and post-contingent state cGeneration of conventional unit g in hour t, W&L sce-nario s, and normal stateLoad shed of bus i in hour t, W&L scenario s, andpost-contingent state cScheduled generation of wind unit w in hour t, W&Lscenario s and normal stateScheduled generation of wind unit w in hour t, W&Lscenario s and post-contingent state cScheduled up and down reserve capacity of conven-tional unit g for hour t, respectivelyUp and down reserve of conventional unit g in hourt, W&L scenario s, and post-contingent state c, de-ployed in addition to , respectively, to

cope with the contingencyDeployed up and down reserve of conventional unit gin hour t, W&L scenario s, and normal state, respec-tively, to cope with the W&L uncertainty

Total deployed up and down reserve of conventionalunit g in hour t of trial scenario s′Up and down generation shift of conventional unit gin hour t of trial scenario s′, respectivelyLoad shed of bus i in hour t of trial scenario s′Generation of conventional unit g in hour t of trial sce-nario s′Wind spillage of wind unit w in hour t, W&L scenarios and normal state to cope with the W&L uncertaintyWind spillage of wind unit w in hour t, W&L scenarios and post-contingent state c, deployed in addition to

to cope with the contingency

Voltage angle of bus i in hour t and W&L scenarios, in normal state and post-contingent state c, respec-tively

1. Introduction

Continuously increasing fossil fuel prices and atmosphere pollu-tion as well as depletion of fossil fuel energy sources causes that re-newable green energy sources have a growing share in supplying hu-man’s energy demands. Wind power is the most widely used renew-able energy source in today’s electricity generation. However, the un-certain and variable nature of wind generation together with tradi-tional load forecasting error dramatically add the uncertainty sourcesand seriously challenge the optimal and secure operation of powersystems [1]. Moreover, load growth, change of generation patterncaused by renewable resources, and increased energy transactions inthe environment of competitive electricity markets increase the prob-ability of overload in the electric network, which in turn jeopardizethe system security [2]. Therefore, power system operators shouldmake secure their systems against different unexpected generation/load patterns and network configurations. However, the uncertaintysources seriously challenge the applicability of traditional optimal

power flow (OPF), which usually determines the operating state ofpower system assuming only its most likely operating conditions [3].

A few approaches have been presented in the literature to han-dle wind/load uncertainties in OPF models. The most important meth-ods include probabilistic techniques [4], chance constrained program-ming [5], robust optimization [6], and stochastic programming [7].Probabilistic methods tend to present the probability density functioncharacteristics of desired outputs instead of one aggregated optimalsolution. In chance constrained programming, constraints are satis-fied with some predetermined probabilities. Chance constrained pro-gramming leads to nonlinear problems, which are not easy to han-dle. While robust optimization (RO) is a professional uncertainty han-dling method for non-deterministic optimization problems involvinglow-frequency and non-random uncertain variables, setting the appro-priate uncertainty sets for RO may be a challenging task. Moreover,when the uncertainty sources of non-deterministic optimization prob-lem are in the form of high-frequency uncertain variables, RO maynot necessarily be able to effectively employ the whole statistical in-formation of these variables. Apart from the foregoing problems, noneof the mentioned research works considers the required preventive/corrective actions together with their associated costs, e.g., when theso-called N − 1 criterion is employed.

Inclusion of security constraints in the OPF problem leads to amore effective operation tool, usually known as the security con-strained OPF (SCOPF). However, SCOPF is a more computation-ally complex optimization problem than OPF as it includes signifi-cantly more constraints. Different numerical optimization approaches,such as iterative-based solution methods [8], Benders decomposition[9], and parallelism techniques [10], have been proposed to handlethe high dimensional SCOPF problem. Although various evolutionaryalgorithms have also been presented for solving SCOPF [11], thesemethods usually suffer from high computation burden to solve SCOPFfor practical power systems. Moreover, the effectiveness of evolu-tionary algorithms depends on the initial population and selected val-ues for their settings, while usually there is no analytical method tofine-tune these settings for solving SCOPF.

The main contributions of this paper can be summarized as:

1) A new multi-period security-constrained stochastic optimal powerflow (MPSC-SOPF) model taking into account load and wind gen-eration uncertainties as well as uncertainties associated with equip-ment unavailability is presented. The proposed model, based ontwo-stage stochastic programming, determines the optimal oper-ating point and required corrective actions considering probabil-ity of different load/wind generation scenarios and contingencystates. By simultaneously handling the corrective actions requiredfor coping with different uncertainty sources (including the uncer-tainties of load and wind as well as the uncertainties of equip-ment unavailability), a more effective reserve procurement strat-egy is achieved. This is the key issue which distinguishes the pro-posed MPSC-SOPF model from the previous stochastic OPF mod-els which determine the corrective actions required for handlingdifferent uncertainty sources separately.

2) A new scenario generation method composed of Latin hypercubesampling (LHS) and rank correlation is proposed. The proposedmethod can model the correlations between wind generations andload demands, correlations between generations of different windfarms, and inter-temporal dependencies.

3) A new out-of-sample analysis to evaluate the performance of sto-chastic models for ‘after-the-fact’ conditions in a sufficiently longrun is presented. In this way, the performance of a stochastic modelcan be better evaluated for unseen scenarios and realizations.

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2. The proposed MPSC-SOPF model

The objective function of the proposed MPSC-SOPF model in-cluding the costs of here-and-now, i.e. first stage decisions, andwait-and-see, i.e. second stage decisions, in the both normal and con-tingency states is as follows:

The objective function includes the cost of energy and up/down re-serve capacity procurement, i.e. the first summation term, plus the ex-pected cost of corrective actions, i.e. the second summation term. Thefirst summation term includes the cost of first stage decisions, i.e. thedecision variables that are independent of the scenarios. The secondpart of Eq. (1) denotes the costs of second stage variables, which de-pend on the realized scenario. The first summation in the second partof Eq. (1) represents deployed reserves’ cost in normal state to copewith the uncertainties stemmed from load/wind forecast errors. Thesecond summation of this part is the cost of corrective actions, includ-ing deployed reserves and load shedding, to eliminate overloads inpost-contingent states of branch outages. In the second part of Eq. (1),the scenarios modeling the wind generation and load demand (W&L)uncertainties are weighted by their probabilities . Similarly, the sce-narios pertaining to branch contingencies are weighted by their proba-bilities . The considered linear offer cost in the objective function isconsistent with the today’s electricity market practices in which partic-ipants submit their offering curves usually in single or multiple linearenergy–price blocks. The presented objective function considers sin-gle energy–price blocks but can be extended to multiple energy–priceblocks.

The constraints of the proposed model associated with the normaland post-contingent states are as follows.

A.) Normal state constraints

- Variables’ bounds

- Ramp rate limits

- Power flow equality constraints

- Branch flow limits

B.) Post-contingent states’ constraints

- Variables’ bounds

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- Power flow equality constraints

- Line flow limits

C.) Non-anticipativity constraints

Constraints (2)–(10) and (16)–(22) bound the variables to their al-lowable ranges in normal and post contingency states, respectively.Wind spillage is considered in Eq. (19) as a down reserve to enhancefeasibility of serious extra-generation scenarios and decrease total op-erating cost by reducing downward reserve procurement [12]. Con-straints (11) and (12) model the ramp rate limits of conventional gen-erators. DC power flow constraints are shown in Eqs. (13) and (14)and (23)–(24) in normal and post-contingent states, respectively. Con-straints (15) and (25) express the branch flow limits in normal andpost-contingent states, respectively. The non-anticipativity constraintsof (26)–(32) avoid from unreasonable decisions made in the stochas-tic program. These conditions state that two scenarios with the samehistory of the realized uncertain variables from hour 1 to hour t musthave the same decisions made in hour t [13].

The proposed MPSC-SOPF includes the first stage decision vari-ables of , and , and second stage decision variables of

, , , , , , and . The corrective

actions of the proposed model, appeared as the second stage decisionvariables, are divided to two categories. The first category includesthe corrective actions required for dealing with the uncertainty sourcesof wind generations and load demands, which consist of , ,

and . The second one contains , , , and

, adopted to cope with the uncertainty source of contingencies (loadshed is only allowed for treating contingencies). An important advan-tage of the proposed MPSC-SOPF model is that both categories of thesecond stage decision variables are simultaneously optimized alongthe first stage decision variables within it leading to lower requiredcorrective actions and their associated costs. For instance, both cate-gories of the deployed reserves, including ( , ) and ( ,

) are simultaneously optimized along the capacity of reserves (

and ) within the proposed MPSC-SOPF model. This leads tolower required total reserve deployment and capacity compared to theother works that separately determine the reserves required for copingwith the uncertainties of W&L and contingencies.

An OPF tool usually determines the final generation levels of com-mitted generation units which their off/on statuses have already beendetermined in the unit commitment (UC) problem [14]. What is crit-ical in the multi-period OPF compared to the single-period OPF isramp rate limits of generating units [6], which have been consideredin the proposed model as the constraints (11) and (12). However, if theon/off states of conventional units are considered, the model changesinto the security constrained UC (SCUC) model which is not the focusof this paper. Additionally, the proposed MPSC-SOPF model is basedon DC-OPF and thus control targets only involve generator and loadaspects instead of switched capacitors/reactors and transformer taps.As reviewed in Ref. [15], none of the ISOs in USA use the AC opti-mal power flow. Thus, DC optimal power flow is more consistent withthe current industry practice. The proposed MPSC-SOPF model canbe used as the sub-problem of the UC problem. Moreover, it can beused as the rescheduling tool to fine tune the UC decisions when moreexact forecasts are available in upcoming hours. Finally, it can be di-rectly used as the clearing mechanism in the OPF-based markets.

3. Proposed scenario generation and reduction

The proposed scenario generation approach produces scenariosbased on two important criteria:

1) The produced scenarios should be consistent with the stochastic be-havior of the uncertain variables.

2) Correlation should be considered in the scenario generation ap-proach to avoid producing infeasible scenarios.

Despite the recent developments in the load and wind power fore-cast methods, prediction error is an indispensable aspect of these fore-cast processes. These errors have important effects on the operationaldecisions of power systems. As the prediction error values are notknown in advance, load demand and wind power generation appear asuncertain variables. In the stochastic programming (SP) framework,the uncertain variables are modeled using their possible realizationscalled scenarios. Owning to the key role of scenarios on the qual-ity of SP model, various researchers have focused on the scenariogeneration techniques. Among these techniques, sampling methodshave some prominent advantages, such as simplicity and the abilityto resemble the probability distribution of stochastic variables. How-ever, generally, they need to a large number of samples to effec

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tively represent the original distribution. Among the sampling-basedmethods, stratified sampling techniques, such as Latin hypercube sam-pling (LHS), have higher efficiency and require lower number of sce-narios with respect to random sampling [16,17]. Accordingly, thismethod is adopted in this paper to generate the scenarios for theMPSC-SOPF model. The main feature of LHS technique is to di-vide the probability space of stochastic variables to some intervals,so-called ‘strata’, and sample from these intervals to efficiently coverthe probability space of stochastic variables. However, despite the pre-vious LHS-based sampling methods [16,17], the proposed scenariogeneration approach does not rely on the predetermined probabilitydistributions for the uncertain variables. Instead, the proposed ap-proach uses the relative frequency of the stochastic variables. In thisway, the proposed approach can avoid from the errors of approximat-ing the uncertain nature of stochastic variables by predetermined prob-ability distributions. Additionally, the performance of LHS methoddepends on the selected strata, which should be consistent with therelative frequency of the uncertain variables. The proposed scenariogeneration approach can directly produce such intervals. On the otherhand, the previous LHS-based sampling methods, which fit the data toa predetermined distribution and then partition it, may produce someintervals that are inconsistent with the relative frequency of the uncer-tain variables.

The proposed LHS-based scenario generation method can be sum-marized as the following algorithm, its flowchart is shown in theAppendix A:

1) Suppose that the number of the uncertain variables is NV. Acounter nv is initialized as: nv = 1. Also, let be the number of re-quired scenarios.

2) The probability domain of the uncertain variable nv, i.e. [0,1], ispartitioned to n equal intervals.

3) The uncertain variable nv is clustered using k-means clusteringtechnique based on its historical data [18]. The relative frequencyof each produced cluster is computed. Using these clusters andtheir relative frequencies, a discrete cumulative relative frequencydistribution is constructed for the uncertain variable nv.

4) Produce random numbers with uniform distribution in therange of [0,1]. Using each produced , probability of the sampleassociated with kth probability interval, denoted by , is deter-mined as follows:

Note that is a random number with uniform distribution in the kthprobability interval, i.e. .

5) The intersection point of each with the discretecumulative relative frequency distribution (constructed in step 3) isdetermined. Suppose that the intersection point belongs to the hthinterval of the distribution. The kth sample of the uncertainvariable nv, denoted by , is generated as below:

where is a random number with uniform distribution in the range[0,1], separate from . In this way, n samples for the uncertain vari-able nv are generated such that each sample is randomly generatedwith uniform distribution within one interval of the probability do-main. Accordingly, the proposed scenario generation method can

effectively cover the uncertainty spectrum of every uncertain variable.

6) Increment nv: nv = nv + 1. If nv > NV go to the next step. Other-wise go back to step 2.

7) The n samples generated for every uncertain variable are randomlyarranged in one column of a matrix, denoted by R, with the dimen-sions of n × NV. Each row of the matrix R presents one scenarioproduced by the proposed approach for the stochastic program. To-tally, n rows or n scenarios are given by R.

Due to the correlation between wind farm generations as wellas the correlation between wind farm generations and power systemloads, mostly because of their dependency on the weather conditions,their associated random variables have statistical correlation. Addi-tionally, loads and wind generations of successive hours have corre-lation. Ignoring these correlations in the scenario generation processmay lead to infeasible generated scenarios [12]. One of thewell-known approaches to consider correlation in scenario generationis based on the Cholesky decomposition [19]. However, as discussedin Ref. [19], this method may distort ‘strata’ made by LHS tech-nique. Moreover, this method cannot effectively reproduce the origi-nal distribution of stochastic variables in non-normal distributions. Tocope with this problem, an effective technique called ‘rank correla-tion’ (having advantages, such as simplicity, independency of distri-bution, and applicability to any sampling approach) is adopted in theproposed scenario generation method. The proposed LHS-based sce-nario generation approach, incorporating rank correlation (RC) tech-nique to consider correlation between stochastic variables, denoted byLHS-RC, can be described as the following steps:

1) Let the correlations between the columns of the matrix , gener-ated by the LHS algorithm, is denoted by the matrix . Also, sup-pose that the matrix illustrates the actual correlation matrix be-tween the uncertain variables. Note that the samples of the uncer-tain variables, i.e. the columns of the matrix R, are independentlygenerated by the LHS algorithm and so their possible correlations,represented by the matrix A, do not necessarily correspond to theactual correlations included in the matrix B.

2) Let and be the lower triangular matrix of and obtainedby Cholesky decomposition, i.e. and .

3) Compute . The columns of the matrix will

have the same correlations represented by the matrix (i.e., the ac-tual correlations) [19].

4) Sort the elements of each column in based on the rank (order) ofthe elements of the corresponding column in the matrix . It canbe mathematically shown that the sorted columns of R will havethe same correlations of the matrix B [20].

The scenarios generated by the proposed LHS-RC can effectivelyrepresent both the individual stochastic behavior of the uncertain vari-ables (based on the uniform distribution of the samples within the un-certainty spectrum of every uncertain variable) as well as the corre-lations between them (based on the RC technique). Note that the RCtechnique does not apply any transformation to the matrix R and onlychanges the sorting of its columns.

The proposed LHS-RC generates scenarios that model the individ-ual and joint stochastic behaviors of W&L uncertainties, i.e. the sce-narios . To model the uncertainty source of branch unavailabil-ity, the scenarios pertaining to branch contingencies or aregenerated based on criterion and reduced using the contingencyfiltering approach of Ref. [9]. This contingency filtering approachfinds a subset of important contingencies among the set of all consid-ered contingencies (here, the set ) and filters out the remaining un

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necessary contingencies. To do this, the performance of the contin-gency filtering approach can be summarized as the following steps:

Step 1) The contingency filtering approach first sets the subset ofimportant contingencies to null set.

Step 2) Solve the MPSC-SOPF problem given in Eqs. (1)–(32)only considering the subset of important contingencies.

Step 3) Check the solution obtained from Step 2 for all consid-ered contingencies. If the obtained solution from Step 2 does not leadto any branch overload for all considered contingencies, the contin-gency filtering approach is terminated and the obtained solution withthe subset of important contingencies is adopted; otherwise go to thenext step.

Step 4) Among the contingencies that lead to branch overloads, se-lect the contingencies that lead to the highest branch overloads for atleast one branch and filter out the other ones. Add the selected contin-gencies to the subset of important contingencies and go back to Step2.

4. Simulation results

Performance of the proposed MPSC-SOPF model and LHS-RCscenario generation method is evaluated on the IEEE 24-bus test sys-tem whose data is presented in Ref. [12]. Two wind farms are alsoconnected to the buses 1 and 2 of the test system. The required datafor load demand and wind generation levels as well as their forecastsare obtained from Ref. [21] and scaled down based on the peak loadratios to fit to the IEEE 24-bus test system. One system and two windpower generations have been considered due to available forecast data[21]. Time steps of the scheduling horizon, the number of scenarios

and number of contingencies are assumed to be ,, and , respectively. The IEEE 24-bus test system has

38 branches. Since the outage of one of these 38 branches leads toislanded network, we have omitted this branch from the contingencyset . Thus, the total number of sample scenarios included in theMPSC-SOPF model is . Considering one system load,two wind generations and , there are (1 + 2) × 6 = 18 correlateduncertain variables. Thus, the correlation matrices A and B of the pro-posed LHS-RC for this test case become 18 × 18. It is noted that theproposed MPSC-SOPF model can be used for longer scheduling hori-zons, such as .

The up and down spinning reserve capacities of conventional unitsare limited to 10% of their output power ranges, i.e.

. The load shedding cost is con-

sidered to be [12]. Cost of up/down deployedreserve of each unit is adopted as its offer cost for energy, i.e.

. Up and down reserve capacity cost is assumed to be10% of the offer cost for energy, namely . The LPproblem of the proposed MPSC-SOPF model is solved using CPLEXsolver within GAMS software package [22]. The run time of LHS-RCand MPSC-SOPF, measured on the simple hardware set of Core i72.20 GHz laptop computer with 8 GB RAM memory, is about 3 sand 11 s, respectively, for this test case. These run times are accept-able within a six-hour-ahead decision making framework for the IEEE24-bus test system. Three solutions have been considered in the pro-posed framework to decrease its computation burden:

1) The proficient method of LHS has been used to generate scenarios[16,17].

2) The generated scenarios are decreased to the desired number usinga “Backward/Forward” process available as the “SCENRED” toolin GAMS [22] to reduce the problem size.

3) An efficient solution approach based on a non-dominated contin-gency filtering concept [9] is implemented to decrease the problemsize and solve the proposed model in an acceptable time.

To show the effectiveness of the proposed model, the following ex-perimental test cases are considered:

Case 0All uncertainty sources are ignored (i.e. deterministic counter-

part).

Case 1Only branch availability uncertainty is included.

Case 2Only load demand uncertainty is considered.

Case 3Only wind generation uncertainty is included.

Case 4Wind generation and load demand uncertainty sources are

taken into account.

Case 5All three uncertainty sources, i.e. wind generation, load de-

mand, and branch availability, are considered.

The results obtained for the objective function, i.e. total cost, andsum of all procured reserve volume for h, i.e.

, in these experimental test cases are shown

in Table 1. It can be seen that considering the uncertainty sources in-creases the operation costs due to changing the operating point and ad-ditional reserves required. Comparison of the objective function val-ues demonstrates an additional incurred cost of 10.55% in case 5, in-dicated by bold font in Table 1, with respect to case 0.

One of the key advantages of the proposed model can be inferredfrom the procured reserve capacity reported in case 5 in which allthree mentioned uncertainty sources are taken into account. If thereserve requirements are determined by three separate optimizationprocess of cases 1–3, the total reserve requirement will be the sum of3.285 + 5.431 + 7.651 = 16.376 p.u. Note that each of case 1, case 2,and case 3 deal with a different uncertainty source including branchavailability uncertainty in case 1, load demand uncertainty in case 2,and wind generation uncertainty in case 3. Each of these uncertaintysources requires its own reserve. In other words, we may encountersome load forecast error and some wind generation forecast error, andat the same time some branches of the system become unavailabledue to contingencies. Thus, we may need the sum of reserves deter-mined for case 1, case 2, and case 3, simultaneously. This is whatwe typically encounter in real-world power systems. This reserve re-quirement is considered as a constraint in the optimization problem.In other words, the total reserve requirement of 16.376 p.u. should beprovided. This is significantly higher than the procured reserve by the

Table 1Results obtained in different experimental test cases.

Test case Objective function ($) Procured reserve (p.u.)

Case 0 83545.49 –Case 1 85991.97 3.285Case 2 85496.37 5.431Case 3 85647.03 7.651Case 4 89634.37 11.822Case 5 92355.56 12.810

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proposed MPSC-SOPF, i.e. case 5 (12.810 p.u.), which simultane-ously optimizes all reserves required for coping with the uncertaintysources. As an alternative approach, if the reserves required for treat-ing the W&L uncertainty sources are determined simultaneously by anoptimization process, as given in case 4, and then added to the branchfailure reserve requirement, obtained in case 1, the total procured re-serve will decrease to 3.285 + 11.822 = 15.107 p.u., which is still con-siderably more than the required reserve in case 5, i.e. 12.810 p.u.

Apart from the procured reserve capacity, the proposedMPSC-SOPF approach determines a more optimal operating pointcompared to the other approaches that rely on predetermined reserverequirements in their formulation or separately handle different uncer-tainty sources. This issue is discussed in the following.

Fig. 1 shows the scheduled generation as well as preventive/cor-rective actions determined by the proposed MPSC-SOPF for the firsthour of the scheduling period in case 5. Preventive actions include ap-plied generation shifts compared to case 0 (without uncertainty) andcorrective actions consist of procured reserve capacities. All of thesecontrol variables are simultaneously optimized within the proposedapproach considering the uncertainty sources.

In order to show the effectiveness of the proposed MPSC-SOPFmodel for determining optimal operating point and reserve require-ments considering the uncertainty sources of power system, it is com-pared with four other deterministic models and one other stochasticmodel. Unlike the proposed stochastic method, deterministic meth-ods cannot model uncertainty sources and determine the required re-serves based on them. Thus, these methods should have a criterionto determine the required reserve. Handling W&L uncertainty sourcesin deterministic approaches is often based on the forecast error stan-dard deviation [23]. Higher forecast error standard deviation meanshigher dispersion of the forecast error probability density function andso higher forecast errors may be occurred in practice. Thus, morereserve is required for higher forecast error standard deviation. Inthis paper, to implement these four deterministic reserve procurementmethods, the standard deviation of the forecast error pertaining tonet load, i.e. total load demand minus sum of the wind generations,is computed and denoted by . Then, the four deterministic ap-proaches, denoted by DA1, DA2, DA3, and DA4, are constructedadopting , , , and as W&L spinningreserve capacity, respectively. Among these four methods, DA1 il-lustrates the least conservative deterministic method and DA4 illus-trates the most conservative deterministic method. In this way, we cancompare the proposed stochastic method with different deterministicmethods with different levels of conservativeness. The comparative

stochastic approach (SA) is implemented based on Ref. [24]. This SAcan only model W&L uncertainty. Thus, the contingency reserve ca-pacity requirement for this SA as well as DA1, DA2, DA3, and DA4 isdetermined using corrective SCOPF [9] such that the system remainssecure for all single branch contingencies. Total reserve capacity ofSA as well as DA1, DA2, DA3, and DA4 is obtained as the sum oftheir W&L and contingency reserve requirements. The objective func-tion value and total reserve capacities obtained by DA1, DA2, DA3,DA4, SA, and proposed MPSC-SOPF are shown in Table 2. From thistable it is seen that while the procured reserve of the proposed ap-proach, indicated by bold font in Table 2, is lower than DA2, DA3,and DA4 as well as SA, its objective function value is higher than allmethods. The reason can be explained as follows.

The deterministic methods only consider the cost of first stagedecision variables, i.e. , , and . In other words, the ob-jective function of these methods only includes the first summationterm of Eq. (1). The objective function of SA also does not con-sider the cost of second stage decision variables associated with thepost-contingent states. On the other hand, the proposed non-determin-istic MPSC-SOPF considers the cost of both the first and second stagedecision variables, i.e. the first and second summation terms of Eq.(1). However, the objective function values reported in Table 2 arerelated to ‘before-the-fact' conditions, i.e., when the uncertain vari-ables have not been realized. As the real system conditions are un-certain, to have a more realistic evaluation for the effectiveness ofdifferent approaches, their performance for ‘after-the-fact’ conditions(i.e. when the uncertainty sources are realized) should be evaluated ina sufficiently long run. This evaluation is implemented here throughan out-of-sample analysis. In the ‘before-the-fact’ analysis reported inTable 2 both the first and second stage decision variables are opti-mized, while only the second stage decision variables are optimized inthe ‘after-the-fact’ or out-of-sample analysis (the first stage decisionshave been fixed).

A large number of trial scenarios (here, 10,000) are generated byLHS-RC. These trial scenarios are different from the 296 sample sce-narios produced by LHS-RC in the previous section and used to im-plement the MPSC-SOPF. In other words, the trial scenarios of the‘after-the-fact’ or out-of-sample analysis are unseen for MPSC-SOPF.Thus, the performance of the proposed model to optimize the firststage decisions is evaluated by unseen scenarios, which gives a betterassessment of the model’s performance. Additionally, by means of alarge number of trial scenarios generated by LHS-RC, the out-of-sam-ple analysis can simulate the individual and joint stochastic behaviorsof the uncertain variables in a long run.

Fig. 1. Scheduled generation and preventive/corrective actions in t = 1, obtained by the proposed MPSC-SOPF (case 5).

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Table 2Results of four deterministic models, one stochastic model, and proposed MPSC-SOPFmodel.

Approach Objective function ($) Procured reserve (p.u.)

DA1 84655.25 9.455DA2 85308.18 13.180DA3 86205.17 16.905DA4 88152.92 20.630SA 89929.88 15.406MPSC-SOPF 92355.56 12.810

For each trial scenario, the following optimization problem shouldbe solved in the out-of-sample analysis:

Subject to

The first summation term in Eq. (35) is a constant term as the firststage decision variables have already been fixed. It is only includedin Eq. (35) to compute the total cost of each trial scenario s′. How-ever, the second and third summation terms include the trial scenariodependent decision variables of , , , , and that

should be optimized to minimize the cost of the trial scenario s′, i.e.given in Eq. (35). Here, / , / , and repre-

sent the deployed up/down reserves, up/down generation shifts, andload sheds, respectively, employed to cope with the realization of theuncertain variables (i.e. deviation of W&L from their forecasts andoccurred contingencies) in trial scenario s′. The up/down generationshifts of / are procured for security purposes [25].

After solving (35)–(46) for all trial scenarios (i.e. ), theiraggregated cost, denoted by AC, is obtained as follows:

The AC results of DA1, DA2, DA3, DA4, SA and proposedMPSC-SOPF (bold font) for different costs of the up/down genera-tion shifts (i.e., when and equal 2–4 times of ) are shownin Table 3. It is seen that SA by considering W&L uncertainty leadsto lower AC results than DA1-DA4 since it more effectively deter-mines W&L reserves. Table 3 shows that the proposed MPSC-SOPFobtains lower AC results than all other deterministic and stochasticmethods for all values of the generation shifts’ costs. Moreover, thehighest AC result of MPSC-SOPF (obtained for generation shifts' costsequal ) is lower than the lowest AC result of DA1–DA4 and SA(obtained for as the generation shifts’ costs). These compar-isons illustrate higher effectiveness of MPSC-SOPF compared to thedeterministic and stochastic approaches to cope with various realiza-tions of the uncertainty sources in a long run. In other words, in af-ter-the-fact conditions, when the uncertain variables are realized, theproposed MPSC-SOPF performs better than DA1 to DA4 as well asSA, since MPSC-SOPF simultaneously optimizes all W&L and con-tingency reserves.

To evaluate the convergence of the out-of-sample analysis, theconvergence coefficient is used [26]:

where and represent the standard deviation of the AC re-sults and number of trial scenarios, respectively. The results obtained

Table 3The results of out-of-sample analysis (AC results).

Approach /

DA1 96486.62 99119.54 100329.9DA2 96155.28 97546.02 97956.58DA3 95958.71 96461.34 96561.25DA4 96569.97 96674.68 96693.35SA 95432.84 96242.111 96477.763MPSC-SOPF 92702.89 92920.35 93101.51

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

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for in all cases of Table 3 are well below 0.01 indicating acceptableconvergence of the out-of-sample analysis [26].

The set of selected scenarios is effective on the performance ofa stochastic model. Usually, by increasing the number of scenar-ios, we can more accurately model the stochastic problem, since wewill have more realizations of the uncertain variables, but at the ex-pense of higher computation burden. To study this aspect, a sensitivityanalysis is performed on the test system, which evaluates the impactof the number of scenarios on the performance of the MPSC-SOPFmodel. The results of this sensitivity analysis are shown in Table 4.In this sensitivity analysis, is changed from 3 to 12. For eachW&L scenario, all non-islanding single-contingencies by the num-ber of are considered. Thus, the number of sample scenar-ios, produced by the proposed method of Section 3 and included inthe MPSC-SOPF model, is changed from to

in this sensitivity analysis. Similarly, all ofthese scenarios are different from the 10,000 trial scenarios of theout-of-sample analysis. For each number of sample scenarios, thecomputation time, before-the-fact results (i.e. the objective functionvalue), and after-the-fact results (i.e. the AC results for 2–4 timesof as the cost of generation shift) obtained by the MPSC-SOPFmodel are shown. Table 4 shows that by increasing the number of sce-narios, the computation time of the MPSC-SOPF model rises, sincethe number of second stage variables and constraints increases. Be-fore-the-fact/after-the-fact costs increases/decreases by increasing thenumber of scenarios, since a more accurate evaluation of the totalcost is obtained by the MPSC-SOPF model when more realizationsof the uncertain variables are considered. In other words, the be-fore-the-fact and after-the-fact costs goes toward each other, when thestochastic model becomes more accurate. For instance, the differencebetween the after-the-fact cost with and the before-the-factcost reaches from 93111.91 − 90112.2 = 2999/71 $ for to92689.05 − 92362.85 = 326.2 $ for , i.e. about ten times de-creases. Additionally, it is seen that after the before-the-factand after-the-fact costs do not change significantly, while the compu-tation time rises rapidly. For this reason, indicated by bold fontin Table 4, which corresponds to 296 sample scenarios, is consideredin the other numerical experiments of this paper.

In Tables 5 and 6, the effects of considering correlation amongthe uncertain variables (here, 18 correlated uncertain variables) areshown. The scenarios without/with considering correlation are gener-ated by LHS and LHS-RC, respectively. Table 5 shows that withoutconsidering correlation, both the required reserve and objective func-tion value decreases in the before-the-fact conditions. However, in theafter-the-fact conditions, considering correlation leads to lower AC re-sults for all values of up/down generation shifts’ costs as shown inTable 6. In other words, considering correlation leads to a more real-istic evaluation of the after-the-fact conditions. This in turn leads to alower incurred cost encountering various realizations of the uncertainvariables.

In Figs. 2 and 3, scenarios generated by LHS and LHS-RC(i.e. without/with considering correlation among the uncertain vari

ables) for system load are shown (in the figure legends ‘Sn’ standsfor scenario). While LHS-RC generates scenarios considering correla-tions among loads of different hours, LHS ignores these inter-tempo-ral correlations. In Figs. 2 and 3, is taken into account, the sameas the previous numerical experiments. In these figures, the forecastand actual loads, denoted by “Forecast” and “Actual”, are shown bythick red and thick blue lines, respectively, in addition to the generatedscenarios. The actual load is not available when the applied scenariosare generated to simulate the likely realizations of the upcoming loadlevel. The actual load has been shown in Figs. 2 and 3 to compare itwith the generated scenarios using LHS and display the quality of thegenerated scenarios.

The eight scenarios of Fig. 3 have a more smooth behavior com-pared to the eight scenarios of Fig. 2, due to considering inter-tempo-ral correlation by LHS-RC for scenario generation. This avoids fromsudden changes. Thus, the generated scenarios by considering corre-lation better simulate the load behavior (i.e. Sn 1–8 in Fig. 3 betterfollow ‘Actual’ load pattern compared to Sn 1–8 in Fig. 2), since theactual system load has inertia and cannot change suddenly. Addition-ally, Sn 1–8 in Fig. 3 better cover the ‘Actual’ load compared to Sn1–8 in Fig. 2. Accordingly, the scenarios generated with consideringcorrelation can more effectively capture the uncertain behavior of loadcompared to the scenarios generated without considering correlation.Similar results have been obtained for the two other uncertain vari-ables of this test case, i.e. generations of the wind farms 1 and 2.

5. Conclusion

To cope with the uncertainty sources of wind farm generations,power system load and transmission equipment unavailability in anoptimal power flow framework, a new MPSC-SOPF model as well asLHS-RC scenario generation approach has been proposed. The pro-posed MPSC-SOPF model simultaneously optimizes preventive andcorrective actions considering the uncertainty sources. The LHS-RCscenario generation method produces scenarios taking into accountcorrelation among wind farm generations, and wind generations andsystem load as well as inter-temporal correlations. The comparativeresults of the proposed MPSC-SOPF model with other determinis-tic models show that the proposed approach may obtain higher be-fore-the-fact costs than other deterministic models. Its reason is thatdeterministic methods only consider the cost of the first stage decisionvariables, while the proposed MPSC-SOPF considers the cost of boththe first and second stage decision variables. However, MPSC-SOPFleads to lower after-the-fact operation costs considering various real-izations of the uncertain variables in a long run. In other words, in af-ter-the-fact conditions, when the uncertain variables are realized, theproposed approach performs better than deterministic models as wellas another stochastic model, since the proposed MPSC-SOPF simul-taneously optimizes all W&L and contingency reserves. Additionally,considering correlation among the uncertain variables may result inhigher before-the-fact costs and procured reserves. However, it canimprove the after-the-fact performance by more realistically model-ing the uncertain behaviors of loads and wind generations. The pro

Table 4Sensitivity analysis with respect to the number of scenarios.

3 4 5 6 7 8 9 10 11 12

Time (s) 5.1 5.8 6.8 8.1 9.6 11.2 13.6 16.5 20.2 25Before-the-fact 90112.2 91832.71 92188.27 92324.33 92341.69 92355.56 92358.91 92360.08 92361.55 92362.85After-the-fact 93111.91 93004.22 92819.46 92739.19 92710.03 92702.89 92698.48 92694.11 92691.62 92689.05

93633.04 93243.97 93095.08 92966.11 92938.27 92920.35 92912.3 92906.09 92903.61 92900.7594120.16 93581.01 93238.72 93197.92 93143.05 93101.51 93084.68 93071.73 93064.25 93058.84

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Table 5Results of before-the-fact conditions without/with considering correlation (results of“with correlation” are indicated by bold font).

Scenario generation Objective function ($) Procured reserve(p.u.)

Without correlation 91237.94 10.06With correlation 92355.56 12.81

Table 6AC results without/with considering correlation for after-the-fact conditions (results of“with correlation” are indicated by bold font).

Scenario generation /

Without correlation 92739.31 93422.54 93942.28With correlation 92702.89 92920.35 93101.51

Fig. 2. Eight scenarios generated by LHS (without considering correlation) with T = 6.(For interpretation of the references to color in the text, the reader is referred to the webversion of this article.)

Fig. 3. Eight scenarios generated by LHS-RC (with considering correlation) with T = 6.(For interpretation of the references to color in the text, the reader is referred to the webversion of this article.)

posed MPSC-SOPF model focuses on the optimization of the expectedvalue of the operation cost. However, if the operators of a power sys-tem tend to hedge operation costs against the worst case scenarios,other methods such as robust optimization can be proposed. Addition-ally, the current study considers the contingencies and their impacts.Because of increasing pressure on transmission network in the somecurrent power systems leading to the risk of instability problems, con-sidering stability constraints such as voltage stability constraints canbe also proposed. Both the mentioned challenges can be considered asthe subjects of future works.

Appendix A.

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Fig. A1. Flowchart of the proposed LHS-based scenario generation method.

References

[1] J.M. Morales, A.J. Conejo, H. Madsen, P. Pinson, M. Zugno, Integrating Re-newables in Electricity Markets: Operational Problems, vol. 205, Springer Sci-ence & Business Media, 2013.

[2] A. Gómez-Expósito, A.J. Conejo, C. Cañizares, Electric Energy Systems:Analysis and Operation, CRC Press, 2008.

[3] F. Capitanescu, J.M. Ramos, P. Panciatici, D. Kirschen, A.M. Marcolini, L.Platbrood, L. Wehenkel, State-of-the-art, challenges, and future trends in secu-rity constrained optimal power flow, Electr. Power Syst. Res. 81 (8) (2011)1731–1741.

[4] X. Li, Y. Li, S. Zhang, Analysis of probabilistic optimal power flow taking ac-count of the variation of load power, IEEE Trans. Power Syst. 23 (August (3))(2008) 992–999.

[5] H. Zhang, P. Li, Chance constrained programming for optimal power flow un-der uncertainty, IEEE Trans. Power Syst. 26 (October (4)) (2011) 2417–2424.

[6] R. Jabr, S. Karaki, J.A. Korbane, Robust multi-period OPF with storage and re-newables, IEEE Trans. Power Syst. 30 (November (5)) (2014) 2790–2799.

[7] C. Hamon, M. Perninge, L. Soder, A stochastic optimal power flow problemwith stability constraints, part I: approximating the stability boundary, IEEETrans. Power Syst. 28 (May (2)) (2013) 1839–1848.

[8] M. Rodrigues, O. Saavedra, A. Monticelli, Asynchronous programming modelfor the concurrent solution of the security constrained optimal power flow prob-lem, IEEE Trans. Power Syst. 9 (November (4)) (1994) 2021–2027.

[9] F. Capitanescu, L. Wehenkel, A new iterative approach to the corrective secu-rity-constrained optimal power flow problem, IEEE Trans. Power Syst. 23 (No-vember (4)) (2008) 1533–1541.

[10] W. Qiu, A.J. Flueck, F. Tu, A new parallel algorithm for security constrainedoptimal power flow with a nonlinear interior point method, IEEE PES GeneralMeeting (2005) 447–453.

[11] N. Amjady, H. Sharifzadeh, Security constrained optimal power flow consider-ing detailed generator model by a new robust differential evolution algorithm,Electr. Power Syst. Res. 81 (February (2)) (2011) 740–749.

[12] A.J. Conejo, M. Carrión, J.M. Morales, Decision Making under Uncertainty inElectricity Markets, vol. 1, Springer, 2010.

[13] J.L. Higle, Stochastic programming: optimization when uncertainty matters, Tu-torials in Operations Research, INFORMS, Baltimore, 200530–53.

[14] A.J. Wood, B.F. Wollenberg, Power Generation, Operation, and Control, JohnWiley & Sons, 2012.

[15] FERC Report, Recent ISO Software Enhancements and Future Software andModeling Plans. [Online], 2017. Available: http://www.ferc.gov/industries/electric/indus-act/rto/rto-iso-soft-2011.pdf.

[16] Z. Shu, P. Jirutitijaroen, Latin hypercube sampling techniques for power sys-tems reliability analysis with renewable energy sources, IEEE Trans. PowerSyst. 26 (November (4)) (2011) 2066–2073.

[17] D. Caiu, D. Shi, J. Chen, Probabilistic load flow computation using Copula andLatin hypercube sampling, IET Gener. Transm. Distrib. 8 (9) (2014)1539–1549.

[18] E. Alpaydin, Introduction to Machine Learning, MIT Press, 2014.[19] R.L. Iman, W. Conover, A distribution-free approach to inducing rank correla-

tion among input variables, Commun. Stat. Simul. Comput. 11 (June (3)) (1982)311–334.

[20] J.C. Helton, F.J. Davis, Latin hypercube sampling and the propagation of uncer-tainty in analyses of complex systems, Reliab. Eng. Syst. Saf. 81 (2003) 23–69.

[21] Available: http://www.elia.be/en/grid-data.[22] Available: http://www.gams.com.[23] E. Ela, M. Milligan, B. Kirby, Operating reserves and variable generation, Con-

tract 303 (2011) 275–3000.[24] F. Bouffard, F.D. Galiana, Stochastic security for operations planning with sig-

nificant wind power generation, IEEE Trans. Power Syst. 23 (May (2)) (2008)306–316.

[25] R. Zárate-Miñano, A.J. Conejo, F. Milano, OPF-based security redispatching in-cluding FACTS devices, IET Gener. Transm. Distrib. 2 (November (6)) (2008)821–833.

[26] A.M.L. Da Silva, L. da Fonseca Manso, J.C. de Oliveira, R. Billinton,Pseudo-chronological simulation for composite reliability analysis with timevarying loads, IEEE Trans. Power Syst. 15 (February (1)) (2000) 73–80.