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Digital Object Identifier: 10.1109/TPWRS.2013.2282286

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IEEE TRANSACTIONS ON POWER SYSTEMS 1

Fast Sensitivity Analysis Approach to AssessingCongestion Induced Wind Curtailment

Yingzhong Gu, Student Member, IEEE, and Le Xie, Member, IEEE

Abstract—Although the installed wind generation capacity hasgrown remarkably over the past decades, percentage of windenergy in electricity supply portfolio is still relatively low. Due tothe technical limitations of power system operations, considerablewind generation cannot integrate into the grid but gets curtailed.Among various factors, transmission congestion accounts for asignificant portion of wind curtailment. Derived from DC powernetwork, an analytical approach is proposed to efficiently assessthe congestion induced wind curtailment sensitivity without iter-ative simulation. Compared to empirical simulation-based windcurtailment studies, the proposed approach offers the followingadvantages: 1) computational efficiency, 2) low input informa-tion requirement, and 3) robustness against uncertainties. Thisapproach could benefit system operators, wind farm owners aswell as wind power investors to better understand the interactionsbetween wind curtailment and power system operations and canfurther help for curtailment alleviation. Numerical experiments ofa modified IEEE 24-bus Reliability Test System (RTS) as well as apractical 5889-bus system are conducted to verify the effectivenessand robustness of the proposed approach.

Index Terms—Integration of renewable energy, sensitivity anal-ysis, transmission congestion, wind generation curtailment.

I. INTRODUCTION

T HE increasing capacity in wind generation resourcesaround the globe has changed the energy supply mix in

many jurisdictions. Over the past decades, wind energy hasbecome one of the fastest growing renewable energy resources.By the end of 2011, installed wind power capacity has reached238 GW globally, which is 20% higher than in 2010 [1]. Suchrapid expansions of wind energy are driven by its environ-mental benefits, regulatory incentives, as well as the cost-competitiveness against conventional generation technologies.However, due to the power system reliability requirements

and technical security limitations [2]–[8], large portions of thewind generation are curtailed and thus wasted [9], [10]. Windcurtailment has already become a global challenge towardhigher penetration of renewable energy. Many areas with highwind capacity (installed) are nowadays encountering curtail-ment issues [11], [12]. Significant wind curtailment is oftenobserved in regions with a rapid development of wind capacity

Manuscript received September 21, 2012; revised February 09, 2013 andMay21, 2013; accepted July 19, 2013. This work was supported in part by VestasTechnology R&DAmericas, and in part by Power System Engineering ResearchCenter (PSERC). Paper no. TPWRS-01067-2012.The authors are with the Department of Electrical and Computer Engineering,

Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2013.2282286

TABLE IWIND CURTAILMENT FACTS [10], [11], [15]–[17]

(e.g., ERCOT curtailed 17.1% of its wind generation in 2009[11]). In 2011, China curtailed wind power equivalent to theenergy generated by 3.3 million tons standard coal, and theeconomic loss reached 833 million US$, which equalled 50%of the profit of the wind industry for that year [13]. The factsof wind curtailments in some areas with high wind penetrationare presented in Table I. The negative impact of wind curtail-ment is three-fold. First, waste of energy: unlike conventionalresources such as coal and natural gas, curtailed wind energyrepresents lost opportunities for power production rather thanunburned fuel. Second, environmental impacts: the curtailmentof wind generation reduces the potential for replacing fossilfuel consumption and results in more carbon emissions. Third,economic loss: the energy revenue of the wind industry canbe reduced significantly, which puts wind projects at risk ofnot meeting their debt obligations or reducing the amount ofdebt the project’s cash flows can support [14]. These impactsnot only affect current wind projects but also undermine futureinvestment in the wind industry. Therefore, timely research andinvestigation of wind curtailment issues are very much needed.

A. Related Work

In recent years, the recognition of wind curtailment issues hasled to a large body of literature. The National Renewable En-ergy Laboratory (NREL) has conducted comprehensive studiesof curtailment instances and factors causing wind curtailment inmultiple areas [9], [10].Valuable efforts have been devoted to searching for solutions

to wind curtailment issues. The majority of the findings can be

0885-8950 © 2013 IEEE


2 IEEE TRANSACTIONS ON POWER SYSTEMS

divided into three categories: 1) network reinforcement, 2) im-proved utilization of the existing network infrastructure, and3) coordination between wind generation and energy storageresources.Transmission reinforcement is a direct option to be con-

sidered. independent system operators (ISO) (e.g., ERCOT)can initiate ambitious plans for network expansion that wouldultimately incorporate more renewable resources [18]. Highvoltage DC (HVDC) link is often proposed to alleviate windcurtailment problems in a cost-effective manner [19].Network reinforcement can effectively relieve curtailment

but it is very time-consuming and expensive, and, furthermore,it might be economically unfeasible to upgrade the network torelieve wind curtailment [20], which may cause curtailmenta long lasting situation. Instead of network reinforcement, abetter utilization of the existing network becomes a reasonablesubstitute path. Owing to the development in power electronics,flexible AC transmission systems (FACTS) can be utilized tomanage the congestion that causes wind curtailment [21]. LikeFACTS, phase shift transformers can also help with curtailmentmanagement [22]. Dynamic thermal rating of overhead linesis another recent advance that can maximize the utilizationof the available transmission capacity [23]. More recently,active network management (ANM) scheme is proposed andimplemented on the Orkney Islands’ power grid in the U.K.,which enables real-time power production and consumptionmanagement and increases local grid connection capacity forrenewable generation [24]–[26].In addition to the previous two options, the use of energy

storage resources is a third alternative option for curtailment al-leviation. Due to the relatively low cost and large capacity, thehydro resource (including pumped hydro storage) is one of themost promising options [27], [28]. Other alternatives, such asplug-in hybrid electric vehicles (PHEV) [29], [30], wave gener-ation [31], a large scale battery energy storage system (BESS),and compressed air storage [27] have been proposed to reducethe transmission requirements of wind generation as well.

B. Scope of the Study

In the existing literature, there are many efforts to assess theamount of wind curtailment under certain conditions [12], [32].Limited work has been dedicated to the variational view of windcurtailment subject to the changes of grid conditions. The windcurtailment sensitivity analysis is thus proposed here to pro-vide insights into: 1) better characterizing the marginal impactson wind curtailment in response to adjustments in the powersystem, 2) understanding the interaction between the system andthe curtailment, and 3) pinpointing the criticality1 of differentparts of the network to the curtailment issues. A well-devel-oped wind curtailment sensitivity analysis is especially helpfulfor the network utilization enhancement approaches [20]–[22]as well as transmission dynamic rating deployment [23] to alle-viate wind curtailment issues.In addition to proposing the wind curtailment sensitivity anal-

ysis, we derive the analytical relationship between the wind cur-

1Criticality refers that a criterion with small change (as a percentage) in itsweight, may cause a significant change in the final solution [33].

tailment sensitivity and the power network branch parameters.Incorporated into wind curtailment analysis, the proposed ap-proach can describe the grid-curtailment interaction with thefollowing advantages:• computational efficiency;• low input information requirements;• robustness against uncertainties.This paper is organized as follows. Section II presents

a review of the conventional simulation-based curtailmentanalysis. Section III introduces the concept of wind curtail-ment sensitivity analysis. The analytical expression of windcurtailment sensitivity is derived using DC power networktheory. Section IV presents the implementation of the proposedapproach in existing EMS. In Section V, numerical experi-ments of IEEE RTS 24-bus system (modified) are conducted todemonstrate the wind curtailment scenarios and to verify theproposed approach. Conclusions and future work are addressedin Section VI.

II. SIMULATION BASED APPROACH

Most of the existing wind curtailment studies [12], [32] andsoftware tools [34] adopt power system scheduling simulation.These approaches estimate the wind curtailment by simulatingthe electricity market (or centralized dispatch if not deregu-lated) scheduling operations including unit commitment (UC)and economic dispatch (ED).

A. Mathematical Formulation

The formulation of the economic dispatch is discussed in[35]–[38]. As a more comprehensive production simulationmodel (incorporating commitment decisions, reserve andstart-up/shut-down ramping), we formulate a security con-strained unit commitment problem from (1)–(14):

(1)Subject to

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)


GU AND XIE: FAST SENSITIVITY ANALYSIS APPROACH TO ASSESSING CONGESTION INDUCED WIND CURTAILMENT 3

Fig. 1. Simulation approach versus analytical approach.

(10)

(11)

(12)

(13)

(14)

where is the marginal generation cost of generator ;is the reserve cost of generator ; is the output level of gen-erator at time step , and are its upper and lowerbounds; is the selected capacity in reserve services, isthe system wide load level at time step , is the systemwide reserve requirement at time step ; is the vector ofbranch flow at the time step and is the vector of trans-mission constraints of branches; is the startup cost of gen-erator ; is the shutdown cost of generator ; andare the ramp up(down) limit of generator ; and are thestartup ramp rate and shutdown ramp rate of generator ;and are the minimum up (down) time of generator ; isthe on/off status of generator at time step , and are thebinary indicators of starting-up and shutting down generator .By comparing the dispatched outputs and the generation po-

tential of the wind farms, the curtailment of wind generation canbe determined.

B. Comparison

A comparison between the simulation based approach(“simulation approach,” for short) and the sensitivity analysisbased analytical approach (“analytical approach,” for short) ispresented in Fig. 1. By using the simulation approach (e.g.,running power system economic dispatch/unit commitment),detailed operational information can be assessed. Wind genera-tion curtailment and transmission congestion can be identifiedand quantified. However, given a practical scale power net-work, conducting power system simulation is a time consumingprocess (20 min to several hours). Numerous iterations andsophisticated procedures are required to solve the complicatedsimulation (especially for large scale mixed-integer program-ming). Besides, in order to gain relatively accurate simulationresults, comprehensive input information is required, such asoffer/bidding functions, generator parameters, load forecast,network parameters, etc. Some of them are not easy for themarket participants (e.g., wind farm operators [39], [40]) tobe estimated. Furthermore, in order to assess the interactive

characteristics between wind curtailment and power networkadjustment, multiple scenarios have to be simulated.The analytical approach, which is discussed in Section III, has

different features. With the application of the derived analyticalrelationship, the analytical approach can directly and efficientlyassess the interaction relationship betweenwind curtailment andthe power network without doing simulation. Due to its com-putational advantages, this mathematical tool can be applied tothe online management of dynamic ratings, FACTS, demand re-sponses, and storage resources for wind curtailment alleviation[21], [23], [28]. Nevertheless, we shall also admit the disadvan-tages of sensitivity analysis. Without scheduling simulation, theanalytical approach cannot easily provide information of accu-mulated quantity such as wind curtailment amounts over spe-cific time periods. Besides, as discussed in Section III-A, the an-alytical approach relies on knowledge of shift factors, and majorISOs do provide such information [41].It is suggested to have a wind curtailment analysis framework

with a combination of the simulation approach and the analyt-ical approach.

III. WIND CURTAILMENT SENSITIVITY ANALYSIS

There is existing literature that uses the DC network relation-ship to conduct short-term predictions and sensitivity analysis ofthe dispatch levels, line flows, line congestions, marginal units,and LMPs. Conejo et al. provide direct analytical expressions tocompute LMP sensitivities with respect to changes in demandthroughout an electric network [42]. Li and Bo propose an ef-ficient algorithm to identify the new binding constraint and thenew marginal unit set [43]. Zhou et al. derive a global piecewiselinear-affine mapping between the distributed load and systemoutcomes for arbitrary load distributions for better short-termpredictions of system variables [44].The sensitivity analysis described in this section mainly ad-

dresses how to characterize the impact on wind curtailment inresponse to changes in transmission congestions.We define the sensitivity of wind curtailment subject to trans-

mission congestion as the marginal alleviation of wind curtail-ment due to relaxing the corresponding congested transmissioncapacity limit. The mathematical definition is provided in (15):

(15)

where is the wind curtailment sensitivity, is the curtailedwind generation, and is the congested branch capacity.

A. Analytical Expression

In this subsection, the analytical relationship between curtail-ment and transmission congestion is derived. Two levels of con-gestion patterns are discussed: 1) curtailments due to single-branch congestion and 2) curtailment due to multiple branchcongestion.1) Single-Branch Congestion: Single-branch congestion de-

picts the scenario when, in a power system, one of the wind units(unit ) gets curtailed, and there is only one congested trans-mission line (branch ) responsible for the curtailment.2 The

2Here “responsible” means if the line capacity is increased, less wind gener-ation will be curtailed.



single-branch congestion pattern is a basic pattern commonlyobserved in power system wind curtailment:

(16)

(17)

This scenario can be characterized by the differential energy bal-ancing equation given in (16) and the differential branch powerflow equation given in (17). As is justified in the authors’ pre-vious work [35], derived from (16) and (17), (18) is yielded,which gives the analytical expression of wind curtailment sen-sitivity under a single-branch congestion scenario. It indicatesthat the curtailment to be relieved by relaxing the branch ca-pacity constraint equals the reciprocal of the difference betweenthe shift factors of the curtailed wind unit and of the compen-sator unit on the congested line:

(18)2) Multi-Branch Congestion: The wind curtailment subject

to single-branch congestion is common and easy to analyze.However, in practical power system operations, the congestionpattern can be more complicated than the single-branch conges-tion pattern, and (18) may not be applicable to those more com-plex congestion patterns. In this subsection, more general casesare discussed. A generic theoretic result is derived to charac-terize wind curtailment sensitivity to most of the scenarios.A general multi-branch congestion case can be described as

follows. In a power network, there is more than one branch con-gested at the same time. Wind generation is curtailed because ofthe congestion. There are multiple branches responsible for thecurtailment. By marginally relieving the capacity limit of anyof these branches, more wind power can be integrated into thepower grid.The multi-branch congestion scenario can be represented by

several important mathematical equations group:

(19)

Equation (19) gives a differential power balance equationwhich indicates that, in the power system, the total change ingeneration should equal the total change in load:

...

(20)Equation (20) gives the differential branch flow equations,

which require the collective impacts of each nodal injection de-viation on one branch should equal the power flow change inthat branch.When one congested branch is relaxed (or tightened), the dif-

ferential power balance (19) requires the relieved (or curtailed)wind generation equals the changes in the outputs of all the com-pensators, and the incremental branch flow (20) then describethe relationship between the changes of the generation outputs

and the changes of each branch. Equations (19) and (20) can becombined and written in the matrix form (21):

......

. . ....

... ...

(21)We define the left-hand side matrix as the Augmented

Curtailment Factor Matrix, which indicates the relationship be-tween the changes in nodal power injection and the changes inbranch flows under the curtailment scenario.Under the multi-branch congestion scenario, the definition of

wind curtailment sensitivity (15) can also be written in vectorform (22):

......

(22)

Substituting (22) for (21) yields

......

. . ....

... ...

(23)In (23), left multiplying on both sides, and right multi-

plying on both sides yields

... ......

. . ....

(24)

The analytical expression of wind curtailment sensitivityunder multi-branch congestion scenarios is then given by (24).The Augmented Curtailment Factor Matrix is invertible inmost of the cases. For a selection of congested branches andmarginal units, Augmented Curtailment Factor Matrix is

a square matrix with full rank. Its invertibility is guaranteed,because the number of marginal units is one more than thenumber of congested branches in DC optimal power flows(DCOPF) [44], [45]. The original reduced shift factor matrix isof rank N-1. With one additional linear independent row of theenergy balancing equation, the Augmented Curtailment FactorMatrix is therefore full rank. In linear programming theory, aslong as the following conditions are satisfied: 1) redundant con-straints are excluded, 2) only basic variables are incorporated,and 3) the considered constraints are binding, the constraintmatrix of the linear programming problem is reduced to a basis(set of basic variables to the linear programming problem) inwhich the linear programming problem is solved [42]. It shouldbe noted that under imperfect knowledge of the system (e.g.,



for market participant), the above conditions are not alwayssatisfied, and there are chances that the Augmented CurtailmentFactor Matrix is not invertible. In this case, corrective steps(e.g. a reselection of branches and units) can be applied toresolve this issue.If all the selected branches are congested, (24) can be simpli-

fied into (25):3

... ......

. . ....

...

...

(25)

We define the inverse of the augmented curtailment factormatrix asWind Curtailment Sensitivity Matrix . The sen-sitivity of wind curtailment regarding branch is then givenby the th column of wind curtailment sensitivity matrix , asshown in (26). The other columns of wind curtailment sensi-tivity matrix assess the sensitivity of the other compensators’outputs subject to the congested branch:

......

(26)

Equation (26) gives the simplified analytical expression forestimating the wind curtailment sensitivity.4

For evaluation of errors in assessing wind curtailment sen-sitivity, the wind curtailment sensitivity assessment error isdefined (27):

(27)

where is the estimated wind curtailment sensitivity and isthe true value.

IV. IMPLEMENTATION OF WIND CURTAILMENTANALYSIS IN EMS OR OTHER APPLICATIONS

This section presents the implementation scheme of the windcurtailment sensitivity analysis developed in Section III. Severalimportant industry-practice issues are discussed.

A. Input Information

The proposed wind curtailment sensitivity analysis approachrequires a certain input information.Required Input Information• Shift factors of marginal units to congested branches• Branch flow perturbation/deviation patterns

3The sign on the right-hand side (RHS) vector depends on the definition ofthe positive direction of the power flow as well as the direction of relaxing/tightening the branch constraints. For simplicity of illustration, in this derivationthe positive direction is defined to be the same as the direction of the congestionand only the relaxing of branch constraints is considered.4The analytical approach works for unit commitment as long as (19) and (20)

hold. However, if line switching decisions change the network typology, thecorresponding shift factors have to be updated during the calculation otherwisethe sensitivity results will be inaccurate.

Fig. 2. Framework of the wind curtailment sensitivity assessment approach.

Depends on the application, the assessment of the inputinformation varies. For system operators (e.g., ISOs), directassessment to the required input information is available.While for a market participant (e.g., wind farm owner, etc.)or a third party organization (e.g., investor, trader, etc.), thenthe above required input information need to be estimated andthe knowledge of the following auxiliary input information ishelpful.Auxiliary Input Information• Shift factors of the network• Historical dispatch levels• Historical market clearing prices (MCP)• Historical branch flows• Historical congestion pattern• Historical shadow prices• Assessment of transmission capacity (thermal limit andstability limit)

In Fig. 2, the implementation scheme of wind curtailmentsensitivity assessment is presented. The branch limit deviationsestimator and shift factor of marginal units estimator generatethe input information for the Sensitivity Assessment Engine.Transmission congestion forecast and branch status estimatorare used to estimate the line limit deviations. Multiple fac-tors can contribute to congestion induced wind curtailmentincluding 1) thermal limit, 2) transient stability, 3) voltage sta-bility, and 4) small signal stability. The heating of conductorsdue to line losses sets the thermal limit which is determinedthrough experiments by the manufacturer [46]. The transientstability limit specifies the maximum capacity of branch flowunder which the system can maintain synchronism whensubjected to a severe disturbance (i.e., contingencies) [47].The voltage stability limit refers to the maximum branch flowwhich allows the power system to maintain steady voltages atall buses in the system after being subjected to disturbances(i.e., contingencies) [47]. Many studies are conducted forpower system voltage stability analysis [48], [49] and assess-ment of transferring capacity limit due to voltage stability[50]. For wind curtailment, the thermal limit is usually not themost significant factor. The stability limit can be much moredemanding than the thermal limit. In our approach, differentline capacity limitation factors (thermal, transient stability,voltage stability, etc.) are considered. These limits can beassessed offline and updated upon changes in the system status.



TABLE IIGENERATOR PARAMETERS

Congestion pattern of branches can be estimated according tohistorical shadow prices. Shift factors of the marginal units tothe selected branches can then be estimated based on publishedshift factors and the estimated congestion patterns. The locationof the marginal units can be estimated upon published networkinformation and patterns of historical LMPs. Based on theupdated information of the network, the augmented shift factormatrix can then be formulated.

V. NUMERICAL EXPERIMENTS

Numerical experiments are conducted in amodified IEEERe-liability Test System (RTS-24) [51]. The simulation duration is24 h. The dispatch interval is 15 min. Load and wind profilesfor 48 h are collected from the ERCOT System [52]. Loadsare scaled and factored out according to the portion of differentareas.The generator parameters are scaled according to [53]. The

generator capacity portfolio (the installed capacity percentageof different technologies) is configured and scaled from the realERCOT system [53]. Ramping rates and marginal costs are ap-plied as shown in Table II.

A. Curtailment Sensitivity Assessment

In this section, the assessment of wind curtailment sensitivityis presented and verified. In our previous work [35], wind cur-tailment sensitivity analysis of single-branch congestion sce-nario has been conducted and discussed. This paper focuses onthe generic level of multi-branch congestion scenario and is in-tended to prove our proposed generic analytical approach.The following list defines the multi-branch curtailment sce-

narios.• Case B: (Base Case) In the system, there are no transmis-sion line capacity limits incorporated.

• Case T: (Congestion Case T) Branch 18 (from Bus 11 toBus 24) has a capacity limit of 100 MW, and Branch 27(from Bus 17 to Bus 22) has a capacity limit of 90 MW.

• Case : (Relaxed Congestion Case ) The capacitylimit of Branch 18 is relaxed from 100 MW to 101 MW,and Branch 27 (from Bus 17 to Bus 22) still has a capacitylimit of 90 MW.

Fig. 3. Power flow of branch 18 (from bus 11 to bus 24) during windcurtailment.

Fig. 4. Generation output of wind farm (ID:9, wind, at bus 15) during thecurtailment.

The branch flows of Branch 18 under the three scenarios arepresented in Fig. 3. Compared with the Case B, the branch flowis reduced to 100 MW (by 25%) due to the incorporation ofline capacity limit. For both Case T and Case , Branch 18 iscongested during most of the time in this period.Fig. 4 shows thewind generation outputs ofWind Farm (ID:9,

Wind, at Bus 15) under the three scenarios. Due to the transmis-sion congestion, the wind generation is curtailed by more than50%.According to our definition in (15), the sensitivity of wind

curtailment under the assumed scenarios are calculated by thesimulation, as presented in Fig. 5. The figure shows the curtail-ment sensitivity is above 1.0 for most of the time, which indi-cates that every MW capacity relaxed will result in 2 or even 3MW wind curtailment being relieved. For those cases in whichthe sensitivity equals zero, it indicates that no congestion in-duced wind curtailment occurs.Based on the benchmark simulation results, the proposed ana-

lytical approach can be verified. According to (26), the wind cur-tailment sensitivity (e.g., time step 4, compensator units: ID:7



Fig. 5. Wind curtailment sensitivity (ID:9, wind, at bus 15).

Fig. 6. Modified IEEE RTS 24-bus system (robust analysis cases are defined).

and ID:11) can be calculated in (28), which gives the same valueas the simulation results in Fig. 5:

(28)

B. Robustness Analysis

In this section, numerical experiments of robustness analysisof the proposed sensitivity approach are conducted. As dis-cussed before, the proposed approach can be used by differententities in an electricity market. For system operators, mostof the information required by our proposed approach (shiftfactor matrix, compensator units, congested lines) is accessible.While, as a market participant such as wind farm operator, theaccess to the information may be limited. Sometimes, estima-tions need to be made. This section shows with a moderateestimation, the proposed approach still works well.1) Selection of Different Branches: Due to the complexity of

the power network, different branches other than the congestedones might be inaccurately selected as the inputs into the ap-proach. Three cases are for instances defined as follows, (high-lighted in Fig. 6)• Case I: Only the information of Branch 18 and of Branch23 is known.

• Case II: Only the information of Branch 27 and of Branch24 is known.

• Case III: Only the information of Branch 20 and of Branch10 is known.

All of the above three cases are based on the same conges-tion scenario defined in the previous Section V-A, that is bothBranch 18 (from Bus 11 to Bus 24) and Branch 27 (from Bus 17to Bus 22) are congested and the outputs of Wind Farm (ID:9,Wind, at Bus 15) have been curtailed. In order to assess the cur-tailment sensitivity of this scenario, the results of the three casesare compared.For Case I, the selected inputs of branches are Branch 18

and Branch 23. Branch 18 is one of the congested branches.The Branch 23, instead, is the line directly connected with thecurtailed wind farm (ID:9, Wind, at Bus 15). Because of thegeographical proximity, the estimation of line shift factors andflow changes could be more accurate. Incorporating the inputsof Branch 18 and Branch 23 into the proposed formula (24)yields

(29)

For Case II, the selected inputs of branches are Branch 27and Branch 24. Branch 27 is one of the congested branches.However, the Branch 24 is a line connected to the curtailed windfarm (ID:9, Wind, at Bus 15). The formula (24) can be applied,which yields

(30)

For Case III, Branch 20 and Branch 10 are selected as theinputs into the model. The selected two lines are located far(geographically and electrically) from the curtailed wind farm.We incorporate their information into the formula (24), whichyields

(31)

According to the results shown in (28)–(31), the different se-lections of input branches give exactly the same solution.5 Eventhough the two selected lines are electrically far from the cur-tailed wind farm, the curtailment sensitivity can still be assessedaccurately. The robustness of the proposed approach over dif-ferent lines selection is high. For any branch with nonzero shiftfactors, the changes in any nodal power injections (including

5The values of the RHS vectors in (25) are the estimated power flow of dif-ferent selections of input branches, where the estimation errors are not consid-ered.



TABLE IIIROBUSTNESS OVER COMPENSATOR BUS SELECTION

wind curtailment) will be reflected in the changes in the powerflow in that branch. Therefore, the information of wind curtail-ment can be reflected by the knowledge of any branch in thesame system with nonzero shift factors no matter where it is.This property justifies the robustness of our proposed approachthat a small piece of regional information can reflect the wholesystem curtailment conditions.2) Selection of Different Compensating Bus: For market par-

ticipants such as wind farm operators, another piece of informa-tion hard to obtain is the compensator’s information. For a windcurtailment scenario, because the load remains unchanged, theMW curtailed from the wind generation should be compensatedfrom elsewhere. The compensating generators are often the lo-cational marginal units. Necessary information of our proposedapproach is the shift factors regarding the compensators’ buses.In order to test the robustness of our proposed approach towarddifferent trials of selecting the compensating bus, the followingexperiments are conducted.We apply formula (24) by assuming every bus in the system

could be a potential candidate for the compensator’s bus. Thecomputation results of the robustness experiments are presentedin Table III.6 The wind curtailment sensitivity assessment erroris calculated (27) andpresented in column“ERROR”.Accordingto the results, as long as the selected bus is within some electricaldistance from the actual compensator bus, the approachperformswell. For example, in the curtailment scenario, the actual com-pensator bus is Bus 8, which gives the correct result. Then, thosebuses located in the same zone as Bus 8 (Houston Zone) alsogive relatively accurate results (with errors 5%). Those buseslocated in the South zone give the solutions with around 20% er-rors. Buses located at theWest zonemay give errors over 3300%.Therefore, as long as the estimation of the compensator bus iselectrically close to the actual one, the algorithm can producerelatively accurate results.

6N: North Zone, S: South Zone, W: West Zone, H: Houston Zone

TABLE IVCOMPUTATION TIME OF WIND CURTAILMENTSENSITIVITY ANALYSIS (UNIT: SECONDS)

Fig. 7. Annual wind curtailment analysis.

C. Large Scale System Analysis

Numerical experiments of a realistic US regional powersystem are conducted to justify the effectiveness of the windcurtailment sensitivity analysis. The system is with 5889 buses,7220 transmission lines, and 523 power plants (including 76aggregated wind farms with a total installed wind capacity of9710.4 MW). For wind curtailment sensitivity analysis, bothsimulation based approach and the analytical approach havebeen applied with a data set of one year with 15-min resolution.In Table IV, the computation time of the two approaches arepresented for each month. As we can see, the analytical ap-proach can be used to calculate the wind curtailment sensitivityin a much more efficient manner than the simulation basedapproach, which saves about 97% of the computation time.Based on the whole year sensitivity analysis results, we

conduct transmission line upgrade experiments. There are threeset of branches are selected: 1) Set A of branch 201 and branch3618 with average sensitivity of 5.11, 2) Set B of branch 452and branch 6170 with average sensitivity of 2.93, and 3) SetC of branch 6426 and branch 6552 with average sensitivityof 0.66. For each set of branches, the transmission capacityis upgraded by 10 MW and a simulation of one-year durationis conducted to assess the annual total wind curtailment. Thesimulation results of accumulated wind curtailment alleviationduring the year are presented in Fig. 7. As we can see, forwind curtailment alleviation, it is more effective to allocate



the 10 MW upgrade capacity to the candidate branch set withhigher sensitivity (Set A) than the ones with lower sensitivity(Set B and C), which enables 500%–600% more wind genera-tion being relieved and integrated to the power grid throughoutthe whole year.

VI. CONCLUSIONS

In this paper, we propose an analysis approach to assessingthe marginal impact of transmission congestion on wind gen-eration curtailment. The proposed simulation-free approach iscomputationally efficient, robust against uncertainties and onlyrequires limited information about the system. Numerical exper-iments in a modified IEEE RTS 24-bus system are conducted tojustify its effectiveness.The major scope of this paper is to describe this simulation-

free approach, prove it through mathematical derivations, anddiscuss its unique features. This is intended to establish the theo-retical foundation for the ongoing and future application studiesof cost-effective wind curtailment alleviations.Future studies may include advanced T&D planning based

on curtailment sensitivity, smart FACTs displacement, powersystem scheduling with dynamic line ratings, ANM basedreal-time distributed generation management, as well as con-trolling and locating storage resources for wind curtailmentrelief.

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Yingzhong (Gary) Gu (S’10) is from Shanghai,China. He received the B.S. degree in electricalengineering in 2009 from Shanghai Jiao Tong Uni-versity, Shanghai, China. He is currently pursuingthe Ph.D. degree in electrical engineering at TexasA&M University, College Station, TX, USA.He had an internship at Alstom Grid (Redmond,

WA, USA) in 2012. He also has an internship at Cali-fornia ISO (Folsom, CA, USA) in 2013. His researchinterests include dynamic optimization approach topower system operation, the assessment of wind gen-

eration curtailment, spatial-temporal wind forecasts and stochastic program-ming for advanced market operation.

Le Xie (S’05–M’10) received the B.E. degree inelectrical engineering from Tsinghua University, Bei-jing, China, in 2004, the S.M. degree in engineeringsciences from Harvard University, Cambridge, MA,USA, in 2005, and the Ph.D. degree from the ElectricEnergy Systems Group (EESG) in the Departmentof Electrical and Computer Engineering at CarnegieMellon University, Pittsburgh, PA, USA, in 2009.He is an Assistant Professor in the Department of

Electrical and Computer Engineering at Texas A&MUniversity, College Station, TX, USA, where he is

affiliated with the Electric Power and Power Electronic Group. His industryexperience includes an internship in 2006 at ISO-NewEngland and an internshipat EdisonMission EnergyMarketing and Trading in 2007. His research interestsinclude the modeling and control of large-scale complex systems, smart gridapplications in support of variable energy integration, and electricity markets.

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