A Sensitivity-based Approach for Optimal Siting of Distributed Energy Resources
Mukesh Gautam, Narayan Bhusal, Mohammed Benidris, Chanan Singh, Joydeep Mitra
AA Sensitivity-based Approach for Optimal Sitingof Distributed Energy Resources
Mukesh Gautam*,
Student Member, IEEE , Narayan Bhusal*,
Student Member, IEEE ,Mohammed Benidris*,
Member, IEEE , Chanan Singh**,
Fellow, IEEE , and Joydeep Mitra***,
Fellow, IEEE *Department of Electrical & Biomedical Engineering, University of Nevada, Reno,(emails: { mukesh.gautam, bhusalnarayan62 } @nevada.unr.edu, and [email protected])**Department of Electrical & Computer Engineering, Texas A&M University, (email: [email protected])***Department of Electrical & Computer Engineering, Michigan State University, (email: [email protected]) Abstract — This paper presents a sensitivity-based approachfor the placement of distributed energy resources (DERs)in power systems. The approach is based on the fact thatmost planning studies utilize some form of optimization, andsolutions to these optimization problems provide insights intothe sensitivity of many system variables to operating conditionsand constraints. However, most of the existing sensitivity-basedplanning criteria do not capture ranges of effectiveness of thesesolutions (i.e., ranges of the effectiveness of Lagrange multipli-ers). The proposed method detects the ranges of effectivenessof Lagrange multipliers and uses them to determine optimalsolution alternatives. Profiles for existing generation and loads,and transmission constraints are taken into consideration. Theproposed method is used to determine the impacts of DERsat different locations, in presence of a stochastic element (loadvariability). This method consists of sequentially calculatingLagrange multipliers of the dual solution of the optimizationproblem for various load buses for all load scenarios. Opti-mal sizes and sites of resources are jointly determined in asequential manner based on the validity of active constraints.The effectiveness of the proposed method is demonstratedthrough several case studies on various test systems includingthe IEEE reliability test system (IEEE RTS), the IEEE 14 and30 bus systems. In comparison with conventional sensitivity-based approaches (i.e., without considering ranges of validityof Lagrange multipliers), the proposed approach provides moreaccurate results for active constraints.
Index Terms — Lagrange multipliers, load variability, powersystem planning, sensitivity analysis.
I. I
NTRODUCTION
In recent years, integration of renewable energy sources aswell as storage devices in the power grid has seen sustainedgrowth. Conventional notions regarding limits on how muchof these resources can be absorbed by the grid have beendispelled by numerous innovative approaches. This trend hasmotivated the development of innovative system planningmethods that can foster and facilitate the integration of theseresources. Determination of the optimal placement and sizesof these devices in terms of operation and ancillary servicesand participation in the electricity market is an importantconsideration in power system planning. Several methodshave been introduced to solve such problems includinganalytical and population-based intelligent search methods.In this work, an analytical method that is based on thesensitivity analysis concept is proposed to jointly determine optimal locations and sizes of distributed energy resources(DERs) with respect to the desired objective function.Several approaches have been presented in the literatureto determine optimal sizes and sites of distributed gen-eration and storage for various purposes. Authors of [1]have proposed a method to determine optimal locationsof virtual synchronous generators to provide an inertialresponse. In [2], an efficient analytical—optimal power flow(EA-OPF) based method—has been proposed for optimallocation of distributed generation in distribution system. In[3], an iterative-analytical method has been proposed todetermine optimal sizes and sites of distributed generationfor radial distribution systems to reduce network losses. In[4], an analytical approach has been proposed to determinethe sizes and sites of distributed generation on distributionsystems for losses minimization. A two-stage sequentialMonte Carlo simulation (MCS)-based stochastic strategyhas been proposed in [5] to determine minimum sizes ofmovable energy resources (MERs) for service restorationand reliability enhancement. In this approach, spanning treesearch algorithm for optimal network configuration, Dijk-stra’s shortest path algorithm for optimal routes to deployMERs, and the traveling time of MERs are incorporated. Amulti-objective optimization framework for sizing and sitingof distributed generation based on genetic algorithm andan ε -constrained method has been proposed in [6]. In [7],a reliability-based method has been used to determine thesize of backup storage units. In [8], optimal locations ofvirtual synchronous generators have been proposed, whichare determined based on an H norm performance metricreflecting network coherency. An optimal size and locationof battery energy storage system for load leveling has beenproposed in [9].A sensitivity analysis-based approach has been used in[10] to determine optimal locations and sizes of DERs. Theresults have been validated using modified genetic algorithm.However, the method proposed in [10] does not considerreactive power consumption nor does it consider dispatchabledistributed energy resources; distributed energy resources areconsidered to produce only real power with “must-take”paradiem.In this paper, a sensitivity-based method is developed and a r X i v : . [ ee ss . S Y ] A ug pplied to determine impacts of DERs placement at differentlocations considering load variability. The proposed approachis based on the fact that most planning studies utilize someform of optimization, and solutions to these optimizationproblems provide insights into the sensitivity of many systemvariables to operating conditions and constraints. However,most of existing sensitivity-based planning criteria do notcapture ranges of effectiveness of these solutions (i.e., rangesof the effectiveness of Lagrange multipliers). The proposedmethod detects the ranges of effectiveness of Lagrangemultipliers and uses them to determine optimal solutionalternatives. Profiles of existing generation and loads, andtransmission constraints are taken into consideration. SinceDERs significantly influence voltage profiles and reactivepower requirements, these too are included in the optimiza-tion framework. Although several objective functions suchas loss reduction, reliability maximization, power qualityimprovement, and cost minimization can be achieved usingthe proposed approach, this paper only considers cost min-imization. Other functions will be included in future work.The effectiveness of the proposed method is demonstratedthrough several case studies on various test systems includingthe IEEE reliability test system (IEEE RTS) and the IEEE14 and 30 bus systems. The proposed approach providesmore accurate results than conventional sensitivity-basedapproaches.The rest of the paper is organized as follows. Section II,III, IV, and V provide, respectively, an overview of sensi-tivity analysis and Lagrange multiplier-based methods; pro-posed approach of considering ranges of validity of Lagrangemultipliers; a solution algorithm to proposed approach; andnetwork modeling and power flow techniques. Section VIpresents case studies and discussions. Section VII providesconcluding remarks.II. S ENSITIVITY A NALYSIS
Sensitivity analysis is an effective tool to assess the effectof optimization problem constraint relaxations on the ob-jective function. Lagrange multiplier-based sensitivity anal-ysis has numerous applications in different areas. Lagrangemultipliers have been first proposed by the economist ContePetrovic in terms of shadow prices. In his work, the linearprogramming has been used to maximize the output of someproducts [11]. Lagrange multipliers have been defined fromdifferent perspectives in various literature. For instance, froma primal-dual perspective, it has been defined as the dualvariables associated with the linear/nonlinear programmingproblem. From optimization point of view, it has beendefined as the rate of change of an objective functionfor an infinitesimal change in the right-hand side of theoptimization problem. From the geometric prospective, it hasbeen explained as the sub-gradients of the objective functionalong the dimension of resource provisioning changes.Several variations of sensitivity analysis have been used todetermine the change in an objective function with respectto problem constraints. For instance, it has been used in [12]to forecast the short-term transmission congestion. In [13], Lagrange multipliers-based sensitivity coefficients have beenused for adaptive load shedding. Lagrange multipliers basedsensitivity analysis has been used in [14] for power systemreliability enhancement. Normalized Lagrange multipliershave been used in [15] for topology error identification.In [16], a Lagrange relaxation technique has been used forscheduling of hydro-thermal power systems. Authors of [17]have used an augmented Lagrange multiplier method to de-termine optimal locations of unified power flow controllers.In [18], Lagrange multipliers have been used to determine themarginal value of spinning reserve and the marginal valueof interruptible load. Lagrange multipliers have been usedfor reliability optimization in [19]. However, using Lagrangemultipliers without determining their range of effectivenesscould produce inaccurate results. This can be attributed inpart to the fact that Lagrange multipliers change with thechange in system conditions. For instance, buses that areranked as the best candidates with respect to generation costsmay not be valid for large energy sources or storage devicessince Lagrange multipliers change with the change in systemloading. In this paper, the Lagrange duality concept withconsidering ranges of validity of Lagrange multipliers hasbeen used as a decision-making tool to determine the optimalsizes and location of DERs.III. T HE P ROPOSED M ETHOD
In this paper, the Lagrange multiplier-based sensitivityapproach has been used as a decision-making tool to deter-mine the optimal sizes and locations of DERs. The proposedmethod detects the ranges of the effectiveness of Lagrangemultipliers and uses them to determine optimal solutionalternatives. Profiles of existing generation and loads, andtransmission constraints are taken into consideration. Apartfrom this, the stochasticity of the load has been includedusing non-sequential Monte Carlo simulation (MCS). Fordetecting the ranges of effectiveness of Lagrange multipli-ers, the proposed method checks the validity of the activeconstraints before finalizing each of the locations for theplacement of DERs.An optimization problem in the standard form can beexpressed as follows. Minimize f ( x ) , (1)subject to g j ( x ) ≤ j = 1 , ...., m , h j ( x ) = 0; j = 1 , ...., p . (2)In (1) and (2), f ( x ) denotes objective function; g j ( x ) denotesinequality constraints; h j ( x ) denotes equality constraints;and x ∈ R n .The basic idea in Lagrangian duality is to take the con-straints in (2) into account by augmenting the objectivefunction of (1) with a weighted sum of the constraintfunctions [20]. The Lagrangian L : R n × R m × R p −→ R associated with the problem of (1) and (2) can be defined as X Constraint 1
Constraint 2
Constraint 3Range of effectiveness of constraint 2Optimal Point P P Fig. 1. Change of Lagrange multipliers with loading and generation follows. L ( x, λ, ν ) = f ( x ) + m (cid:88) j =1 λ j g j ( x ) + p (cid:88) j =1 ν j h j ( x ) (3)where λ j refers to the Lagrangian multiplier associated withthe j th inequality constraint: g j ( x ) ≤ ; and ν j is Lagrangemultiplier associated with the j th equality constraint: h j ( x ) .The Lagrange dual function θ : R m × R p −→ R is denotedas the minimum value of the Lagrangian over x : for λ ∈ R m , ν ∈ R p . Mathematically, it can be expressed as follows. θ ( λ, ν ) = inf { L ( x, λ, ν ) } (4)The Lagrange multipliers used in this paper are associatedwith the power balance equations which are also affected bythe loading level of the system. To illustrate the change invalues of Lagrange multipliers with the change in systemconditions, consider a linear system with two variables ( X and X ) and three constraints as shown in Fig.1.It can be seen from Fig.1 that initially when the system isoperating at the optimal point P , constraint is inactive,thus the Lagrange multiplier associated with constraint is zero. During this condition, the optimal operating pointis constrained by constraint and constraint only. Theconstraint can still be relaxed up to point P withoutmaking it inactive. If constraint is further relaxed beyondpoint P , it becomes inactive and constraint becomesactive. Here, P to P is the range of effectiveness ofconstraint .In the proposed method, the weighted average valuesand range of validity of Lagrange multipliers obtained af-ter solving the optimization problem for several iterationsunder variable loading conditions are used to determine thelocations that are more sensitive to load variation. Due tothis reason, the DER with the highest capacity is placed inthe location that has the highest weighted average Lagrangemultiplier. IV. S OLUTION A LGORITHM
This section describes the solution algorithm to the pro-posed Lagrange multiplier-based sensitivity approach for placement of DERs. In the proposed method, MCS is usedfor determining the average values of Lagrange multipli-ers. In MCS, the states are sampled from the state spaceproportional to their probabilities [21]. Moreover, it is easyto implement and requires less computation time. A stop-ping criterion is required to stop the simulation after theconvergence of Lagrange multipliers [22]. In this paper,the stopping criterion applied on Lagrange multipliers iscalculated as follows. σ = max (cid:32) (cid:112) Var ( λ k ) E [ λ k ] (cid:33) ; k = 1 , ...., N b (5)where E [.] is the expectation operator; Var(.) is the varianceoperator; λ k is Lagrange multiplier of bus k ; and N b is thenumber of buses in the system.The solution algorithm to system planning using theproposed method can be explained as follows.1) Read system data (such as bus data, branch data andgenerator data) along with their hourly load profile.2) Using the hourly load profile for a year, cluster loadlevels into clusters (any number of clusters can beused based on the required accuracy) along with theircumulative probability distribution.3) Start MCS and generate a random number. Selectthe load level corresponding to the generated randomnumber.4) Using the sampled value of the load level, solve theoptimization problem. For the dual solution of theoptimization problem, compute Lagrange multipliersfor each node of the test system associated withpower flow equations. Rank Lagrange multipliers ina descending order, which are used to select the nodesfor DERs to be added.5) According to the number of DERs to be added, theirsizes and ranked Lagrange multipliers, determine thelocations and sizes for DERs placement. For example,if three DERs of different sizes are to be placed,the highest three Lagrange multipliers are used fordetermining the sizes and locations. This implies thatthe highest capacity DER is placed in the location withthe highest value of Lagrange multiplier.6) The optimization problem is again solved after DERsplacement and new values of Lagrange multipliersare computed. Check the inactivity of power flowconstraints at each location. Penalize the bus(es) inwhich power flow constraints are inactive, since thoselocations are no longer responsible for optimality ofthe solution. Store these new values of the Lagrangemultipliers.7) Check whether the convergence criterion is met. Ifyes, go to the next step. Otherwise, generate a newrandom number, select the load level corresponding tothe generated random number and go back to step .8) Compute the weighted average value of the Lagrangemultipliers from the stored values of the Lagrangemultipliers during each iteration of MCS. Based on ead system data and hourly load profile StartGenerate 50 sets of clustered load along with their probability distributionInitialize Monte Carlo simulation and start iterationsSolve the optimization problem and determine Lagrange multipliers
Check inactivity of power limit constraints at each location
Penalize the bus(es) in which power limit constraints are inactiveDetermine average value of Lagrange multipliers of all iterations
Place DERs with higher capacity at the buses with higher values of Lagrange multipliers
Place DERs at the locations with high values of Lagrange multipliers
Solve the optimization problem to determine new values of Lagrange multipliersStopping criteria satisfied?
Stop No
Yes
Fig. 2. Flow chart of proposed solution algorithm these values of Lagrange multipliers, determine thelocations of DERs to be integrated.The flow chart of the proposed solution algorithm is shownin Fig. 2.V. N
ETWORK M ODELING AND P OWER F LOW
In this paper, the AC optimal power flow model is usedto solve the optimization problem for the determinationof Lagrange multipliers. For solving the optimal powerflow, generation cost minimization is used as the objectivefunction. The objective function is subjected to equality andinequality constraints of power system operation limits. The equality constraints include power balance equations at eachbus, and the inequality constraints include capacity limitsof each generating unit, voltage limits at each bus andreactive power capability limits. The network model can beformulated as follows. C = min N g (cid:88) j =1 C j ( P Gj ) , (6)subject to P ( V, δ ) − P D = 0 , Q ( V, δ ) − Q D = 0 , P minG ≤ P ( V, δ ) ≤ P maxG , Q minG ≤ Q ( V, δ ) ≤ Q maxG , (7) V min ≤ V ≤ V max , S ( V, δ ) ≤ S max , δ unrestricted,where C : the generation cost function; N g : total number of generating units; C j : the generation cost function of generating unit j; P D : the active power demand vector ( N d × Q D : the reactive power demand vector ( N d × V : the bus voltage magnitude vector ( N b × δ : the bus voltage angles vector ( N b × V min & V max : the bus voltage limits vector ( N b × V : the bus voltage magnitude vector ( N b × P ( V, δ ) : the active power injection vector ( N b × Q ( V, δ ) : the reactive power injection vector ( N b × S ( V, δ ) : the line flow vector ( N t × S max : the line flow limits vector ( N t × N b : total number of buses ; N d : total number of load buses ; N t : total number of transmission lines ; P minG : minimum active power generation limit ; P maxG : maximum active power generation limit ; Q minG : minimum reactive power generation limit ; Q maxG : maximum reactive power generation limit . VI. C
ASE S TUDIES AND D ISCUSSIONS
The proposed method and solution algorithm is demon-strated on several test systems including the IEEE 14 bussystem, IEEE 30 bus system, and IEEE RTS. These systemshave been extensively used for optimal placement and sizingof DERs for various research objectives [10]. The IEEE14 bus system consists of 14 buses, 5 generators, and 11loads with total generation capacity of . MW and totalpeak load of
MW. The IEEE 30 bus system consists of30 buses, 6 generators, and 20 loads with total generationapacity of
MW and total peak load of . MW.The detailed data of IEEE 14 and IEEE 30 bus system aregiven in [23] and [24], respectively. The IEEE RTS consistsof 24 buses, 33 transmission line, 5 transformers, and 32generating units on 10 buses with total generation capacityof
MW and peak load of
MW. The detailed data(e.g. size and type of generators, failure rate of the varioussystem components, and load profile) of IEEE RTS are givenin [25].Before performing MCS, the system load is modeled bya clustering technique where 50 clusters are generated alongwith their cumulative probability distribution. By drawing auniformly distributed random number, the load level can bedetermined according to the position of the random number.This load level is used to determine actual load at each busfor all test systems.The optimal AC power flow is solved using MATPOWER(open source, version 7.0) [26]. Four case studies are per-formed for different operation paradigms of DERs (dis-patchable and non-dispatchable real and reactive power) onthe aforementioned test systems. We performed these casestudies to test the proposed approach and ranges of validity ofLagrange multipliers with change in real and reactive powercontrol.
A. Case I: Without considering the range of validity of activeconstraints
In this case, the DERs with non-dispatchable active andreactive power are considered, and the range of the validityof active constraints is not considered. This case study is per-formed to compare the proposed approach which considersranges of validity of Lagrange multipliers with conventionalsensitivity-based methods. Table I shows the results for thiscase. The results for the IEEE bus system show that bus is most sensitive to the change in load. So, the DER withthe highest capacity is placed at bus . At buses and ,small DERs are placed in descending order of their averageLagrange multipliers. TABLE IO
PTIMAL L OCATIONS AND S IZES OF
DER
S WITHOUT CONSIDERINGTHE RANGE OF VALIDITY OF ACTIVE CONSTRAINTS
Systems Optimal Locations and SizesIEEE 14 bus Bus 3 10 9Size*
30; 10 20; 6 .
66 10; 3 . IEEE 30 bus Bus 8 21 17Size*
30; 10 20; 6 .
66 10; 3 . IEEE RTS Bus 4 5 2Size*
60; 20 50; 16 . . *: P (MW) ; Q (MVar)
B. Case II: The proposed method with non-dispatchableactive and reactive power
In this case, the DERs with non-dispatchable active andreactive power are considered and the proposed method is applied for all of the aforementioned test systems. Table IIshows the results of the proposed method for DERs withnon-dispatchable active and reactive power. The results showthat when the range of validity of the active constraints isconsidered, the optimal locations of DERs may change forthe locations which have short range of validity of their activeconstraints. The optimal locations of DERs for IEEE bussystem have not changed but the optimal locations for IEEE bus system and IEEE RTS have changed. This impliesthat the optimal locations of DERs for IEEE bus systemhave large range of validity of their active constraints. TABLE IIO
PTIMAL L OCATIONS AND S IZES OF
DER
S WITH NON - DISPATCHABLEACTIVE AND REACTIVE POWER
Systems Optimal Locations and SizesIEEE 14 bus Bus 10 9 7Size*
30; 10 20; 6 .
66 10; 3 . IEEE 30 bus Bus 8 21 17Size*
30; 10 20; 6 .
66 10; 3 . IEEE RTS Bus 14 7 8Size*
60; 20 50; 16 . . *: P (MW) ; Q (MVar)
C. Case III: The proposed method with dispatchable reactivepower and non-dispatchable active power
In this case, optimal locations and sizes are determined forDERs with dispatchable reactive power but non-dispatchableactive power. Table III shows the results of the proposedmethod for DERs with dispatchable reactive power but non-disptchable active power. The results show that the differentoptimal locations may be obtained when DERs with dis-patchable reactive power is considered. Compared to caseII, the optimal locations of DERs for IEEE bus systemhave not changed but the optimal locations for IEEE bussystem and IEEE RTS have changed. Again, this is becauseof the large range of validity of the active constraints of IEEE bus system. TABLE IIIO
PTIMAL L OCATIONS AND S IZES OF
DER
S WITH DISPATCHABLEREACTIVE POWER AND NON - DISPATCHABLE ACTIVE POWER
Systems Optimal Locations and SizesIEEE 14 bus Bus 10 9 7Size*
30; 10 20; 6 .
66 10; 3 . IEEE 30 bus Bus 19 20 18Size*
30; 10 20; 6 .
66 10; 3 . IEEE RTS Bus 14 7 18Size*
60; 20 50; 16 . . *: P (MW) ; Q (MVar)
D. Case IV: The proposed method with dispatchable activeand reactive power
In this case, optimal locations and sizes are determinedfor DERs with dispatchable active and reactive power. TableV shows the results of the proposed method for DERs withdispatchable active and reactive power. The results of thiscase show that the optimal locations differ when DERs withdispatchable active and reactive power are considered. Thisis mainly because of the decrease in the range of validity ofactive constraints when both active and reactive power arecontrollable.
TABLE IVO
PTIMAL L OCATIONS AND S IZES OF
DER
S WITH DISPATCHABLEACTIVE AND REACTIVE POWER
Systems Optimal Locations and SizesIEEE 14 bus Bus 14 10 9Size*
30; 10 20; 6 .
66 10; 3 . IEEE 30 bus Bus 30 29 19Size*
30; 10 20; 6 .
66 10; 3 . IEEE RTS Bus 8 4 5Size*
60; 20 50; 16 . . *: P (MW) ; Q (MVar)
VII. C
ONCLUSION
This paper has introduced a sensitivity-based method todetermine optimal locations and sizes of DERs. By analyz-ing the impacts of DERs at different locations, sizes andlocations of DERs is determined. In developing the proposedapproach, the following variables have been taken into con-sideration: impacts of different DERs; profiles for existinggeneration and loads; and transmission constraints. More-over, the load variablity was also considered for analyzingthe impacts of load variations on Lagrange multipliers. Theproposed method was demonstrated on several test systemsincluding the IEEE RTS, the IEEE 14 and 30 bus systems.As the objective of the proposed approach is to minimize thegeneration cost, the optimal locations and sizes will resultis the saving of the generation cost. In our future workwe will implement this approach for other objectives suchas reliability maximization, loss minimization, contributionin the electricity markets, and providing other ancillaryservices. R
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