Dual Decomposition-Based Privacy-Preserving Multi-Horizon Utility-Community Decision Making Paradigms
11 Dual Decomposition-Based Privacy-PreservingMulti-Horizon Utility-Community Decision MakingParadigms
Vahid R. Disfani,
Student Member, IEEE,
Zhixin Miao,
Senior Member, IEEE,
Lingling Fan,
SeniorMember, IEEE,
Bo Zeng,
Member, IEEE
Abstract —Two types of privacy-preserving decision makingparadigms for utility-community interactions for multi-horizonoperation are examined in this paper. In both designs, communi-ties with renewable energy sources, distributed generators, andenergy storage systems minimize their costs with limited informa-tion exchange with the utility. The utility makes decision basedon the information provided from the communities. Throughan iterative process, all parties achieve agreement. The authors’previous research results on subgradient and lower-upper-boundswitching (LUBS)-based distributed optimization oriented multi-agent control strategies are examined and the convergence anal-ysis of both strategies are provided. The corresponding decisionmaking architectures, including information flow among agentsand learning (or iteration) procedure, are developed for multi-horizon decision making scenarios. Numerical results illustratethe decision making procedures and demonstrate their feasibilityof practical implementation. The two decision making architec-tures are compared for their implementation requirements aswell as performance.
Index Terms —AC OPF, dual decomposition, moving horizonoptimization, convergence property, spin reserve
I. I
NTRODUCTION S IGNIFICANT increase in penetration of private agentshaving their own microgrids to the power network ishighly expected in the future smart grid, which makes controland optimization of the power network more challengingthan before. These microgrids may have different types ofdistributed energy resources (DER) such as renewable en-ergy systems (RES) and energy storage systems (ESS) tomitigate renewable energy intermittency. Storage systems areconstrained by temporal limitations which makes decisionmaking to consider multiple time horizons instead of just onesnapshot. In addition, the private agents have their own specificinterests and do not like to share their private informationwith the system operator due to economical or privacy-relatedreasons. From the operator’s point of view, it is desired tooperate the entire power network including the microgridsin its optimal operation. In such optimization problems, thenetwork constraints such as network congestion and voltagelimits must be considered. Furthermore, the spinning reserveis required, which guarantees the system be ready to respondto contingencies that affect power balance.
V.R. Disfani, Z. Miao, and L. Fan are with the Department of ElectricalEngineering, University of South Florida, Tampa, FL 33620 (Email: [email protected]). B. Zeng is with the Department of Industrial Management andScience at University of South Florida.
The objective of this paper is to develop privacy-preservingmulti-horizon utility-community decision making paradigmsconsidering AC power flow, network constraints, and reserverequirements.The mathematic foundation of privacy-preserving decisionmaking is distributed optimization. In the literature, there areseveral approaches to develop distributed algorithms for bothAC OPF [1]–[5] and DC OPF [6]–[9]. These algorithms de-compose an optimization problem to smaller subproblems anddesign an iterative process to seek the optimal solution, whereone agent is responsible to each one of these subproblems.After the agents solve their own OPF problem (e.g, using usinginterior point method), these algorithms performs an updateprocess to push the solution toward the optimal solution. Theupdate process requires agents to share information. Therefore,each algorithm corresponds to an information flow structure.A modified subgradient-based method has been proposed inthe authors’ previous work [5], where microgrids and a utilityname their desired levels of power transactions and a priceupdating scheme pushes the solution toward the optimal solu-tion. Another method proposed in [5] is named lower-upper-bound-witching (LUBS), where the microgrids update theirdesired prices based on the power demand of the main grid.The main grid defines its optimal power export/import levelfrom each microgrid based on the prices. The convergence ofthe upper-bound and the lower-bound indicates the optimalityof the solution.To handle the time-correlated constraints of ESS, OPF needto consider multiple time horizons instead of just one snapshot.Several approaches has been addressed in the literature totackle multi-horizon OPF problem. Bender’s decompositionmethod is developed in [10] to tackle energy constraints ofhydroelectric plants integrated in irrigation systems. Bender’decomposition focuses on decomposition of integer decisionvariables and continuous decision variables and cannot betranslated into a multi-agent control structure. Another threadof research [11], [12] develops a KKT-based solution to con-sider battery storage systems. An optimization model is alsoproposed in [13] for multi-horizon OPF with wind generationand battery storage. The capability of ESS to provide ancillaryservices such as reserve is taken into account to tackle reserveconstraints in an OPF problem without considering their time-correlated constraints [14], [15].To the authors’ best knowledge, the proposed researchprivacy-preserving decision making for multi-horizons while a r X i v : . [ c s . S Y ] M a r considering AC OPF and spinning reserve, has not been seenin the literature. This paper extends the research in [5] to ap-ply the privacy-preserving decision making to multi-horizonswhile considering spinning reserve requirements. Spinningreserve adequacy is a constraint which is considered not onlyin unit commitment problem [16] but also in OPF problem[17], [18]. A reserve-constrained OPF problem is developedin [17] to incorporate the expected security costs in the systemusing reserve marginal value (reserve price) which indicateshow much cost is imposed to the system operation if onemore unit of reserve is needed. The extension of MATPOWER4.1 [19] can solve OPF with co-optimization of energy andreserve. This capability is employed in our research to solvereserve-constrained OPF problems.The contribution of this paper has threefold. • Two privacy-preserving decision making paradigms basedon our previous research are developed for multi-horizonswhile considering reserve constraints; • Convergence properties for the paradigms are analyzedand convergence enhancement measures are proposed andexplained. • Realistic implementation of the decision makingparadigms at real time is investigated using movinghorizon optimization technique and demonstrated in casestudies.The rest of the paper is organize as following: the twodual decomposition-based paradigms for a multi-horizon OPFproblem are presented in Section II. Implementation for real-time operation is investigated and moving horizon optimiza-tion technique is adopted. The algorithms’ convergence prop-erties are investigated in Section III. Section VV presents acase study to demonstrate the decision making process. Theconclusion is presented in Section V.II. D
UAL D ECOMPOSITION -B ASED P RIVACY -P RESERVING D ECISION M AKING P ARADIGMS
Consider a power network consisting of a set N of busesand a set E of branches. The utility is responsible to operatethe power grid, its generation units and transactions with trans-mission systems. As shown in Fig. 1, community microgridsmay have energy storage systems, renewable energy resources,and internal loads. These microgrids are connected to thenetwork and behave as private agents who will share onlylimited information with the utility. The set A includes allbuses which the communities are connected to.The reserve-constrained 24-hour OPF problem considering G B
NETWORK
COMMUNITY i
UTILITY 𝑷 𝒖 𝒊 , 𝒕 𝑷 𝒈 𝒊 , 𝒕 𝑷 𝑷 𝑽 𝒊 , 𝒕 𝑷 𝒃 𝒊 , 𝒕 𝑷 𝒅 𝒊 , 𝒕 Fig. 1. A community microgrid including a generator, a renewable energysystem (RES), an energy storage system (ESS) and internal load. community microgrids is defined as below: min (cid:88) t =1 (cid:88) i ∈N C i ( P g i ) (1a)s. t. ∀ i ∈N , ∀ j ∈E , ∀ t ∈T P g i,t − P L i,t + P P V i,t − P b i,t − P i,t ( V, θ ) = 0 (1b) Q g i,t − Q L i,t + Q P V i,t − Q b i,t − Q i,t ( V, θ ) = 0 (1c) E b i,t +1 = E b i,t + P b i,t (1d) V mi ≤ V i,t ≤ V Mi (1e) P mg i ≤ P g i,t ≤ P Mg i (1f) Q mg i ≤ Q g i,t ≤ Q Mg i (1g) S j,t ( V, θ ) − S Mj ≤ (1h) P mb i ≤ P b i,t ≤ P Mb i (1i) E mb i ≤ E b i,t +1 ≤ E Mb i (1j) E b i, = E b i, (1k) (cid:88) i R g i,t + R b i,t ≥ R d t (1l) R g i,t ≤ R Mg i (1m) R g i,t ≤ P Mg i − P g i,t (1n) R b i,t ≤ − P mb i + P b i,t (1o)where C ( · ) is the cost function, superscripts M and m denoteupper and low limits. Subscripts i ∈ N and t ∈ T refer to thevariables corresponding to bus i and hour t where the set T = { t ∈ Z | ≤ t ≤ } denotes the time horizon. P g , Q g , P L and Q L are the vectors of bus real and reactive power generations,and real and reactive loads. P b and P P V denote the batterycharging power and PV output power respectively. P ( V, θ ) and Q ( V, θ ) are the active and reactive power injections in termsof bus voltage magnitude and phase angles, and S ( V, θ ) is thevector of line complex power flow. E b also determines theenergy stored in the storage systems. The parameters R g and R b denote the reserve provided by the generators and batterysystems respectively.In the optimization problem (1), the objective function (1a)is defined so as to minimize the total generation cost in the entire system. Active and reactive power balance constraintsare defined in (1b) and (1c) for all buses at all hours. Thedependency between battery charging power and energy storedin the battery is stated in (1d). The constraints (1e)-(1d)describe minimum and maximum limits of the correspondingvariables. The constraint (1k) mandates that the ultimate levelof energy stored in the battery to be equal to that at startingpoint. The minimum level of total reserve from both utilitygenerators and the community energy sources is guaranteedby (1l) while the constraints (1m)-(1o) take care of the reserveconstraints.Let P imp i , t denotes the utility’s power import from commu-nity connected to bus i ∈ A at time t while P exp i , t denotesthe same community’s power export to the utility. Therefore,the power balance equations at Bus i will be replaced by thefollowing four equations. P imp i , t = P i,t ( V, θ ) P exp i , t = P g i,t − P L i,t + P P V i,t − P b i,t Q imp i , t = Q i,t ( V, θ ) Q exp i , t = Q g i,t − Q L i,t + Q P V i,t − Q b i,t (2)A community now only relates to P exp i , t , not P imp i , t .Global constraints that relate the communities to the utilityare imposed as λ pi,t : P imp i , t = P exp i , t ,λ qi,t : Q imp i , t = Q exp i , t ,µ t : R d t − (cid:88) i ∈N −A R g i,t − (cid:88) j ∈A R j,t ≤ . (3)with the corresponding Lagrangian multipliers also notated.In the reserve constraint, we have separated reserve providedby the utility generators R g i,t and the reserve provided bya community j R j,t . Note there are multiple real power andreactive power prices at t . However there is one reserve priceat t .Applying Lagrangian relaxation to relax the three globalconstraints in (3), the main problem is divided into severalsubproblems each of which is solved by the utility or one ofthe communities. The subproblem assigned to the utility is: min (cid:88) t =1 (cid:88) i ∈N −A [ C i − µ t R g i,t ] + (cid:88) j ∈A [ λ pj,t P imp j,t + λ qj,t Q imp j,t ] (4)over P g i,t , Q g i,t , P imp i , t , Q imp i , t , R g i,t (5)s. t. P mg i,t ≤ P g i,t ≤ P Mg i,t Q mg i,t ≤ Q g i,t ≤ Q Mg i,t R i,t ≤ R Mg i , R i,t ≤ P Mg i − P g i,t (6) R i,t ≥ , R imp i , t ≥ . The community i solves the following problem at each hour t . min (cid:88) t =1 [ C i ( P g i,t ) − λ pi,t P exp i , t − λ qi,t Q exp i , t − µ t R i,t ] over P g i,t , Q g i,t , P b i,t s. t. P exp i , t = P g i,t − P L i,t + P P V i,t − P b i,t Q exp i , t = Q g i,t − Q L i,t + Q P V i,t − Q b i,t P mg i,t ≤ P g i,t ≤ P Mg i,t Q mg i,t ≤ Q g i,t ≤ Q Mg i,t Q mP V i ≤ Q P V i,t ≤ Q MP V i P mb i ≤ P b i,t ≤ P Mb i Q mb i ≤ Q b i,t ≤ Q Mb i E b i,t +1 = E b i,t + P g i,t E mb i ≤ E b i,t +1 ≤ E Mb i E b i, = E b i, R i,t = R g i,t + R b i,t R g i,t ≤ R Mg i R g i,t ≤ P Mg i − P g i,t (7)Fig. 2 gives an illustration to show the decomposed systemswhere the utility will treat every community as a generatorwith prices of energy and reserve given and the communitywill treat the main grid as a controllable load with prices ofenergy and reserve also given. Main Grid ~ Community i 𝜇𝐺 𝜇𝐺 𝜇𝐺 𝑛 𝑃 𝑖𝑚𝑝 𝜆 ,𝜇 𝑡 MG 𝑃 𝑒𝑥𝑝 𝑖,𝑡 𝑅 𝑖,𝑡 𝜆 𝑖,𝑡 ,𝜇 𝑡 𝑅 𝑃 𝑖𝑚𝑝 𝑛,𝑡 𝑅 𝜆 ,𝜇 𝑡 𝑃 𝑖𝑚𝑝 𝑅 𝑛,𝑡 𝜆 𝑛,𝑡 ,𝜇 𝑡 Fig. 2. Power networks in utility and community optimization problems.
A. Subgradient-Based Method
In subgradient method, the energy price and reserve pricesignals are first specified by the price update center (PUC)as illustrated in Fig. 3. The price signals are used by theutility and communities to define their power import andexport levels as well as reserve for all t ∈ T . For the nextiteration, the energy price values λ i,t for the next iterationare then updated using the mismatch between correspondingpower import and export levels. The reserve price value µ t is also updated to reflect the mismatch of import and exportlevel. Fig. 3 also depicts the information flow between thesethree blocks through the algorithm.
1) Price Updating:
Updating the energy prices at each busand the reserve price are the functions assigned to Price Update C o mm un i t y C o mm un i t y n C o mm un i t y UTILITYPrice Update Center 𝜆 𝑖,𝑡 𝑃 𝑖𝑚𝑝 𝑖,𝑡 𝑃 𝑒𝑥𝑝 𝜇 𝑡 𝑅 𝑢,𝑡 𝑅 𝜆 𝜇 𝑡 𝑃 𝑒𝑥𝑝 𝑅 𝜆 𝑛,𝑡𝑝 𝜇 𝑡 𝑃 𝑒𝑥𝑝 𝑛,𝑡 𝑅 𝑛,𝑡 𝜆 𝜇 𝑡 Fig. 3. Information flow of multi-horizon subgradient algorithm. In thisfigure, information related to reactive power is not notated.
Center, utilizing the following equations: λ pi,t ( k + 1) = λ pi,t ( k ) + α i,t ( k ) · ( P imp i , t − P exp i , t ) (8) λ qi,t ( k + 1) = λ qi,t ( k ) + α i,t ( k ) · ( Q imp i , t − Q exp i , t ) (9) µ t ( k + 1) = µ t ( k ) + β t ( k ) · ( R d t − (cid:88) i ∈N −A R g i,t − (cid:88) j ∈A R j,t ) (10)where k is the index of updating step, α i,t ( k ) and β t ( k ) arepositive coefficients.
2) Community Optimization:
In order to define its powerexport level at different hours, each community must performits optimization based on the given price from the PUC. Inthis paper, it is assumed that each community has a generatorand a defined power load as well as a PV-battery package.The optimization problem corresponding to the communityconnected to bus i ∈ A is presented in (7).Each community receives the updated price for energy andreserve, solves its own subproblem (7) and determines theamount of power to sell or to export to the main grid ( P exp i , t for all t ∈ T ) as well as the level of reserve it may provide.
3) Utility Optimization:
For each hour t , the utility updatesthe import level. In this paper, MATPOWER 4.1 [19] isemployed to solve the reserve-constrained OPF problem wherethe the communities are assumed as traders selling their powerby the price announced by PUC. These values create the powerimport matrix P imp from different communities at differenthours as well as the reserve vector R , which are sent to thePUC for price updating process for the next iteration. B. LUBS Method
In this structure, communities are responsible to determinethe optimal price vectors at different hours to be fed into theutility’s OPF problems. At each iteration, communities alsoupdate utility about the maximum and minimum power andreserve they can provide. The information flow structure basedon LUBS algorithm is illustrated in Fig. 4.
1) Community Optimization and Price Updating:
Once thecommunities receive the utility’s power import information for24 hours, they must determine their local generator and batterydispatch four 24 hours and announce the corresponding pricesignals back to the utility. C o mm un i t y C o mm un i t y n C o mm un i t y UTILITY 𝑃 𝑒𝑥𝑝 𝑛,𝑡 𝑃 𝑒𝑥𝑝 𝑃 𝑒 𝑥 𝑝 , 𝑡 𝜆 𝑃 𝑖𝑚𝑝 𝑚 𝑃 𝑖𝑚𝑝 𝑀 𝑅 𝑖𝑚𝑝 𝑀 𝜆 𝑃 𝑖𝑚𝑝 𝑚 𝑃 𝑖𝑚𝑝 𝑀 𝑅 𝑖𝑚𝑝 𝑀 𝜆 𝑛,𝑡 𝑃 𝑖𝑚𝑝 𝑛,𝑡 𝑚 𝑃 𝑖𝑚𝑝 𝑛,𝑡 𝑀 𝑅 𝑖𝑚𝑝 𝑛,𝑡 𝑀 Fig. 4. Information flow of multi-horizon LUBS algorithm. Informationrelated to reactive power is not indicated in the figure.
In general, the community connected to the bus i ∈ A de-termines its Lagrange multipliers corresponding to the powerbalance constraints ( λ pi,t , λ qi,t ) . When the generator limits arenot binding, the Lagrangian multipliers or the prices are sameas the marginal prices of the generators.One of the requirements in LUBS method is that the solutionsought by the utility for power and reserve must be feasiblefor communities. If Community i ’s marginal cost is cheaperthan the minimum marginal cost of the utility and othercommunities, the utility will demand its maximum generationand battery power at all hours. This will be infeasible forthe battery due to the energy constraints of the battery.Communities are then required to update the maximum andminimum power limits at each iteration in order to preventsuch issues.Community i also updates the maximum reserve it canprovide at each iteration according to its dispatched batterypower as below and sends it to utility, R Mi,t ( k + 1) = R Mg i − P mb i + P b i,t ( k )
2) Utility Optimization:
At iteration k , the utility solves itsown subproblem given price values λ i,t ( k ) as well as powerlimits to determine the power import vector P imp for 24 hourswhile considering maximum and minimum limits of powerand reserve which are announced by communities. The powerimport information is then announced to the communities. Theprocedure to solve utility OPF problem is the same as that inthe subgradient method. The utility also computes the utilitycost C u ( k ) at k -th iteration. C. Implementation and Moving Horizon Optimization
We assume a 24-hour PV profile and a 24-hour load profilebased on forecast. For the current hour, the load and PVoutputs are known and the future 24 hours are predicted.The decision making procedures described above are thencarried out for 24-hour iteratively. For real-time operationimplementation, each iteration will be carried out for a time
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Multi-Horizon Problem 1Multi-Horizon Problem 2Multi-Horizon Problem 3Multi-Horizon Problem 4Multi-Horizon Problem 5 Time (hour)
Fig. 5. Time schedule for moving-horizon problems solving. period, say two minutes. Every two minutes, the utility and thecommunities exchange information and update their decisionvariables. If an algorithm can converge in ten steps, theoptimal operating condition can be realized in 20 minutes.This “learning process” is achieved by continuous informationexchange and updating.Suppose at Hour k , the learning takes 20 minutes and for therest of the 40 minutes, both the communities and the utilitiesare keeping the process however the generator power outputand the battery power output will be constant. Moving to Hour k + 1 , the load profile now has a change. The communitiesand the utility need to find the optimal battery dispatch andgenerator dispatch again through “learning.”The load profile and the PV profile at Hour k +1 at real-timeare different from the prediction made at Hour k . Therefore,the load and PV profiles should be adjusted to reflect the k +1 hour information. Prediction for the future 24 hours needs tobe made again. Another 24-hour optimization problem has tobe solved iteratively.The above mentioned procedure is termed as moving hori-zon optimization. To take the underlying power system dy-namics into consideration, the power command decided by thedecision making layer should better be continuous. Therefore,instead of assuming a same initial price for all time horizons(e.g, $0 /M W h ), it is more desirable to start the next horizon“learning” process with the information from the previous timehorizon. III. C ONVERGENCE A NALYSIS
In this section, convergence properties of the two iterativemethods are examined. For simplicity, the analysis is limitedto energy price and real power only.
A. Subgradient Method
In this method, the price λ k is announced to both theutility and a community at iteration k , then they decide howmuch power they would like to exchange with each other.Regarding the power level of both sides, the price updatesector updates the price for the next iteration. If the slopes ofmarginal cost functions corresponding to utility and microgridare respectively a > and a > . Then the import and export power can be expressed as: P k imp = P − a λ k (11) P k exp = P + 1 a λ k . (12)(11) also indicates the economic behavior of the utility. Thecheaper the price, the more power will be purchased. On theother hand, the greater the selling price, the more power willbe sent out by the community as shown in (12).Then we can obtain the iteration of the price. λ k +1 = (cid:20) − α (cid:18) a + 1 a (cid:19)(cid:21) λ k + α ( P − P ) (13)The iteration problems can be viewed as discrete domaindynamic problems. Hence, the convergence of the algorithmis the same as the stability of the corresponding discretedynamic system. (cid:12)(cid:12)(cid:12) − α (cid:16) a + a (cid:17)(cid:12)(cid:12)(cid:12) should be less than 1.Therefore, when α is very small, convergence is guaranteed.The critical value of α which is notated as α cr = a a a + a . When α < α cr , subgradient method converges. When α > α cr , thealgorithm will not converge. For an α equal to critical value,the iterations will circulate in a loop. Fig. 6 illustrates thepossible regions for λ k +1 in a case where P k imp > P k exp whichis divided by the line λ k +1 = λ k + α cr ( P k imp − P k exp ) toconverging and diverging areas. Price ($/MWh) P(MW) 𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑡𝑦 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝝀 𝒌 𝝀 𝒌+𝟏 𝑷 𝒆𝒙𝒑𝒌 𝑷 𝒊𝒎𝒑𝒌 𝛼 < 𝛼 𝑐𝑟 Converging Area 𝛼 = 𝛼 𝑐𝑟 𝛼 > 𝛼 𝑐𝑟 Diverging Area Loop 𝛼 = 0
No update
Fig. 6. Graphical presentation of different regions of subgradient methodconvergence for various values of α . The algorithm converges if α < α cr ,diverges if α > α cr , and circulate in a loop if α = α cr . B. LUBS Method
In this method, a community sets price based on the powerdemand by the utility. The utility then sets its import powerbased on the price. Therefore, the iterative procedure can beexplained as follows. P k imp = P − a λ k (14) P k imp = P + 1 a λ k +1 (15)(14) indicates that the utility sets its k -th step power based ona price λ k . (15) indicates that the community updates its pricebased on the power demand from the utility.(14) and (15) lead to the following iteration. λ k +1 = − a a λ k + a ( P − P ) (16)The convergence criterion becomes a < a , or the slope ofmarginal cost of a community should be less than that ofa utility. This condition is very restrictive. The algorithm ismodified to achieve better convergence.The scenarios are shown graphically in Fig. 7 where LUBSdiverges when the community has more convex generation costthan microgrid does, i.e. , a > a (Fig 7-a) and it convergesotherwise when a < a (Fig 7-b). The case of a = a isalso considered as non-converging situation since the solutionswill be trapped in a loop.The price computed in (16) is treated as a prediction ( (cid:101) λ k +1 ).Based on this prediction, σ percentage of the update willbe made. The k + 1 step price will be: λ k +1 = λ k +1 + σ ( (cid:101) λ k +1 − λ k ) (17) = (1 − σ ) λ k + σ (cid:18) a ( P − P ) − a a λ k (cid:19) (18) = (cid:18) − σ a + a a (cid:19) λ k + σa ( P − P ) (19)The convergence criteria is then identified as whether (cid:12)(cid:12)(cid:12) − σ a + a a (cid:12)(cid:12)(cid:12) < or not. The critical value of σ can be ex-plicitly derived as σ cr = 2 a a + a . LUBS algorithm convergesif σ < σ cr , diverges when σ > σ cr . Iteration will betrappedin a loop if σ = σ cr . 𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑡𝑦 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑡𝑦 utility M C ( $ / M W h ) P(MW) (a) M C ( $ / M W h ) P(MW) (b)
Fig. 7. Graphical presentation of LUBS convergence for different cases, a)LUBS diverges when the slope of the utility’s marginal cost is less than thatof the community’s, b) LUBS converges if the slope of the utility’s marginalcost is greater than that of the community’s.
Fig. 8 shows σ modification makes the algorithm convergeto the optimal point for the case where a < a (Fig 7-a). M C ( $ / M W h ) P(MW) 𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑡𝑦 𝑢𝑡𝑖𝑙𝑖𝑡𝑦 𝜎 x100% (1 − 𝜎 )x100% Fig. 8. Graphical presentation of LUBS convergence for different cases, a)LUBS diverges when agent 1’s generation cost is less convex than generator2’s, b) LUBS converges if agent 1’s generation cost is more convex thangenerator 2’s.
IV. C
ASE S TUDY
The proposed two decision making paradigms have beenimplemented in a case study of a radial 42-bus test feederintroduced in IEEE Standard 399 [20]. One community isconnected on Bus 50. The generator cost functions, powerand reserve limits are described in Table I.
TABLE IP
ARAMETERS AND COST FUNCTIONS OF GENERATORS IN
IEEE S TD . 399 TEST FEEDER C ( P g ) = 0 . αP g + βP g + γ Bus P mg P Mg R mg α β γ
100 Utility 0 2 0 0.1 55 04 Utility 0 12 2 0.3 50 09 µG µG µG µG The temporal demand profile of the entire test feeder isobtained by multiplied by a time-variant scaling factor ρ ( t ) asshown in Fig. 9. S c a li ng F a c t o r ( % ) Fig. 9. Demand scaling factor in 24-hour horizon, applied to obtain powerdemand profile.
The utility optimization problem for 24 hours is solved inMATPOWER for 24 snapshots. The community optimizationproblem is carried out by CVX [21]. Simulation resultsare presented in the following figures Figs. 10-15. The 24-hour profiles or 8-hour profiles clearly demonstrate how the“learning” processes took place.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2449.649.85050.2 (cid:3) a ) λ ( $ / M W ) L U B S
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2446485052 (cid:3) a ) λ ( $ / M W ) S ubg r a d i e n t N od a l P r i ce ( $ / M W ) B o t h m e t hod s SubgradientLUBS
Fig. 10. Comparison of community nodal price behavior through iterations inLUBS and subgradient for 24 consecutive moving-horizon simulations, whereeach iteration is assumed to take two minutes. a) The price in LUBS methodthroughout an entire day, b) the price in subgradient method throughout anentire day, c) a closer look on the price in both methods during a two-hourhorizon. P g ( M W ) Q g ( M VA R ) Fig. 11. A utility generator’s power output profile. P g ( M W ) Q g ( M VA R ) P b ( M W ) E b ( M W h ) Fig. 12. The community’s generator and battery profile. (cid:3) P i m p ( M W ) L U B S (cid:3) P i m p , P e xp ( M W ) S ubg r a d i e n t (cid:3) Time (hour) P i m p , P e xp ( M W ) B o t h m e t hod s LUBS P imp
SG P imp
SG P exp P imp P exp Fig. 13. 8-hour importing/exporting power profiles. (cid:3) a ) µ R ( $ / M W ) L U B S (cid:3) b ) µ R ( $ / M W ) S ubg r a d i e n t c ) µ R ( $ / M W ) B o t h M e t hod s SG µ R LUBS µ R Fig. 14. 8-hour reserve price profile.
Remarks: the comparison of LUBS and subgradient-basedmethod shows that for real-world implementation, LUBS ismore favorable due to the smoother change in the price signalsand power levels. V. C
ONCLUSION
Dual decomposition-based privacy preserving decision mak-ing paradigms for utility-community interaction for multiplehorizons are developed. The developed paradigms enableutility and communities to make their own decisions based onlimited information exchange. The paradigms are suitable forradial connected utility and communities. Implementation forreal-time operation scenarios takes care of forecast and real-time discrepancy by using moving horizon optimization tech-nique and by initiating the next horizon variables based on theinformation from the previous horizon. The information flowarchitectures are explained and the convergence properties R e s e r v e ( M W ) L U B S R e s e r v e ( M W ) S ubg r a d i e n t R e s r v e ( M W ) B o t h m e t hod s SubgradientLUBS
Fig. 15. 8-hour reserve profile. are investigated. The proposed decision making architectureswere tested by a case study and demonstrate the “learning”processes taken by a utility and a community.R
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Vahid R. Disfani received his B.S. degree from Amirkabir University ofTechnology (Tehran, Iran) in 2006, and his M.S. degree from Sharif Universityof Technology (Tehran, Iran) in 2008. Currently, he is pursuing a Ph.D. degreeat University of South Florida, Tampa, FL. His research interests include SmartGrid, integration of renewable energy resources to microgrids, and electricitymarkets.
Zhixin Miao (S’00 M’03 SM’09) received the B.S.E.E. degree from theHuazhong University of Science and Technology,Wuhan, China, in 1992, theM.S.E.E. degree from the Graduate School, Nanjing Automation Research In-stitute, Nanjing, China, in 1997, and the Ph.D. degree in electrical engineeringfrom West Virginia University, Morgantown, in 2002.Currently, he is with the University of South Florida (USF), Tampa. Priorto joining USF in 2009, he was with the Transmission Asset ManagementDepartment with Midwest ISO, St. Paul, MN, from 2002 to 2009. His researchinterests include power system stability, microgrid, and renewable energy.
Lingling Fan received the B.S. and M.S. degrees in electrical engineeringfrom Southeast University, Nanjing, China, in 1994 and 1997, respectively,and the Ph.D. degree in electrical engineering from West Virginia University,Morgantown, in 2001. Currently, she is an Associate Professor with theUniversity of South Florida, Tampa, where she has been since 2009. She was aSenior Engineer in the Transmission Asset Management Department, MidwestISO, St. Paul, MN, form 2001 to 2007, and an Assistant Professor with NorthDakota State University, Fargo, from 2007 to 2009. Her research interestsinclude power systems and power electronics. Dr. Fan serves as a technicalprogram committee chair for IEEE Power System Dynamic PerformanceCommittee and an editor for IEEE Trans. Sustainable Energy.