Stochastic Optimization For Residential Demand Response With Unit .
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available stic Optimization for Residential DemandResponse with Unit Commitment and Time of UseZhanle Wang, Member, IEEE, Usman Munawar, Student Member, IEEE, and Raman Paranjape, SeniorMember, IEEEAbstract—As compensation to power generation dispatch, demand response (DR) enables demand controllability by changingthe consumers’ electricity usage patterns, which can be used toreduce electricity cost, integrate renewable energy, and provideancillary services. To reveal the benefits from residential DR,this study develops two approaches: 1) optimal load aggregationunder augmented time-of-use (TOU) pricing; and 2) active DRparticipation in unit commitment (UC) under rewards. We haveshown that plain TOU pricing is not a promising DR policy if residential customers are equipped with home energy managementsystems (EMS). We, therefore, propose an augmented TOU byradial basis functions (RBF). With a 60% participation level, theproposed optimal load aggregation model under the augmentedTOU can reduce the power generation cost by 24% and decreasethe standard deviation of the load profile by 42%. However,these results can be affected by the customer’s participationlevel, which is also quantitatively studied. Specifically, whenthe participation level exceeds 80% this method becomes lessefficient. The second proposed approach, a two-stage stochasticUC model with DR flexibility, reduces the power generation costby 20% and decreases the standard deviation of the load profileby 77%. In addition, the inconvenience of DR participationis quantitatively evaluated, and a Pareto surface is developed,which can be used as a baseline for residential customersto set up the home EMS for DR implementation. Both theproposed mechanisms can be used to improve energy efficiencyby uncovering the residential DR potential.Index Terms—Residential demand response, Residential microgrid, Time-of-use, Unit commitment, Energy management.N OMENCLATURESetsAPBAPCAPTΩAppliancesBackground AppliancesControllable AppliancesTime horizonSet of scenariosThis paper is an expanded paper from the 2020 IEEE InternationalConference on Power Electronics, Smart Grid and Renewable Energy heldon June 2-4, 2020 in Le Meridien Kochi, Kerala, India [1]. This project waspartially supported by SaskPower and Graduate Student Base Funding at theUniversity of Regina.Z. Wang is with Electronic Systems Engineering, University of Regina,Regina, SK, Canada, S4S 0A2, and College of Electrical Engineering, NorthChina University of Science and Technology, Tangshan, China, 063210 (email:zhanle.wang@uregina.ca).U. Manuwar is with Electronic Systems Engineering, University of Regina,Regina, SK, Canada, S4S 0A2 (e-mail: umx648@uregina.ca).R. Paranjape is with Electronic Systems Engineering, University of Regina,Regina, SK, Canada, S4S 0A2(e-mail: raman.paranjape@uregina.ca).Symbols and VariablesλHPenalty coefficient for a home H at time ttµjRadial basis function centersωIndex of a scenarioPtjMaximum power the unit j can generateQHRate power of the power panelf latπflat-rateπtElectricity Price{of f,mid,on}πtPrice in off-peak, mid-peak and on-peak demand periodσBandwidth of radial basis functionMinimum power the unit j can generatePtjtof fMinimum down timeMinimum up timetonapdEPredicted energy consumptionap,ωldPredictedload in the scenario ωtapcltPredicted loadξωProbability of the scenario ωapAppliancecj (·)Operation cost of unit jcm(·)Marginal cost of reserve power jjAActual generation costCtElectricity market-based generation costCtMcUCost of starting up unit j at time tj,trateddRated driving distancef (·)Augment functionLtAggregated load at tLAGAggregated load at time t of a aggregator AGtltap,ωLoad of ap at t in the scenario ωltapLoad of ap at tpjtAmount of power generationpj,ωAmount of power generation of scenario ωtapqratedRated power of apj,ωrt,DGeneration down-reserve of the scenario ωj,ωrt,URjDRjUtap0tap1Generation up-reserve of the scenario ωRamping down limitRamping up limitStart time of of operating appliance in loadforecast modelCompleting time of of operating appliance inload forecast modelCopyright (c) 2021 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available tAcronymsDREMSEVRBFSOCTOUUCMust completing time of operating applianceOn/off status of unit j at time tDemand ResponseEnergy Management SystemElectric VehicleRadial Basis FunctionState of ChargeTime of UseUnit CommitmentI. I NTRODUCTIONHE balance of power generation and demand is crucialand challenging for power system operation. Increasing penetration of distributed energy resources, intermittentrenewable energy and electric vehicles (EV) makes systemoperation more challenging. As an essential compensation topower generation control, DR enables demand controllabilityby changing the electricity usage patterns of consumers [2–9], which includes 1) shifting demand from on-peak to offpeak periods; 2) consuming power based on renewable energyavailability; 3) responding to price signals; and 4) respondingto control signals.Accordingly, DR can 1) save significant capital cost forutilities by reducing peak demand; 2) reduce greenhouse gasemission by integrating renewable energy [3], [4]; 3) costeffectively balance power generation and demand in electricitymarkets [5], [6]; and 4) provide economic ancillary servicesfor power systems [7–9].Residential customers consume about one-third of the electricity, and the residential sector has the most uncovered DRpotential compared with commercial and industrial sectors[10], [11]. Furthermore, the adoption of EVs and vehicle-togrid applications are bringing more DR potential to the residential sector. It is greatly beneficial to unlock the residentialDR potential and transform residential homes/buildings intoactive DR participants. Residential DR has therefore receivedconsiderable attention from academia and industries [12].Among others, optimization plays an important role inDR implementation and various deterministic optimizationmodels [13] and stochastic optimization models [3], [5], [14]have been developed to aggregate controllable loads. Sinceresidential electricity consumption is random in nature, usingstochastic and robust optimization models has become a newtrend in DR studies. It appears that robust optimization tends toprovide more desirable results while stochastic programmingrequires less computational power [15]. Residential DR shouldalso consider customers’ comfort level in addition to energycost [16–18]. Furthermore, since household loads are smallbut numerous, load aggregators can be introduced [19–21].In addition to these DR technologies, to encourage DRparticipation, various policies have been developed and canbe categorized into two groups: price and incentive-based[22]. More specifically, direct load control, ancillary servicesand market programs are considered as the incentive-basedpolicies. On the other hand, price-based policies includes realtime pricing, critical peak pricing and TOU [23–27]. TOUTis used worldwide in the residential sector since its structureis clear and easy to track [28–31]. For example, in Ontario,Canada, TOU is applied to 60% of buildings with the smartmeters in place [32].However, peak demand rebounding in the lowest priceperiod has been reported under TOU since the home EMStends to shift the load to the earliest time with the lowestprice [33], [34]. To deal with this issue, augmented TOUscan be used, e.g., multiple TOUs [35]. However, this mayraise fairness issues since customers are charged with different electricity prices. More appropriate augmenting methodsshould be developed.A deeper reason behind the peak demand rebounding isthat the electricity pricing and load aggregation are coupled.More specifically, the electricity price directly depends onthe magnitude of the load; hence, the plain predeterminedprice structures may not work as expected. To incorporate thisrelationship, attempts have been made to employ game theory[36], [37], mechanism design [38] and bi-level optimization[39–41]. However, these methods require intense computation,which may not be available in embeded home EMSs.Alternatively, demand flexibility can be incorporated intoUC models, which are solved by a utility or an independentsystem operator. UC models can economically schedule various power generation units such as coal, gas and diesel tomeet a predicted load profile. The traditional UC model can bemodified to incorporate incentive-based DR [42]. For instance,modified UC models under DR were presented for renewableenergy integration [43] and minimize the operational cost inthe presence of controllable loads, fuel cells, and solar energy[44].In this study, to reveal the residential DR benefits, weproposed two approaches: 1) optimal load aggregation underaugmented TOU pricing; and 2) active DR participation inUC under rewards. In the first approach, the load aggregationproblem is formulated as a stochastic optimization model andthen reformulated as a deterministic linear programming (LP)model. The LP model can be efficiently solved by a homeEMS. In the second approach, a two-stage stochastic UCmodel with demand flexibility is developed, which can besolved efficiently by a utility. A reward mechanism is thendeveloped to encourage DR participation.This study is extended from our earlier work in [1]. Thedifference and new contributions are: 1) Augmented TOU isdeveloped. 2) UC model with demand flexibility is developed.3) The time interval/resolution is enhanced from 1 hourto 5 minutes, which greatly improves the accuracy of theproposed models. 4) The inconvenience of DR participationis quantitatively evaluated, and a Pareto surface is developed.The contributions of this work are summarized as follows:1) A simple and effective augmented TOU pricing structureis proposed. An optimal load aggregation model is further developed to incorporate the proposed augmentedTOU for DR applications.2) A two-stage stochastic UC model with demand flexibility is developed, based on which a reward mechanism isdeveloped to encourage DR participation. This methodis simple, robust, and practical.Copyright (c) 2021 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available athttp://dx.doi.org/10.1109/TIA.2020.3048643Fig. 1.System architecture of demand response in a residential microgrid3) The inconvenience of DR participation is quantitativelyevaluated and a Pareto surface is developed, which canbe used as a baseline for residential customers to set upthe home EMS for DR implementation.4) Both the proposed mechanisms can be used to improveenergy efficiency by uncovering the residential DR potential.The rest of the paper is structured as follows. Section IIpresents the problem formulation. The simulation results arepresented in Section III followed by a discussion in SectionIV. Section V concludes this study.II. P ROBLEM F ORMULATIONFig. 1 shows the system architecture of the two proposedDR approaches in a residential microgrid. The microgrid mayhave multiple load aggregators and each aggregator contractswith a number of homes to provide DR services. Each homehas a set of controllable loads such as dishwasher, clothesdryer and EV. The load aggregators can forecast and aggregatedemand flexibility of the residential loads and participate forDR applications under TOU or rewards.This section presents the residential load forecast model, thestochastic optimal load aggregation model, the deterministicUC model, and the stochastic UC model with DR flexibility.where ltap is the power consumption of appliance ap at time t.We assume that the appliances are operated at the rated powerapapqratedand the operating period is [tap0 , t1 ] shown in Eq. (1).The standby power of the appliances is assumed to be 0 shownin Eq. (2). AP is the set of appliances in a home, which isfurther classified as a set of background appliances BAP anda set of controllable appliances CAP. The background appliances (e.g., light) cannot be rescheduled in a DR application.The controllable appliance can be rescheduled, e.g., EV. T isthe time horizon of load forecasting and aggregation.The load profile will be determined by the parameters ofapapqrated, tap0 and t1 . These parameters are stochastic in nature.For instance, the operation of most of the appliances is basedon human activity. The rated power of appliances is differentfrom home to home. The statistical information on how peopleuse appliances are from the UK Time Use Survey [45].Except for EVs, the parameters of the other appliances (e.g.,light, refrigerator, oven, etc.) are obtained from [46], whichwas validated by a comprehensive comparison with actualelectricity measurements in the UK.Now, we discuss the method to determine the parametersfor the EV charging (load) model. The rated power is basedon the charger, which is usually a level 1 or level 2 charger inthe residential sector. In this study, we only consider chargingthe EV at home since we focus on residential DR. Thecharging period is based on the battery capacity and its stateof charge (SOC) when an EV arrives at home. Since the SOCapproximately linearly depends on EV driving distance, theSOC can be calculated from the daily driving distance asfollows [47], [48].soc drated ddrated(3)where drated is the rated driving distance, i.e., the drivingdistance of a fully charged EV. drated can be found in thedatasheet of EVs. d is the daily driving distance. The averageand standard deviation of d can be found in [49].The initial charging time tap0 is the same as EV homearriving time assuming that people plug-in the EV and startthe charging on arrival. This time can be assumed as a normaldistribution [47].The aggregated load forecasting is defined as follows. XX apX ap LAG lt lt (4)tH AGap BAPap CAPwhere LAGis the aggregated load prediction at time t of atload aggregator AG.A. Stochastic Residential Load ModelsB. Stochastic Optimal Aggregation ModelIn this study, we use a set of stochastic residential loadmodels developed in our earlier study [33], [34]. These modelsinclude 19 appliances in which dishwasher, clothes dryer andEV are considered as controllable loads.Although the residential power consumption is random innature, the aggregation of a number of household loads canbe statistically stable. Therefore, we develop an optimizationmodel based on expected values. X X X apminimizeE lt f (πt )(5)apapltap qrated,ltap 0,ap ap AP, t [tap0 , t1 ](1)ap[tap0 , t1 ](2) ap AP, t T \ltH AG t T ap CAPCopyright (c) 2021 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available t to:C. Deterministic Unit Commitment Modelap0 ltap qrated,ltap 0,ap ap CAP, t [tap0 , t2 ]ap ap CAP, t T \ [tap0 , t2 ]Xap ,dltap E ap CAP(6)(7)(8)t TWe use a standard deterministic UC model [50] to developthe TOU structures. The deterministic UC model is also usedto calculate the generation cost under the plain TOU andaugmented TOU. The demand flexibility is not incorporatedinto this model. XX jcj (pjt ) cU(12)minimizej ytpjt ,ytjXltap QH , t T(9)ap APsubject toXwhere the objective is to minimize the total electricity cost.Since the set of background appliances BAP are not considered as schedulable, they are not included in the objectivefunction.πt is the electricity price. In this study, we use TOU asthe price structure, which is discuss in the next section. f (·)is an augmenting function to augment the TOU since an unaugmented TOU can cause peak demand rebound. The RBFis used to augment the TOU pricing.RBF et T j J kt µj k22σ 2f (πt ) πt (RBF µRBF )(10)(11)where σ is the bandwidth. j 1, . . . , k and µj is the a RBFcenter. The value of an RBF depends on the distance betweenthe input t and µj , given a value σ. µRBF is the average valueof the RBF. By taking away the average value, the impact ofRBF on TOU is minimized while the variation is still added tothe TOU. Eq. (10) and (11) are discussed in detail in sectionIII.Eq. (6) and (7) are the appliance operation constraints. Thedifference from the forecasting model shown in Eq. (1) and (2)is the completing time tap2 . In the forecasting model, the completing time tapdependson tap10 and the operation period. Inapthe optimization model, t2 is the must completing time, whichis more flexible. For instance, tap2 for EVs can be the homeleaving time in the morning. In other words, when peoplearrive home and plug-in the EV, the optimization model willdetermine when to charge the EV based on electricity price πt .In addition, the power consumption of the controllable loadsare considered as interruptible.Eq. (8) shows that the rescheduled load should consumeap . We focus on thedthe same amount of energy as predicted EDR effect for peak demand reduction and load flattening ratherthan energy reduction. Eq. (9) shows that the rated power ofthe power panel cannot be exceeded, which limits the totaldemand of a single household.In the optimization model, the decision variables areltap , ap CAP. The objective function is the summationof decision variables times predetermined TOU; therefore, theobjective function is linear. In addition, all the constraintsare affine. Therefore, the model is a stochastic LP model.By minimizing the expected electricity cost, the stochasticoptimization model is transformed into a deterministic LPmodel.pjt LAGt , t T(13)j Jytj {0, 1}, t T , j JPtj ytj pjt Ptj ytj , t T , j J RjD pjt pjt 1 RjU ,jytj yt 1 ykjjyt 1 ytj 1 ykj t T , j J j J , t {2 . . . T 1}, k min t ton 1, T(14)(15)(16)(17) j J , t {2 . . . T 1}, (18)k min t tof f 1, TEq. 6 9The objective is to minimize the operational cost of powerunits. cj (pjt ) is the operation cost of unit j as a function ofthe amount of power generation pjt . cUj,t is the cost of startingup unit j.Eq. (13) is the balance constraint of power generation andis the aggregated demand shown indemand, where LAGtEq. (4). In Eq. (14), yjt is the on/off status of unit j at time t,which is a binary constraint. In Eq. (15), Ptj and Ptj are themaximum and minimum amount of power that the unit j cangenerate, respectively. Eq. (16) shows the unit’s ramping upand down constraint, where RjU and RjD are the ramping upand down limit, respectively.Eq. (17) represents that the power generation unit needs tostay on for a minimum amount of time ton after it turns on dueto mechanical design limits or economic reasons. Similarly,as shown in Eq. (18), a unit needs to stay off for a minimumamount of time tof f after it turns off before it can be turnedback on.D. Stochastic Unit Commitment ModelA two-stage stochastic UC model is also developed. In thefirst stage, the generation units are dispatched to meet theexpected demand. The uncertainty is realized in the secondstage by various scenarios: ω Ω, where ω is the indexof a scenario, and Ω is the set of scenarios.Each scenarioPoccurs with a probability of ξω , and ω Ω ξω 1. Also, thegeneration will match the demand in each scenario by varyingj,ωj,ωgeneration up-reserve (rt,U) and down-reserve (rt,D). We alsoincorporate DR flexibility into this stochastic UC model soCopyright (c) 2021 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available athttp://dx.doi.org/10.1109/TIA.2020.3048643that the householders can use DR flexibility as a virtual powerplant to participate in power economic dispatch. XX jcj (pjt ) cUminimizej ytt T j J XXXj,ωj,ωrt,U rt,Dξω cmj ω Ω t T j J Xω ΩξωXλHtH AGXap,ω(ltap,ω ld)tap CAP(19)subject toX pjt E LAG, t Tt(20)j Jj,ωj,ωpj,ω pjt rt,U rt,D, t T , ω ΩtX j,ωpt LAG,ω, t T , ω ΩtTABLE IPARAMETERS OF THE CONTROLLABLE LOADS [54], [55]EVtap2apqratedtap0aptap1 t01.7 kWµ 17σ : 2.8 hµ 4.39 h µ 7σ 0.61 h σ 1 hDishwasher µ 1.13 kW Some usagesσ 0.12 kW µ 10 : 25σ 3hOther usagesµ 18 : 15σ 1.6 hDryerµ 2.52 kW Some usagesσ 0.26 kW µ 9 : 25σ 1.5 hOther usagesµ 16 : 00σ 3.1 hµ 1.41 h t Tσ 0.72 hµ 1.41 h t Tσ 0.35 h(21)(22)j JPtj ytj pj,ω Ptj ytj ,t t T , ω Ω, j JU RjD pj,ω pj,ωtt 1 Rj ,(23) t T , ω Ω, j J (24)Eq. 6 9, 14 18The objective is to minimize the operational cost of powerunits and the inconvenience of residential customers in the UCfor DR purposes. The first term is the expected cost in the firststage, and the second term is the generation cost in the secondstage. cmj (·) is the marginal generation cost function. The thirdterm is the 1 norm of the difference between predicted loadand rescheduled load from participating in the UC. λHt is thepenalty coefficient for a home H at time t, representing theunwillingness or inconvenience to participate in the UC forDR applications. This value can be different from home tohome and from time to time.Eq. (20) is the power-demand balance constraint in thestage 1. Eq. (21) is the realized power generation unit j in allthe scenarios. Eq. (22) is the power-demand balance constraintin all the scenarios. Eq. (23) and Eq. (24) are the power unitoperation constraints in all the scenarios.j,ω j,ωThe decision variables include pjt , ytj , rt,U, rt,D andap,ωlt , ap CAP. The stochastic variables are householdload ltap . For instance, one home may use a dishwasher andanther home may not use a dishwasher in the simulationday. One homeowner may need to drive their EV at 8 am(tap2 8) and another homeowner may need it at 8:30 am.To realize the stochastic variables, sampling is applied for the100 homes. More precisely, each home is one scenario with1% probability. The Monte Carlo method is used to conductthe simulation.III. S IMULATION R ESULTS AND A NALYSISWe considered one load aggregator and 100 homes inthe residential microgrid. Dishwashers, Dryers and EVs wereconsidered as controllable loads, and 20% of homes wereassumed to have EVs. The Nissan Altra-EV with LithiumIon Battery was considered in this study, and the capacity was33 kW [47], [48]. We also assumed the DR participant had aFig. 2.Forecasted load profiles. Left: 4 individual homes (H); Right:aggregated load profile of 4 groups (G) of 25 homeshome EMS to schedule the controllable appliances. The timehorizon was 24 hours, and the time resolution was 5 minutes.The CVX [51], [52] and Gurobi solver [53] were used to solvethe proposed optimization models. Case studies and simulationwere conducted in three scenarios:1) Residential loads were predicted and aggregated withoutDR application. This is our reference scenario.2) Optimal load aggregation under TOU and augmentedTOU.3) DR applications in UC.A. Scenario #1: Residential load forecastWe first simulated an aggregated load profile where therewere no DR applications. The key parameters includedapapqrated, tap0 and t1 . These parameters for controllable loadswere summarized in Table I, which were derived from ourearlier research [54], [55].Fig. 2 shows forecasted load profiles for individual homes,and an aggregated load profile of 25 homes. As can be seenfrom the left plot, the load profile is very random fromhome to home. However, the aggregated load profile canhave high correlation with 25 home profiles shown on theright. Therefore, although the homes are heterogeneous, theaggregated load profile of a large number of homes can bestatistically stable due to similar daily routines.Copyright (c) 2021 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available athttp://dx.doi.org/10.1109/TIA.2020.3048643Fig. 3.Aggregated load profile of 100 homesFig. 4.TABLE IIPARAMETERS IN THE UNIT COMMITMENT MODELCoalGasDieselajbj0.0140.1100.220Ptj60 kW60 kW60 kWPtjRjU , RjD10 kW0 kW0 kW1 kW/min Power generation and load profile in Scenario 1The simulation results from the UC are shown in Fig. 4. Ascan be seen, the units are dispatched in the merit order, i.e.,the units with the lowest cost are dispatched first followed bymore expensive ones. The are two types of generation costs:actual generation cost defined in Eq. (27) and market-basedgeneration cost defined in Eq. (28). X jCtA (27)cj (pjt ) cUj,t ytj JFig. 3 shows the aggregated load profile of 100 homes andthe areas show load from different type of appliances. Thebackground loads are non reschedulable loads such as lighting,TV, and Oven. To simulate this aggregated load profile, we ranthe load forecast model with 1000 homes. The load profile ofthese 1000 homes are divided into 10 groups and each grouphas 100 homes. We then take the average load profile as theexpected load aggregation of 100 homes as shown in Fig. 3.We now discuss the method to use the deterministic UCmodel to calculate the generation cost. We also developed aflat-rate and a TOU structure based on the generation cost.The load profile shown in Fig. 3 was used as the demand inthe deterministic UC model. For demonstration purposes, onlythree types of generation units were considered, including coal,gas, and diesel units. Also, since the simulated grid is muchsmaller than a real-world grid, the generation units’ capacitieswere scaled down, which does not imply any real-worldapplications. We did not constrain the minimum on or offperiod for these units. The starting up cost was also assumedto be 0. The quadratic function was used for power generationcost shown in Eq. (25), and its first order derivative functionat 80% of the capacity was used as marginal generation costof reserve power shown in Eq. (26). j,ωcmrt,Ujcj (pjt ) aj (pjt )2 bj pjt j,ωj,ωj,ω rt,D 1.6 aj Ptj rt,U rt,D(25)(26)The key parameters of the UC model are summarized in TableII. Note that all these parameters can be readily tuned. jCtM max cj (pjt ) cUj,t ytj J(28)where CtA is the actual generation cost and CtM is thegeneration cost in an electricity market.If all the units are owned by a utility or the microgridoperator, the actual generation cost should be considered. Onthe other hand, in a market context, the generation cost isdetermined by the generation cost of most expensive unitsat time t. The electricity market is cleared when the powergeneration units are dispatched to meet the demand. All thedispatched generation units are paid the same as the generationunit with the highest generation cost. For instance, at 20:00,all the units are operating and they are paid by the cost ofDiesel unit since the Diesel unit has the highest generationcost.Since dynamic pricing such as TOU is a market incentivemeasure, we use the market generation cost to calculate theTOU. For a comparison, we also calculate a flat-rate. Tocalculate the equivalent TOU and flat-rate based on marketgeneration cost, we assume a budget balanced market, i.e., thepower generation units do not yield profit. This does not losegenerality since the price can be easily modified to incorporateprofits.Based on the TOU in Ontario Canada [56], we define theTOU with three tiers as follows. of f πt Tof f tπt πtmid t Tmid(29) onπtt TonCopyright (c) 2021 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available athttp://dx.doi.org/10.1109/TIA.2020.3048643Fig. 5.The market-based generation cost, time-of-use price and flat-rateFig. 6. Load profile before and after application of optimal load aggregationwith TOU{of f,mid,on}where πt is the TOU. πtare the price in off-peak,mid-peak and on-peak demand period respectively. Similarly,T{of f,mid,on} are the corresponding time periods.We assume πtmid 1.5πtof f and πtpeak 2πtof f . Then, theof fπt is calculated from the following equation.X of fXXπt Lt 1.5πtof f Lt 2πtof f Ltt Tof f Xt Tmidt TonCtM(30)t Twhere Lt is the aggregated load at t. In this equation, theonly unknown is πtof f .The flat-rate π f lat is calculated from the following equation.Xt Tπ f lat Lt XCtM(31)t TFig. 5 shows the market generation cost, TOU and flat-rate.The electricity market is cleared when the power generationunits are dispatched to meet the demand. All the dispatchedgeneration units are paid the same as the generation unit withthe highest generation cost regardless their actual generationcost. The dots
power generation control, DR enables demand controllability by changing the electricity usage patterns of consumers [2- 9], which includes 1) shifting demand from on-peak to off-peak periods; 2) consuming power based on renewable energy availability; 3) responding to price signals; and 4) responding to control signals.
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Jul 09, 2010 · Stochastic Calculus of Heston’s Stochastic–Volatility Model Floyd B. Hanson Abstract—The Heston (1993) stochastic–volatility model is a square–root diffusion model for the stochastic–variance. It gives rise to a singular diffusion for the distribution according to Fell
are times when the fast stochastic lines either cross above 80 or below 20, while the slow stochastic lines do not. By slowing the lines, the slow stochastic generates fewer trading signals. INTERPRETATION You can see in the figures that the stochastic oscillator fluctuates between zero and 100. A stochastic value of 50 indicates that the closing
