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Global Energy Interconnection
Volume 4, Issue 1, Feb 2021, Pages 28-38
Optimal flexibility dispatch of demand side resources with high penetration of renewables:a Stackelberg game method
Keywords
Abstract
To promote the utilization of renewable energy,such as photovoltaics,this paper proposes an optimal flexibility dispatch method for demand-side resources (DSR) based on the Stackelberg game theory.First,the concept of the generalized DSR is analyzed and flexibility models for various DSR are constructed.Second,owing to the characteristics of small capacity but large-scale,an outer approximation is proposed to describe the aggregate flexibility of DSR.Then,the optimal flexibility dispatch model of DSR based on the Stackelberg game is established and a decentralized solution algorithm is designed to obtain the Stackelberg equilibrium.Finally,the actual data are utilized for the case study and the results show that,compared to the traditional centralized optimization method,the proposed optimal flexibility dispatch method can not only reduce the net load variability of the DSR aggregator but is beneficial for all DSR owners,which is more suitable for practical applications.
0 Introduction
Vigorously developing renewable energy,such as wind and solar energy,is an inevitable choice to address energy shortages and environmental problems [1-3].However,affected by weather,environment,and other factors,the output of renewable energy generation has significant intermittency,volatility,and uncertainty [3-6],which sets a higher requirement for the flexibility of power system operation and control.For example,the high penetration of photovoltaics (PV) in California in the United States leads to a “duck curve” in net load [7-8],which significantly increases the demand for the rapid ramping capability(i.e.,flexibility) of the power system.Thus,improving the flexibility of the power system has attracted widespread attention from experts and scholars at home and abroad.
From the perspective of flexibility balance,the flexibility demand of high-proportion renewable energy power systems is mainly derived from the volatility and uncertainty of renewable energy generation and electric load [9-10].Correspondingly,the flexibility supply refers to the resources that can be utilized to respond to the changes in volatility and uncertainty of the power system [11-13].Since renewable energy can be integrated into power systems in both centralized and distributed manners [14-16],the flexibility supply of power systems can be divided into two aspects:conventional power generation-side flexibility supply and generalized demand-side flexibility supply.The former generally refers to a large-scale thermal power plant or fast-ramping gas plants while the latter commonly refers to the flexibility resources at the distribution level.Note that the generalized demand-side flexibility supply not only contains the flexibility of electricity users motivated by the demand response mechanism but also includes the flexibility of the distributed generators (DG),energy storage(ES) devices,etc.,which can usually be divided into two categories:active flexibility resources and passive flexibility supply.The former usually includes controllable DG (such as micro-turbines,diesel generators (DEG),fuel cells,etc.),ES devices,and shiftable loads (such as electric vehicles,temperature-controlled loads,etc.).For the latter,renewable energy curtailment and load shedding are two typical passive flexibility supply methods.Note that,in this paper,the above demand-side resources (DSR) at the distribution level are considered as the generalized demand-side flexibility supply.However,for promoting the utilization of renewable energy and ensuring the reliability of power supply,the passive flexibility resources mentioned above are not considered in this study.
As shown in existing research,DSR plays a significant role in promoting renewable energy utilization and alleviating the variability and uncertainty of the net load of power grids [17].For example,the effectiveness of ES in reducing operating cost cycling and improving the efficiency of the system is illustrated in [18]and the feasibility and benefits of the large-scale charge load for reducing the peak-valley difference of the power grid are verified in [19].Generally,the flexibility dispatch of DSR needs to solve the following two key issues:1) owing to the characteristics of small capacity but large-scale,it is necessary to build flexibility models for large-scale DSR.2) There are multiple differentiated entities (such as DG owner,electricity user,ES owner,etc.) in the dispatch process;thus,the key to optimizing the various DSR is to coordinate the different entities.To model the DSR,existing studies have established flexibility models of controllable DG,shiftable load,and ES,such as the operating flexibility of controllable DG,as illustrated in [11],the demand flexibility model of the power system depicted in [20],and the upward flexibility supply and downward flexibility supply of ES shown in [21].For the flexibility dispatch of DSR,a twostage optimization model for the co-optimization of energy and flexibility reserve provided by ES devices was studied in [22]and a multilevel optimization methodology for the provision of multiple system services was proposed in [23]by harnessing the flexibility of thermal demand.Overall,existing research on the optimal scheduling of demand-side resources has the following shortcomings:first,the existing modeling methods are suitable for scenarios in which the scale of DSR is small.For large-scale DSR,there will be problems of increased variables and complex calculations.Second,the existing optimization scheduling methods are mostly centralized,which fails to focus on the personal privacy of different decision-makers.
In this regard,this paper concentrates on the flexibility dispatch problem of DSR and proposes a Stackelberg game-based optimal dispatch method for various DSR.With respect to DSR modeling,the flexibility model of the single DSR is presented and the outer approximation method is employed to qualify the aggregate flexibility of the large-scale DSR.With respect to DSR dispatching,a Stackelberg game framework is constructed,in which the DSR aggregator works as the leader while various DSRs are followers.Moreover,a decentralized solution method is designed to obtain the Stackelberg equilibrium (SE) in a distributed manner.
The remainder of this paper is organized as follows.Section 1 introduces the proposed optimization framework based on the Stackelberg game.The flexibility models and economic models of DSR are presented in Sections 2 and 3,respectively.The Stackelberg game model and decentralized solution algorithm are presented in Section 4.Section 5 analyzes the numerical results,followed by concluding remarks in Section 6.
1 Optimization framework
1.1 Generalized demand side resources
In general,the DSR is considered a load resource that can proactively respond to price or incentive signals from the power grid or market.In the context of the smart grid,large-scale DG and ES devices (or similar to ES devices,such as electric vehicles) are integrated into the distribution network through advanced information and communication technology.Therefore,the scope of traditional DSR can be expanded to generalized DSR,which includes DG,load resource,and ES [24-25].
(1) DG resource
This type of generalized DSR can be divided into two categories:dispatchable DG (such as fuel cells,microturbines,etc.) and un-dispatchable DG (such as wind power,PV,etc.).Note that “un-dispatchable” indicates that the output of DG cannot be fully controlled.However if necessary,the power shedding of un-dispatchable DG can also be implemented.
(2) Load resources
There are large numbers of electric loads that can cooperate with the grid on the demand side,which present great potential for power regulation.Electric loads can generally be divided into fixed loads and shiftable loads[26].In a smart grid environment,a power grid or DSR aggregator can guide users to proactively participate in optimal operation through incentives or price signals.
(3) ES or similar-to-ES resources
This resource generally includes static ES and electric vehicles.With the development of advanced information and communication technology,such as vehicle-to-grid technology,ES or similar-to-ES resources can serve as both power generation and power demand,which has great flexibility for power regulation.
1.2 Optimization framework
Considering the characteristics of multiple-type,smallcapacity,and large-scale,DSR can provide potential adjustment capability for power grids.To fully explore the flexibility of large-scale DSR,a flexibility optimal dispatch DSR framework is constructed,as shown in Fig.1.The DSR aggregator serves as a coordinator,meeting the power grid regulation needs or participating in the power market bidding by aggregating the flexibility of largescale DSR (only the former situation is considered in this paper).Specifically,the DSR aggregator acts as the leader,formulating internal price signals to guide the behavior of various DSR while all kinds of DSR serve as followers,arranging their own power generation or electricity consumption plans in response to price signals from the aggregator.It should be noted that this article assumes that the DSR aggregator receives signals from power grid demand and its impact on the price of the power grid is ignored [27].
Fig.1 Optimal dispatch framework of DSR
2 Flexibility modeling of DSR
2.1 Flexibility description of DSR
The DSR flexibility can be described by the power regulation capability or range.According to [4]and [5],flexibility can be defined as the ability of a system to deploy its resources to respond to changes in the net load.Thus,the flexibility quantification process of the DSR considered in this study is as follows.
For simplicity,only diesel generation (DEG),PV,electric load,and ES are considered as DSR.Let T ≡{1,2,…,T},T-1 ≡{2,3,…,T},where T is the number of optimization time slots.Then,the flexibility of the DEG can be calculated by:
where G is the set of DEG and Pgmax and Pgmin are the upper and lower limits of DEG g,respectively; Rgup and Rgdown are the upward and downward ramping ratios of DEG g,respectively; Δt is the length of time slot t; and Pg,t is the output of DEG g at time slot t.
Note that when necessary,PV can provide flexibility for the system by reducing its output,which can be calculated by:
where J is the set of PV and Pmax j,t is the upper limit of PV j at time period t,which is taken as the predicted value.Pj,t is the actual output of PV j at time period t.
We assume that the electric load comprises a fixed and shiftable load.The fixed load is generally considered not flexible.The shiftable load flexibility is described by the following:
where I is the set of electricity users,Li,t,
and 
are the electric load,fixed load,and shiftable load of user i at time period t,respectively.
and 
are the upper and lower limits of the shiftable load of user i at time period t,[ti,min,ti,max]is the time range that user i can adjust its shiftable load,and Qi is the demand amount of the shiftable load of user i.
For ES,taking the battery as an example,its flexibility model is as follows:
where K is the set of ES devices.
and
are the power and cumulative energy,respectively,of ES k at time period t.
and
are the upper and lower power limits,respectively,of ES k,and 
and
are the upper and lower energy limits,respectively,of ES k.
2.2 Aggregate flexibility model with outer approximation
DSR has the characteristics of small capacity but large-scale,which are affected by many factors,such as environmental conditions and human habits.In addition,with the increasing DSR scale,traditional flexibility modeling methods based on a single type of DSR face the problem of dimensionality disaster.Therefore,it is necessary to aggregate large-scale DSR and the outer approximation method is employed in this study to quantify the aggregation flexibility of large-scale DSR,through which the complexity of calculation can be reduced.
The aggregation flexibility of DEG can be described by the following:
where 
is the aggregate output of DEG at time slot t.
and
are the aggregate upper and lower limits of DEG,respectively.
and
are the aggregate upward and downward ramping ratios of the DEG,respectively.
Similarly,the aggregate flexibility of PV,shiftable load,and ES are shown in the following formulas.
where 
is the upper limit of the aggregate PV at time period t.
and 
are the upper and lower limits,respectively,of the aggregate shiftable load at time period t.
and 
are the upper and lower power limits of aggregate ES,respectively,and
and 
are the upper and lower energy limits of aggregate ES,respectively.
3 Economic model for flexibility dispatch of DSR
3.1 Price model of power grid
The DSR aggregator acts as the receiver of power grid price signals and subsequently optimizes the power generation or consumption behavior of various DSR.It is assumed that the price signals of the power grid are given in this paper.
where
is the selling price of the power grid and 
is the buying price,which is constant within the optimization range.
3.2 Cost model of DSR aggregator
The DSR aggregator guides the behaviors of various DSR by formulating internal price signals.The detailed price is defined as follows:
where 
is the selling price to the DSR and 
is the buying price from DSR,both of which are determined by the DSR aggregator and:
This study assumes that the DSR aggregator has a double role:1) It can respond to the power regulation demand of the power grid by aggregating the flexibility of various DSR,such as to smooth the net load volatility caused by a high proportion of PV.2) It should minimize the aggregate cost of various DSR.Therefore,the cost model of the DSR aggregator can be described by:
where the cost of the DSR aggregator includes two main parts:operating cost CAGG and fluctuation of the aggregate power fAGG.ω is the weight coefficient of the above two parts.Cgrid,CDG,Cload,and CES are the interactive cost with power grid,DG,electric load,and ES,respectively.NLt is the net load of the aggregate DSR at time period t and Lave is the average of the net load during the operating range.
3.3 Profit models of DSR
(1) Profit model of the DG owner
The benefit of the DG owners can be described by the following:
where the first item denotes the revenue of the DG owner selling electricity to the aggregator,and the second item is the power generation cost of controllable DG,in which aG,bG,and cG are the cost coefficients of DEG.It should be noted that the power generation cost of the PV system is ignored.
(2) Profit model for electricity users
The aggregate cost of electricity users comprises the utility profit and expenditure of purchasing electricity,which can be defined as:
where 
denotes the aggregate utility that the electricity users achieve from consuming power
and κ is the preference parameter [26].
(3) Profit model of the ES owner
The cost of an ES owner comprises profit from interaction with the DSR aggregator and power loss penalty cost,which can be defined as:
where 
is the power loss coefficient of ES.
4 Stackelberg game model and solution
4.1 Stackelberg game model
In this study,the optimal dispatch problem of aggregate DSR is modeled as a Stackelberg game,where the DSR aggregator serves as a leader,who sets the internal price signal to guide the behavior of various DSR,while the DG owner,aggregate users,and ES owner are the followers.Thus,the strategic form of the proposed Stackelberg game can be defined as follows [28-29]:
where N is the set of the DG owner,aggregate user,and ES owner while M refers to the DSR aggregator.The above formula contains the following parts:
1) The players.In the game L,the DSR aggregator is the leader while the DG owner,aggregate user,and ES owner are the followers.
2) The strategies.For the DSR aggregator,its strategies are prices
For the DG owner,aggregate user,and ES owner,the optimal strategies are
respectively.
3) The payoffs.For the DSR aggregator,its payoff is to minimize the aggregate cost and net load fluctuation (i.e.,FAGG).For the DG owner,aggregate user,and ES owner,the payoffs are to maximize their utilities (i.e.,FDG,Fuser,FES respectively).
In the game L defined above,
is the strategy set of all players if this strategy set satisfies the following constraint:
The strategy 
termed as the Stackelberg equilibrium (SE).For the proposed Stackelberg game model L,the existence and uniqueness of the SE can be guaranteed and detailed information for proof can be found in [26]and [27].
4.2 Decentralized solution algorithm
For the proposed Stackelberg game model L,there are many different decision-makers (i.e.,DSR aggregator and DG owner,electricity users,and ES owner),which impede obtaining the SE in a centralized manner because of the privacy protection of various entities.Thus,the traditional solution method cannot be used to solve the problem in this paper (such as transforming the followers’ optimal problems into the equivalent Karush-Kuhn-Tucker conditions,which are added into the leader’s optimal model as the additional constraints).Therefore,an iterative-based decentralized solution algorithm is designed in this section to obtain the SE.The detailed solution process is shown in Algorithms 1-4 and the flowchart of the proposed decentralized solution algorithm is illustrated in Fig.2.First,based on the price signals
and
of the power grid,the DSR aggregator formulates its internal prices
and
solves the cost minimization problem defined by formulas (18-27) and releases the optimized price signals to the followers.Then,motivated by the price signals from the DSR aggregator,each follower optimizes its own strategy by solving the corresponding utility maximization problem separately and returns the obtained strategy back to the DSR aggregator.After receiving the feedback strategies from all followers,the DSR aggregator adjusts the internal price signal and repeats the above interactive process until the convergence condition is reached.
Fig.2 Flowchart of the proposed decentralized solution algorithm
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5 Case study
5.1 Basic data
In this study,a set of DSR is selected as the study case,which comprises DG,PV,shiftable load,and ES devices.The optimization time range T = 24,with Δt is set to 1 h.In addition,we assume that one DSR aggregator is in charge of various DSR and optimizes their power generation or consumption in response to the power grid requirement.For various DSR,we assume the DSR number in sets G,J,I,and K are 300 and the parameters of the single DEG,PV,and ES are listed in Table1 [26,30].The electricity prices are derived from practical data in Henan province,China[26,31].Thus,the predicted output of the aggregate PV and original load consumption on a typical day is shown in Fig.3.Note that all numerical tests were conducted on a laptop with an Intel(R) Core(TM) i7-4790 CPU at 3.60 GHz and 8 GB RAM and the optimal problems were solved using MATLAB software (ver.R2018b) by calling the CPLEX solver (ver.12.5).
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Fig.3 The power curves of DSR in a typical day
Table1 Parameters of the proposed model
Techno-economic parameters Value aG (¥/MWh2) 0.183 bG (¥/MWh) 14.64 cG (¥/h) 48.8 κ 10 mloss ES 0.013
5.2 Validation of the proposed aggregation method
For the DSR aggregator,it is computationally intractable to optimize the output of each type of DSR,especially since the number of DSR is large.Thus,the flexibility aggregation method based on the outer approximation is employed.To verify the effectiveness of the proposed aggregation method,this section considers the aggregation shiftable load as an example.The calculation accuracy and calculation time based on the accurate shiftable load model (i.e.,defined by formula (4)) and the proposed aggregate shiftable load model (i.e.,defined by formula (12)) are analyzed under different shiftable load numbers and the comparative results are shown in Fig.4 and Fig.5.
From Fig.4 and Fig.5,we can conclude that the proposed aggregation method achieves good calculation accuracy.In addition,as the number of shiftable loads increase,the calculation time of the proposed aggregate shiftable load modeling method can be significantly reduced compared with the accurate modeling method.This is because the variables and constraints of the accurate shiftable load model increase dramatically with the increase in the number of shiftable loads,which increases the solution complexity of the optimal model.Therefore,it is not difficult to conclude that the proposed aggregation modeling method is suitable for large-scale shiftable loads;this conclusion is also applicable to other types of DSR.
Fig.4 Change of calculation accuracy with shiftable load number
Fig.5 Change of calculation time with shiftable load number
5.3 Flexibility dispatch results
(1) Results of the DSR aggregator prices
The proposed Stackelberg game-based optimization model is applied to the DSR aggregator and the optimization iterative processes of the DSR aggregator cost and followers’ profits are shown in Fig.6.We note that convergence is reached after approximately 48 iterations.The changing tendency of the DSR aggregator cost is different from that of the followers’ profits.The DSR aggregator cost gradually diminishes while the followers’profits increase with the growth of the iteration number.
The optimized results of the DSR aggregator selling and buying price are shown in Fig.7,from which we can conclude that the DSR aggregator’s selling prices are lower than the power grid’s selling prices (i.e.,the TOU price),and the DSR aggregator’s buying prices are higher than the power grid’s buying prices.Thus,the proposed model is more beneficial for the DSR,which is helpful in reducing the electricity expenditure for users (the cost of purchasing electricity from the DSR aggregator is lower than that from the power grid directly).In addition,the proposed method can increase the revenue for the owners of DS and ES as the power selling price to the DSR aggregator is higher than that to the power grid.
Fig.6 Optimization iterative process
Fig.7 Optimized prices of the DSR aggregator
(2) Results of the DSR strategies
For the DG owner,this paper assumes that the PV power curtailment is not considered; thus,the optimal strategy of the DG owner is the optimal output of aggregate DEG,as can be seen in Fig.8.For electricity users,this study assumes that its electric load comprises a fixed and shiftable load and its strategy is the optimal result of the shiftable load,which is depicted in Fig.9.For the ES owner,its strategy is the optimal output of the aggregate ES,which is illustrated in Fig.8 (a) and (b).In addition,the aggregate net load of the DSR aggregator (i.e.,NL defined by formula(25)) through optimization is shown in Fig.10.Note that the original net load of the DSR aggregator indicates the original load consumption minus the PV output.
As shown in Fig.8-10,we can determine that compared with the original net load,the proposed flexibility dispatch of DSR can reduce the variability of the net load of the DSR aggregator,which is beneficial for the power grid.This is due to the motivation of the proposed internal prices.From Figs.6,8,and 9,we can conclude that during the valley period of the original net load curve (i.e.,12:00-16:00),the optimized internal selling price is lower (which is helpful in promoting power consumption) and the buying price is also lower (which is utilized to limit the discharge power of ES).During the peak period of the original net load curve(i.e.,19:00-23:00),the optimized internal selling price is higher (which is utilized to limit the power consumption) and the internal buying price is also higher (which is helpful to promoting ES discharge).Therefore,the fluctuation of the DSR net load is significantly reduced through the guidance of the DSR aggregator’s internal price signals.
Fig.8 Optimized results of DEG and ES
Fig.9 Optimized power consumption of the shiftable load
Fig.10 Optimized net load of DSR aggregator
(3) Comparison with centralized optimization
To demonstrate the effectiveness of the proposed decentralized solution method,this section presents the centralized method for comparison.In a centralized optimization paradigm,the personal information of the DG owner (e.g.,cost coefficients),electricity user (e.g.,preference parameter),and ES owner (e.g.,loss coefficient)should be provided to the DSR aggregator.Thus,the optimal results in centralized and decentralized paradigms are shown in Table1.
As illustrated in Table2,in the centralized paradigm,the cost of the DSR aggregator is much lower (i.e.,19%lower than that in a decentralized paradigm) while the prosumer profits are all lower than those in the decentralized paradigm.Therefore,we can conclude that in the centralized paradigm,the cost reduction of the DSR aggregator is premised on reducing the profits of the DG owner,electricity user,and ES owner.Fortunately,in the proposed decentralized paradigm,the autonomy and independence of the DG owner,electricity user,and ES owner in decision making is considered and their profits can be guaranteed,which is closer to the actual situation.
Table2 The optimal results in different paradigms
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5.4 Sensitive analysis for the penetration of PV power
From Section 5.3,we can see that the proposed optimization method is helpful in reducing the peak-tovalley difference of the net load of the DSR aggregator (as illustrated in Fig.10).To analyze the effectiveness of the proposed method under different PV penetration rates,this section takes the PV penetration described in Section 6.1 as the benchmark to analyze the impact of the PV penetration ratio on the net load fluctuation of the DSR aggregator,which is qualified by the difference between the maximum and minimum values of the net load defined by formula (25).It is not difficult to draw from Fig.11 that when the change rate of PV penetration varies from -0.2-0.2,the proposed optimization method has a significant effect on reducing the fluctuation of the net load compared with the original value.In addition,the maximum fluctuation of the net load in the proposed method shows an upward trend with the increase in PV penetration but the trend is relatively flat compared with the original case,indicating that the proposed method has better robustness against the change in PV penetration ratio.
Fig.11 Optimized net load of DSR aggregator
6 Conclusion
This paper proposes an optimal flexibility dispatch method for demand-side resources based on the Stackelberg game.An iterative solution algorithm is designed to obtain the Stackelberg equilibrium in a distributed manner.It can be concluded from the simulation results that the proposed method can significantly reduce the solution time while ensuring high modeling accuracy,especially when the scale of the demand-side resources is large.Moreover,compared with the traditional centralized optimization method,the proposed flexibility dispatch method can not only reduce the net load variability of the DSR aggregator but also benefits the DG owner,electricity user,and ES owner,which is suitable for practical applications.It should be noted that the volatility and uncertainty of renewable energy and electric load are the sources of the flexibility demand.For simplicity,this article assumes that the PV output and electric load are predicted values.Thus,the flexibility optimization of DSR with uncertainties of renewables and demand should be studied further.
Acknowledgements
This work was supported by Science and Technology Project of State Grid Hebei Electric Power Company(SGHE0000DKJS2000228).
Declaration of Competing Interest
We declare that we have no conflict of interest.
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Fund Information
supported by Science and Technology Project of State Grid Hebei Electric Power Company (SGHE0000DKJS2000228)；
supported by Science and Technology Project of State Grid Hebei Electric Power Company (SGHE0000DKJS2000228)；
Author
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Peng Lu
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Hao Lv
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Nian Liu
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Tieqiang Wang
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Jianpei Han
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Wenwu Zhang
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Li Ma