Optimization operation model of electricity market considering renewable energy accommodation and flexibility requirement

Jinye Yang,Chunyang Liu,Yuanze Mi,Hengxu Zhang,Vladimir Terzija,2

1.Key Laboratory of Power System Intelligent Dispatch and Control of Ministry of Education(Shandong University),Jinan 250061,Shandong,P.R.China

2.The Center for Energy Science and Technology,Skolkovo Institute of Science and Technology,Moscow,Russia

Abstract: The renewable portfolio standard has been promoted in parallel with the reform of the electricity market,and the flexibility requirement of the power system has rapidly increased.To promote renewable energy consumption and improve power system flexibility,a bi-level optimal operation model of the electricity market is proposed.A probabilistic model of the flexibility requirement is established,considering the correlation between wind power,photovoltaic power,and load.A bi-level optimization model is established for the multi-markets; the upper and lower models represent the intra-provincial market and inter-provincial market models,respectively.To efficiently solve the model,it is transformed into a mixed-integer linear programming model using the Karush-Kuhn-Tucker condition and Lagrangian duality theory.The economy and flexibility of the model are verified using a provincial power grid as an example.

Keywords: Renewable energy accommodation,Renewable portfolio standards,Flexibility requirement,Optimization operation,Mixed-integer linear programming.

0 Introduction

The renewable portfolio standard (RPS)by law requires the power system to increasingly accommodate renewable energy (RE)[1].However,large-scale RE penetration intensifies the uncertainty of power system operation,which presents great challenges to the power balance [2-7].Flexibility is defined here as the ability of a system to deploy its resources to respond to changes in net load,where the net load is defined as the remaining load not served by variable generation [8,9].

To promote the consumption of RE and ensure the safe and stable operation of the power system,we can consider two aspects: improving the prediction accuracy of RE and improving the flexibility of the power system [10].In recent years,with the rise of artificial intelligence technology,neural network,machine learning,and other forecasting methods have become effective means to improve the prediction accuracy of wind power and photovoltaic [11-16].The copula function can be applied to consider its spatial and temporal correlation so that the predicting results are more in line with the actual situation [17,18].To improve the flexibility in power systems with a high percentage of RE,the flexibility supply and requirement models must first be studied.The basic characteristics of a high-proportion RE power system were analyzed in[19],and power system flexibility was defined.Based on the flexibility characteristics of the power system,probabilistic models of flexibility supply and requirement were established in reference [20],and an evaluation index for flexibility balance in multi-time scales was proposed.A flexibility index to evaluate the operation scheme of a power system was defined in reference [21].It was concluded that flexibility acquisition has a certain economic cost.These studies focused on the evaluation of power system flexibility,but did not discuss how to improve the flexibility.

Previous research on flexibility supply capacity mainly considered the physical constraints and uncertainty of RE,defined as technical flexibility [22],while ignoring the market influence,defined as market flexibility [23-30].Considering the different ratios of power sources in different regions,an optimal operation model of the bi-level electricity market considering inter-provincial transactions of RE was proposed to promote the accommodation of RE and realize optimal allocation in a wider range [31].By combining and analyzing the content of China’s RPS framework,reference [32] established a multi-agent optimization operation model including the RE market,the conventional energy (CE)market,and the real-time market.Reference [33] established an electricity market model including the transaction of tradable green certificates(TGC)and analyzed the promoting effect of RPS on RE accommodation.Although market flexibility has been studied,few types of electricity transactions have been considered,and the market models must still be improved.

RPS can effectively promote RE accommodation in China.As RE penetration increases,the flexibility requirement for power systems is bound to increase.To increase the power system flexibility,it is necessary to promote RE accommodation and improve the existing electricity market model.Thus,this study proposes a bilevel optimization operation model of the electricity market.The contributions of this study are described as follows.First,a probabilistic model of flexibility requirement is established considering the correlation between wind power,photovoltaic power,and load.Second,inter-provincial RE and CE transactions are considered to improve the bi-level optimization model for the multi-markets.Third,the model is transformed into a mixed-integer linear programming(MILP)model using the Karush-Kuhn-Tucker (KKT)condition and Lagrangian duality theory.

The remainder of this paper is organized as follows.Section 2 presents the models of flexibility supply and requirement for different time scales.Section 3 presents the electricity market framework under the RPS.On this basis,Section 4 presents the inter-provincial and intra-provincial bi-level electricity market models.Section 5 presents case studies,which include wind power,photovoltaic power,and energy storage.The conclusions are presented in Section 6.

1 Flexibility supply and requirement models

1.1 Multi-dimensional stochastic model considering correlation

Assuming that the joint distribution function of random vectorX=(x1,x2,…xn)isH(·),and themarginaldistributions areF1,F2,…,Fn,a Copulafunction isdefined as

whereρcis the linear correlation coefficient.

AsF1,F2,…,Fnare continuous,C(·)is uniquely determined.Thus,H(·)is an n-dimensional joint distribution function with marginal distributionsF1,F2,…,Fn.

By calculating partial derivatives of Eq.(1),the joint probability density function of random vectorXcan be obtained:

wherec(·)is the Copula probability density function;fi(xi)is the probability density function ofxi.

1.2 Flexibility requirement model

Because the imbalance power comes mainly from the load and RE output fluctuations,the power system flexibility requirement model is established considering the load and RE fluctuations.

Using normal distribution fitting,the probability density function of load fluctuation at timetis

where ΔQtis the load fluctuation;μandσare the mean and standard deviation of the load fluctuation probability distribution; their values can be obtained from historical data.

The corresponding normal distribution function can be obtained by integrating the probability density function.Similarly,the probability distribution functions of photovoltaic and wind power output fluctuations can be fitted.Using the Copula function,the joint probability density function of wind power,photovoltaic power,and load fluctuation at timetis

where ΔWtand ΔStare wind power and photovoltaic power output fluctuations,respectively;FΔw,t,FΔs,t,andFΔq,tare the probability distribution functions of wind power,photovoltaic power,and load fluctuation,respectively;fΔw,t,fΔs,t,andfΔq,tare the probability density functions of wind power,photovoltaic power,and load fluctuation,respectively;ρwsqis the linear correlation coefficient,which can be estimated from the historical wind power,photovoltaic power,and load output data.

The probability distribution function of the power system flexibility requirement at timetcan be calculated by triple integration of Eq.(4):

wherexr,tis the power system flexibility requirement at timet;V3={(ΔWt,ΔSt,ΔQt)|ΔWt+ ΔSt+ΔQt≤xr,t} represents an integral region.

The probability distribution function of the flexibility requirement iswhen the time scale is 15 min andwhen it is 30 min.

1.3 Flexibility supply model

The adjustment direction and the time scales are the two key characteristics describing the flexibility of CE units and energy storage devices.The flexibility supply capacity is divided into 15-min and 30-min intervals,corresponding to the flexibility requirement for different time scales.The flexibility supply types considered in this study are shown in Table 1.

Table 1 Flexibility supply types

The flexibility supply capacities of CE units in 15-min and 30-min intervals are shown in Fig.1.

Fig.1 Schematic of conventional unit output

The flexibility supply capacity of conventional units is related to the time scales and the operating points.For simplification,the average output power at adjacent times is used as the operating point.The flexibility model of conventional units at 15-min and 30-min intervals is shown in Eq.(6).

Unlike conventional units,energy storage devices have a fast climbing rate.Assuming that the charging and discharging state of energy storage is constant in each period,the flexibility supply capacity is related to the time scales,the charging and discharging power,and the remaining energy (or state of charge):

2 Electricity market system under RPS

2.1 Electricity market considering RE interprovincial trades

China’s RPS stipulates that the responsibility weight of RE accommodation is set according to provincial administrative regions.Due to the uneven renewable resources in different provinces,research on the interprovincial trade of RE is of great significance for promoting its accommodation.

The “inter-provincial and intra-provincial” electricity market schematic diagram under RPS is shown in Fig.2.

Fig.2 Schematic of the bi-level electricity market

2.2 Bi-level optimization operation framework

The clearing price of the inter-provincial RE market affects the demand for electricity purchase,thereby affecting the clearing results of the electricity market.The demand for inter-provincial electricity purchases also affects the clearing results [34].Thus,we proposed a bi-level optimization model in which the upper model is the intraprovincial market model and the lower model is the interprovincial market model,as shown in Fig.3.There is data transfer and mutual influence between the upper and lower models.To show the bi-level optimization model more clearly,a schematic is presented in Fig.4.

Fig.3 System structure of bi-level electricity market system

Fig.4 Diagram of bi-level optimization model

· Upper-level model

The upper-level model aims at minimizing the total operating cost in the intra-provincial market,consisting of the intra-provincial CE and RE day-ahead market costs,the extra-provincial CE and RE purchasing costs,the auxiliary service market cost,and the TGC market cost.The upperlevel model is subject to the constraints of power balance,unit output,unit climbing,unit startup and shutdown time,reserve,energy storage power and capacity,RE output,the RPS requirement,and flexibility requirements.The upperlevel model outputs inter-provincial power demand as input data for the lower-level model.

· Lower-level model

The lower level model aims at minimizing the cost of inter-provincial electricity purchase.It is subject to the constraints of inter-provincial transmission capacity limit,inter-provincial power balance,and the maximum extraprovincial CE and RE power.The clearing price of the inter-provincial market is transmitted to the upper-level as a known condition.

3 Electricity market optimization operation model

3.1 Intra-provincial electricity market model

· Objective function

(1)Cost in intra-provincial CE market

whereNGis the number of CE units;Ciga ndare t he cost and output power of unitiin periodt,respectively;CiupandCidnare the startup cost and shutdown cost of uniti,respectively.

(2)Cost in intra-provincial RE market

whereNRis the number of RE power stations;Crj,,andare the cost,output power,and prediction power of RE power stationjin periodt,respectively;ρris thepenalty costof abandoningwind power and photovoltaic power.

(3)Cost in inter-provincial RE and CE market

whereCtErandPtErare the clearing price and clearing electricity quantity of the extra-provincial RE market,respectively;CtEgandPtEgare the clearing price and clearing electricity quantity of the extra-provincial CE market,respectively.

(4)Cost in auxiliary service market

whereNBis the number of energy storage devices;CkchandCkdcare the charging and discharging costs,of the energy storage devicekin periodt,respectively.

(5)Cost in TGC market

whereCzis the trade price of TGC;PtLis the load demand in periodt;βis the weight of RPS.

Considering these costs synthetically,the objective function is expressed as

· Deterministic constraints

(1)Power balance constraint

(2)Unit climbing constraints

(3)Unit output constraints

(4)Energy storage device power constraints

whereandare the charging decision variable (1 for charging and 0 otherwise)and discharging decision variable(1 for discharging and 0 otherwise)of energy storage devicekin periodt,respectively.

(5)Energy storage device capacity constraints

whereδkis the self-discharge rate of energy storage devicek;ηkchandηkdcare the charge and discharge efficiency of energy storage devicek,respectively;Ek′,tis the remaining energy of energy storage devicekin periodtconsidering flexibility supply in a 30-min scale.

(6)Charge and discharge state constraint

(7)Equal beginning and ending capacity

(8)Reserve constraints

wherehis the reserve coefficient;ui,tis the operating state variable of conventional unitiin periodt(1 for operating and 0 otherwise).

(9)Unit startup/shutdown cost constraints

whereHiandJiare the single startup cost and shutdown cost of conventional uniti,respectively

(10)Unit startup/shutdown time constraints

whereTdis the minimum downtime andTois the minimum uptime.

(11)RE output constraint

· Uncertainty constraints

At a certain confidence level,the operation result is allowed to fail to meet the flexibility requirements so that the operation result can consider flexibility and economy.As the flexibility requirement model is stochastic,the flexibility supply and requirement constraints can be represented by chance constraints.

(1)Flexibility supply and requirement constraints for 15- min time scale

whereis the flexibility requirement for a 15-min time scale;αis the given confidence level.

(2)Flexibility supply and requirement constraints for 30-min time scale

whereis the flexibility requirement for a 30-min time scale.

3.2 Inter-provincial electricity market model

· Objective function

whereNerandNegare the number of types of extraprovincial RE or CE,respectively;andare the clearing price and quantity of REmin periodt,respectively,andare the clearing price and quantity of CEnin periodt,respectively,andCLis the transmission cost for the inter-provincial channel.

· Constraint conditions

(1)Inter-provincial power balance constraints

whereξLis the line loss rate for the inter-provincial channel;λt1andλt2are the Lagrange dual variables.

(2)Inter-provincial transmission capacity constraint

whereandare the lower and upper limits of channel capacity;andare the Lagrange dual variables.

(3)Maximum power constraints in extra-provincial RE and CE market

whereandare the lower and upper limits of RE trading for delivery;andare the lower and upper limits of CE trading for delivery; and,andare the Lagrange dual variables.

3.3 Model simplification

· Deterministic equivalence of chance constraints

Theorem 1[35]: Letzbe a decision vector,ξbe a random variable with a distribution functionψ,andg(z,ξ)=h(z )-ξbe a random constraint function.The chance constraint condition satisfies: Pr{g(z,ξ)≤ 0} ≥αis true if and only ifh(z)≤Kα,whereαis the confidence level andKα= sup{K|K=ψ-1(1 -α)}.

According to Theorem 1,the deterministic equivalent forms of Eqs.(25)and (26)can be obtained as shown in Eqs.(31)and (32).

· Bi-level optimization model simplification

In the bi-level optimization model,there is data transfer and mutual influence between the upper and lower models.If the upper and lower models are solved separately,reasonable boundary conditions should be assumed in advance; the optimal results can be obtained by iterative calculation.Using Lagrangian duality theory and KKT conditions,the KKT conditions of the lower model can be used as the linear constraint of the upper model,and the bi-level optimization model can be transformed into a single-level model to obtain optimal results without iterative calculation.

(1)Lagrange gradients equal zero

(2)Dual variable non-negative constraints

(3)Relaxation complementary conditions

Using the KKT conditions and the relaxation complementary conditions of the lower model as the constraints of the upper model,the single-level optimization operation model can be obtained.The single-level objective function is expressed by Eq.(13),and the constraints are expressed by Eqs.(14)-(24)and (28)-(35).

· Linearization

There are multiplier terms of two decision variables in the objective function,indicating a quadratic programming model.Using Lagrangian duality theory,the model can be transformed into an MILP model.

The duality of the lower problem is expressed as

According to Lagrangian duality theory,the optimal solution of the objective function of the original problem and the dual problem are the same; minC3′ = maxC3′,and we can obtain:

After linearization,the original bi-level model is transformed into a single-level MILP model.

4 Case study

4.1 Case introduction

Considering one provincial power grid as an example,the unit data are shown in Table 2.The conventional units are categorized as coal-fired and gas-fired.It is assumed that the intra-provincial wind power price is 210 yuan/(MW·h)and the photovoltaic power price is 280 yuan/(MW·h)[27].The flexibility requirements for different time scales are shown in Fig.5.the confidence level of the flexibility requirement is 95%.

Table 2 Test system unit data

Fig.5 Load curve and flexibility requirement for different time scales

The predicted intra-provincial and extra-provincial RE outputs are shown in Fig.6.The TGC price is 200 yuan/(MW·h)[30],and the RPS requirement is 15%.The interprovincial channel capacity is 20000 MW,the line loss rate is 5%,and the trade cost is 50 yuan/(MW·h).The power capacities and prices in the extra-provincial RE and CE market are shown in Table 3.

Fig.6 Predicted typical daily intra-provincial and extraprovincial RE output

4.2 Simulation results and analysis

To demonstrate the advantages of the proposed model,(Model 3),it is compared with two other commonly used models.

(1)Model 1 optimizes the operation only in the intraprovincial electricity market,without considering the flexibility requirements.

(2)Model 2 optimizes the operation only in the intraprovincial electricity market considering the flexibility requirements.

The simulation results of Model 1 and Model 2 are shown in Fig.7 and Fig.8.respectively.

Fig.7 Optimization results in intra-provincial electricity market without considering flexibility requirements (Model 1)

Fig.8 Optimization results in intra-provincial power market considering flexibility requirements (Model 2)

Figs.7 and 8 show that the gas-fired units supply more electricity when considering the flexibility requirements than without considering the flexibility requirements.This is because the lower cost unit is preferred without considering factors such as the climbing rate when the flexibility requirements are not considered.When considering the flexibility requirements,different time scales are satisfied in priority; more gas-fired units start up to meet the flexibility requirement.With flexibility requirements,the throughput energy of the energy storage devices is reduced.

When the inter-provincial electricity market is considered for optimal operation,the RE purchased in the inter-provincial RE market can also be used for RPS.To reduce the total cost and satisfy the RPS requirement,a bilevel operation model is proposed; the simulation results are shown in Fig.9.

Fig.9 Optimization results in inter-provincial and intraprovincial bi-level power markets (proposed Model 1)

Compared with Model 2 and Model 3,the proposed model promotes more RE accommodation through interprovincial trade.As the consumption of RE increases and less expensive CE is purchased outside the province,the electricity supplied by conventional thermal units decreases.Compared with Model 2,the proposed model schedules the coal-fired units to supply much less electricity; however,the output of gas-fired units is greater because the gas-fired units can supply more flexibility.

Table 4 shows the total costs,accommodation of RE,and purchase of TGC using different models.The cost comparison between Model 1 and Model 2 shows that the flexibility requirements increase operating costs.To meet the flexibility requirements for different time scales,flexibility supply is reasonably scheduled with the economic cost.Compared with Model 2 and Model 3,the proposed model decreases the total cost and increases RE accommodation.The proposed model performs well because it makes full use of the energy complementarity between inter-provincial and intra-provincial power grids.

Table 4 Optimization results with different models

4.3 TGC price sensitivity analysis

Table 5 shows the operating costs of the single intraprovincial electricity market and the bi-level electricity market with different RPS requirements and TGC prices.As the RPS requirements and TGC prices increase,the total cost also increases.With the same RPS requirement and TGC price,the proposed model considering inter-provincial electricity trading can significantly reduce operating costs.

Table 5 Operating costs with different TGC prices and RPS requirements

continue

When the RPS requirements are 10% and 15%,the RE electricity in the inter-provincial RE market with a lower price is traded to reduce the total cost and meet RPS requirements.Thus,there is no need to purchase additional TGC,and the total cost is not affected by the TGC price.When the RPS is 20%,the total cost increases,and the consumption of RE decreases with an increase in TGC price.When the TGC price is lower,TGC are purchased in priority to meet the RPS requirement; otherwise,extraprovincial RE is purchased in priority.However,TGC are also obtained through the consumption of RE; overall,TGC still increase the consumption of RE.When the RPS requirement is 25%,it cannot be fulfilled in the intraprovincial and inter-provincial RE markets,and additional TGC are required.

5 Conclusions

With further implementation of RPS,RE accommodation gradually increases.However,large-scale RE penetration presents great challenges to the power balance,which requires more flexibility.This study proposes an electricity market optimization operation model considering RE accommodation and flexibility requirements.The supply and requirement models of power system flexibility for different time scales are established considering the unit climbing constraints and output limits.Considering the inter-provincial RE market and the flexibility requirement,a bi-level operation model is established.The transformed single-level MILP model improves the calculation efficiency.

To meet the flexibility requirements for different time scales,more flexibility units must be started up,which increases the operation cost.The total cost decreases by making full use of the energy complementarity between inter-provincial and intra-provincial power grids.Together,the RPS requirement and TGC price have a great impact on the total operation cost.

Acknowledgements

This work is supported by the National Key R＆D Program of China(2018YFA0702200)and Science and Technology Project of State Grid Shandong Electric Power Corporation (52062518000Q).

Declaration of Competing Interest

We declare that we have no conflict of interest.