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A Distributionally Robust Bi-level Optimization Model for Power Market Considering Source–Load Interaction and Carbon Permit Allocation

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Abstract

To control the uncertain risk of source–load interaction, this paper proposes a distributionally robust equilinear model of the power market clearing considering the carbon permit allocation. Firstly, a novel bi-level market-clearing framework that considers the allocation of carbon emission rights is proposed to coordinate energy trading between system operators and load aggregators in the market trading mechanism. Then, based on the conditional value at risk (CVaR) and distributionally robust (DR) theories, a risk avoidance expression and its equivalent convex form for the power constraint under the load aggregators are designed to address the risk of load uncertainty in load aggregators, revealing the underlying mechanism of risk avoidance under the DR CVaR power constraint. By leveraging the characteristics of a normal distribution and the Karush–Kuhn–Tucker condition, the proposed nonlinear bi-level DR CVaR market-clearing model is transformed into an efficient single-level linear model, resulting in reduced model solving difficulty, computational time, and resource consumption. Finally, the simulation of the case shows that the DR CVaR market-clearing model and the equilinear model can realize efficient allocation and complementary optimization of flexible resources in the market environment and improve the operation economy and stability of the power system.

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Abbreviations

CVaR:

Conditional value at risk

DR:

Distributionally robust

LA:

Load aggregator

KKT:

Karush–Kuhn–Tucker

KL:

Kullback–Leibler

RO:

Robust optimization

SP:

Stochastic programming

|⋅|:

Number of elements in “⋅”, if “⋅” is a set

[⋅] + :

Max (0, ⋅)

i :

Index of conventional units

t :

Index of time slot

EQ :

The expectation of probability measure Q

N:

Set of all units

\({\text{N}(\text{E}}_{0},{\Sigma }_{0})\) :

Empirical probability distribution of historical data

T:

Set of all hours

S TP :

Set of periods, and |STP|= T

\({\mathbb{D}}\) :

Set of all probability distributions

\({\mathbb{Q}}\) :

The ambiguity set is based on the KL divergence

\({a}_{i},{b}_{i},{c}_{i}\) :

Coefficients of the production cost function of unit \(i\)

\({a}_{\text{DG}},{b}_{\text{DG}},{c}_{\text{DG}}\) :

Cost coefficient of distributed generation

\({\sigma }_{i}\) :

Carbon intensity of unit \(i\)

\(\eta \) :

KL divergence radius

\(1-\upbeta \) :

Confidence level

\({y}_{\text{LA}}\) :

The load in the load aggregator (MW)

\({\rho }_{\text{Em}}\) :

The carbon emission right price (¥/MWh)

\({\rho }_{\text{EVD}}\) :

Discharge price of energy storage equipment (¥/MWh)

\({\rho }_{\text{EVG}}\) :

Charge price of energy storage equipment (¥/MWh)

\({\eta }_{\text{EVD}}\) :

Discharge efficiency of energy storage equipment

\({\eta }_{\text{EVG}}\) :

Charging efficiency of energy storage equipment

\({\text{Q}}_{i,t}\) :

The carbon emission quota is assigned to the unit \(i\) at time \(t\) (10KT/ \(t\))

\({\underline{\text{P}}}_{\text{DG}}\) :

The lower limit of distributed generation (MW)

\({\overline{\text{P}}}_{\text{DG}}\) :

The upper limit of distributed generation (MW)

\({\text{P}}_{\text{D}}\) :

The system demand load (MW)

\({\overline{\text{P}}}_{\text{EVD}}\) :

Maximum discharge power of energy storage equipment at time \(t\) (MW)

\({\overline{\text{P}}}_{\text{EVG}}\) :

Maximum charge power of energy storage equipment at time \(t\) (MW)

\({\overline{\text{P}}}_{l}\) :

Maximum transmission capacities of the tie-line power (MW)

\({\underline{{\text{P}}}}_{i}\) :

Minimum power output of unit \(i\) (MW)

\({\overline{\text{P}}}_{i}\) :

Maximum power output of unit \(i\) (MW)

\({\text{P}}_{\text{up},i}, {\text{P}}_{\text{down},i}\) :

Ramp-up/ramp-down costs of unit \(i\)

\({\text{P}}_{\text{start},i},{\text{P}}_{\text{shut},i}\) :

Start-up/shut-down costs of unit \(i\)

\({\text{P}}_{\text{LA}}\) :

The power sum of the load aggregator (MW)

\({\underline{\text{S}}}_{\text{OC}}\) :

The lower limit of the available state of the residual charge

\({\overline{\text{S}}}_{\text{OC}}\) :

The upper limit of the available state of the residual charge

\({\underline{\text{T}}}_{\text{on},i}\) :

Minimum uptime of unit \(i\)

\({\underline{\text{T}}}_{\text{off},i}\) :

Minimum downtime of unit \(i\)

\({u}_{i,t}\) :

Binary decision variable to indicate the start-up status of unit \(i\) in period \(t\)

\({\lambda }_{t}\) :

The market clearing price in period \(t\) ($/MWh)

\(\xi \) :

The random fluctuation factor of the load in the load aggregator

x :

The vector of the variables for all the units

y * :

The tie-line power variables

z :

The variables for load aggregator, marginal price \(\lambda \) and \({y}^{*}\)

\({\text{E}}_{i,t}\) :

The carbon emission of the unit \(i\) in period \(t\) (\(10KT/t\))

\({\text{F}}_{\text{DG},t}\) :

Distributed generation cost function in period \(t\)

\({\text{F}}_{\text{EVD},t}\) :

Discharge cost of energy storage equipment in period \(t\)

\({\text{F}}_{\text{EVG},t}\) :

Charge cost of energy storage equipment in period \(t\)

\({\text{F}}_{\text{F},i,t}\) :

Generation cost of the unit \(i\) in period \(t\)

\({\text{F}}_{\text{Em},i,t}\) :

The carbon emission cost of the unit \(i\) in period \(t\)

\({\text{P}}_{\text{DG},t}\) :

The actual output power of distributed generation in period \(t\)(MW)

\({P}_{\text{EVD},t}\) :

The discharge power of energy storage equipment in period \(t\) (MW)

\({P}_{\text{EVG},t}\) :

The charge power of energy storage equipment in period \(t (\text{MW})\)

\({P}_{i,t}\) :

System load demand on unit \(i\) in period \(t\) (MW).

\({P}_{l,t}\) :

The tie-line power in period \(t\) (MW)

\({S}_{\text{OC},t}\) :

The charged state of energy storage devices

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Acknowledgements

This work was sponsored by the Shanghai Sailing Program (21YF1430300), Guangxi Science and Technology Program (AD23023001), and partially supported by the National Natural Science Foundation of China (72361003).

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Authors and Affiliations

  1. School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Chen Zhang & Ying Lai

  2. School of Computer Electronics and Information, The Guangxi Key Laboratory of Multimedia Communication and Network Technology, Guangxi University, Nanning, 530004, China

    Linfeng Yang

  3. Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning, 530006, China

    Chen Zhang

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Zhang, C., Lai, Y. & Yang, L. A Distributionally Robust Bi-level Optimization Model for Power Market Considering Source–Load Interaction and Carbon Permit Allocation. J. Electr. Eng. Technol. (2024). https://doi.org/10.1007/s42835-024-01928-2

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