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Optimal Trading Strategy of Data-Center Prosumer in Multiple Local Energy Markets

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Local Energy Markets

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Abstract

The local energy market (LEM) has emerged as an effective way to automatically regulate the power balance and promote distributed energy utilization of the distribution system. With the increasing integration of data centers (DCs) with high load-shifting flexibility into the distribution network, they have gradually become significant market players in the LEMs. However, limited research has been done on the optimal trading strategy of DCs within LEMs. A bi-level optimization model for networked DCs across multiple LEMs is proposed to bridge the research gap. Firstly, since DCs with substantial power consumption can directly impact the market clearing prices, LEM participants are classified into DC-type and conventional-type prosumers, both analytically modeled. Secondly, a LEM model incorporating DCs is created, employing the alternating direction method of multipliers (ADMM) to clear the market in a distributed method, thereby ensuring end-user privacy. Thirdly, a bi-level optimal trading strategy for interconnected DC prosumers across various LEMs is proposed to distribute workloads among DCs optimally. Fourth, the differential evolution and meta-model address the bi-level optimization problem efficiently. The simulation results indicate that this strategy enhances networked DCs’ benefits, boosts renewable energy utilization, and improves overall social welfare.

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Author information

Authors and Affiliations

  1. Southeast University, Nanjing, Jiangsu, China

    Meng Song

  2. Southeast University, Nanjing, Jiangsu, China

    Ciwei Gao

  3. Huazhong University of Science and Technology, Wuhan, Hubei, China

    Mingyu Yan

  4. Nanjing Normal University, Nanjing, Jiangsu, China

    Yunting Yao

  5. Southeast University, Nanjing, Jiangsu, China

    Tao Chen

Authors
  1. Meng Song
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  2. Ciwei Gao
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  3. Mingyu Yan
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  4. Yunting Yao
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  5. Tao Chen
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Corresponding author

Correspondence to Meng Song .

Appendices

Appendix 4.1

First, the LEM clearing problem is converted into the standard form of the ADMM algorithm and then the iterative format of the optimization problem is obtained. Finally, the termination conditions of the iteration process are given.

  1. 1.

    Transformation of the optimization problem

Let \({\varvec{x}}_{\xi }^{\text{T}} = [{\varvec{x}}_{\xi ,1}^{\text{T}} ,\ldots,{\varvec{x}}_{\xi ,k}^{\text{T}} ,\ldots,{\varvec{x}}_{{\xi ,K_{\xi } }}^{\text{T}} ]\) represent the decision variables of all prosumers, where the decision variables of the k-th prosumer are described as follows:

$$x_{\xi ,k}^{T} = \left\{ {\begin{array}{*{20}l} {[m_{\xi ,1,t}^{{\text{IW}}} ,p_{\xi ,1,t}^{{\text{DC}}} ,p_{\xi ,1,t}^{{{\text{grid}} ,s}} ,p_{\xi ,1,t}^{{{\text{grid}} ,b}} ,p_{\xi ,1m,t}^{{\text{LEM}}} ]_{{m \in K_{\xi } ,t \in T}} } \hfill & {k = 1} \hfill \\ {[p_{\xi ,k,t}^{{\text{DG}}} ,p_{\xi ,k,t}^{{{\text{ESS}} ,ch}} ,p_{\xi ,k,t}^{{{\text{ESS}} ,dis}} ,p_{\xi ,k,t}^{{\text{FL}}} ,p_{\xi ,k,t}^{{{\text{grid}} ,s}} ,p_{\xi ,k,t}^{{{\text{grid}} ,b}} ,p_{\xi ,km,t}^{{\text{LEM}}} ]_{{m \in K_{\xi } ,t \in T}} } \hfill & {k = 2,3, \ldots ,K_{\xi } } \hfill \\ \end{array} } \right.$$
(4.40)
  1. (a)

    Convert the maximization problem to the minimization problem.

Define the function \(f_{\xi ,k}\) as follows:

$$f_{\xi ,k} \left( {x_{\xi ,k} } \right) = \left\{ {\begin{array}{*{20}l} {\sum\limits_{t = 1}^{T} {( - B_{\xi ,1,t}^{{\text{DC}}} - B_{\xi ,1,t}^{{\text{grid}}} + C_{\xi ,1,t}^{{{\text{net}}}} )} ,} \hfill & {k = {1}} \hfill \\ {\sum\limits_{t = 1}^{T} {(C_{\xi ,k,t}^{{\text{DG}}} + C_{\xi ,k,t}^{{{\text{ESS}}}} - U_{\xi ,k,t}^{{{\text{FL}}}} - B_{\xi ,k,t}^{{{\text{grid}}}} + C_{\xi ,k,t}^{{{\text{net}}}} )} ,} \hfill & {k = 2, \ldots ,K_{\xi } } \hfill \\ \end{array} } \right.$$
(4.41)

The optimization problem of LEM clearing is equivalent to

$$\begin{aligned} & \min \, \sum\limits_{k = 1}^{{K_{\xi } }} {f_{\xi ,k} } (x_{\xi ,k} ) \\ & {\text{s.t.}}\;\;(4.15) {-} (4.31) \\ \end{aligned}$$
(4.42)
  1. (b)

    Remove inequality constraints.

Define the extended real-valued function of \(f_{\xi ,k}\)

$$\tilde{f}_{\xi ,k} \left( {x_{\xi ,k} } \right) = \left\{ {\begin{array}{*{20}l} {f_{\xi ,k} (x_{\xi ,k} ),x_{\xi ,k} } \hfill & {{\text{constraints}}\;{\text{satisfied}}} \hfill \\ { + \, \infty ,x_{\xi ,k} } \hfill & {{\text{constraints}}\;{\text{not}}\;{\text{satisfied}}} \hfill \\ \end{array} } \right.$$
(4.43)

Putting the prosumers’ constraints implicitly in the objective function, the market optimization problem is equivalent to

$$\tilde{f}_{\xi ,k} \left( {x_{\xi ,k} } \right) = \left\{ {\begin{array}{*{20}l} {f_{\xi ,k} (x_{\xi ,k} ),x_{\xi ,k} } \hfill & {{\text{constraints}}\;{\text{satisfied}}} \hfill \\ { + \, \infty ,x_{\xi ,k} } \hfill & {{\text{constraints}}\;{\text{not}}\;{\text{satisfied}}} \hfill \\ \end{array} } \right.$$
(4.44)
  1. (c)

    Split local variables and global variables.

The \(p_{\xi ,km,t}^{{\text{LEM}}}\) in the above model is the local variable of the k-th prosumer. In the constraints, it is coupled with the local variable \(p_{\xi ,mk,t}^{{\text{LEM}}}\) of the m-th prosumer. In order to decompose the problem, the global variable \(h_{\xi ,km,t}\) is introduced, and the market power balance constraint are rewritten as:

$$\frac{{h_{\xi ,km,t} - h_{\xi ,mk,t} }}{2} - p_{\xi ,km,t}^{{\text{LEM}}} = {0}$$
(4.45)

By multiplying Δt, the market power balance constraint is changed into the electric energy balance constraint. Then the market optimization problem is finally expressed as follows:

$$\begin{aligned} & \min \, \sum\limits_{k = 1}^{{K_{\xi } }} {\tilde{f}_{\xi ,k} } (x_{\xi ,k} ) \\ & {\text{s.t.}}\;\;\left( {\frac{{h_{\xi ,km,t} - h_{\xi ,mk,t} }}{2} - p_{\xi ,km,t}^{{\text{LEM}}} } \right)\Delta t = {0} \\ \end{aligned}$$
(4.46)
  1. 2.

    Iterative format for optimization problems.

Define \({\varvec{H}}_{\xi }^{\text{T}} = [h_{\xi ,km,t} ](m \in K_{\xi } ,t \in T,k \in K_{\xi } )\).

Let \({\varvec{\Pi }}_{\xi }^{\text{T}} = [{\varvec{\Pi }}_{\xi ,1}^{\text{T}} , \ldots,{\varvec{\Pi }}_{\xi ,k}^{\text{T}} , \ldots,{\varvec{\Pi }}_{{\xi ,K_{\xi } }}^{\text{T}} ]\) denote the dual variable of all constraints, where \({\varvec{\Pi }}_{\xi ,k}^{\text{T}} = [\pi_{\xi ,km,t} ](m \in K_{\xi } ,t \in T,k \in K_{\xi } )\).

The augmented Lagrange function of the optimization problem is

$$L_{\rho ,\xi } ({\varvec{x}}_{\xi } ,{\varvec{H}}_{\xi } ,{\varvec{\Pi }}_{\xi } ) = \sum\limits_{k = 1}^{{K_{\xi } }} {L_{\rho ,\xi ,k} } ({\varvec{x}}_{\xi ,k} ,{\varvec{H}}_{\xi } ,{\varvec{\Pi }}_{\xi ,k} )$$
(4.47)

where

$$\begin{aligned} L_{\rho ,\xi ,k} ({\varvec{x}}_{\xi ,k} ,{\varvec{H}}_{\xi } ,{\varvec{\Pi }}_{\xi ,k} ) & = \tilde{f}_{\xi ,k} ({\varvec{x}}_{\xi ,k} ) + \sum\limits_{t = 1}^{T} {\sum\limits_{m = 1}^{{K_{\xi } }} {\left( {\pi_{\xi ,km,t} \left( {\frac{{h_{\xi ,km,t} - h_{\xi ,mk,t} }}{2} - p_{\xi ,km,t}^{{\text{LEM}}} } \right)} \right.} }\Delta t \\ & \quad + \left. {\frac{{\rho_{\xi } }}{2}\left( {\frac{{h_{\xi ,km,t} - h_{\xi ,mk,t} }}{2} - p_{\xi ,km,t}^{{\text{LEM}}} } \right)^{2}\Delta t^{2} } \right) \\ \end{aligned}$$
(4.48)

The iterative process of the ADMM algorithm is as follows

$$\begin{aligned} {\varvec{x}}_{\xi ,k}^{l + 1} & = \mathop {\arg \min }\limits_{{{{x}}_{\xi ,k} }} L_{\rho ,\xi ,k} ({\varvec{x}}_{\xi ,k} ,{\varvec{H}}_{\xi }^{l} ,{\varvec{\Pi }}_{\xi ,k}^{l} )h_{\xi ,km,t}^{l + 1} \\ & = \frac{{p_{\xi ,km,t}^{{{\text{LEM}} ,l + 1}} - p_{\xi ,mk,t}^{{{\text{LEM}} ,l + 1}} }}{2} - \frac{{\pi_{\xi ,km,t}^{l} - \pi_{\xi ,mk,t}^{l} }}{{2\rho_{\xi }\Delta t}}\pi_{\xi ,km,t}^{l + 1} \\ & = \pi_{\xi ,km,t}^{l} + \rho_{\xi } \left( {\frac{{h_{\xi ,km,t}^{l + 1} - h_{\xi ,mk,t}^{l + 1} }}{2} - p_{\xi ,km,t}^{{{\text{LEM}} ,l + 1}} } \right)\Delta t \\ \end{aligned}$$
(4.49)

Combining the 2nd and 3rd of the above formulas gives

$$\pi_{\xi ,km,t}^{l + 1} = - \rho_{\xi } \frac{{p_{\xi ,km,t}^{{{\text{LEM}} ,l + 1}} + p_{\xi ,mk,t}^{{{\text{LEM}} ,l + 1}} }}{2}\Delta t + \frac{{\pi_{\xi ,km,t}^{l} + \pi_{\xi ,mk,t}^{l} }}{2}$$
(4.50)

Thus

$$\pi_{\xi ,km,t}^{l + 1} - \pi_{\xi ,mk,t}^{l + 1} = 0$$
(4.51)

Choose an appropriate initial value so that \(\pi_{\xi ,km,t}^{0} - \pi_{\xi ,km,t}^{0} = 0\), then

$$\pi_{\xi ,km,t}^{l} - \pi_{\xi ,mk,t}^{l} = 0$$
(4.52)
$$h_{\xi ,km,t}^{l + 1} = \frac{{p_{\xi ,km,t}^{{{\text{LEM}} ,l + 1}} - p_{\xi ,mk,t}^{{{\text{LEM}} ,l + 1}} }}{2}$$
(4.53)

Therefore, the iterative process of ADMM algorithm can be expressed as

$$\begin{aligned} {\varvec{x}}_{\xi ,k}^{l + 1} & = \mathop {\arg \min }\limits_{{{\varvec{x}}_{\xi ,k} }} \left( {\tilde{f}_{\xi ,k} \left( {{\varvec{x}}_{\xi ,k} } \right)} \right. \\ & \quad + \sum\limits_{t = 1}^{T} {\sum\limits_{m = 1}^{{K_{\xi } }} ( } \pi_{\xi ,km,t}^{l} (\frac{{p_{\xi ,km,t}^{{{\text{LEM}} ,l}} - p_{\xi ,mk,t}^{{{\text{LEM}} ,l}} }}{2} - p_{\xi ,km,t}^{{\text{LEM}}} )\Delta t \\ & \quad + \left. {\frac{{\rho_{\xi } }}{2}\left( {\frac{{p_{\xi ,km,t}^{{{\text{LEM}} ,l}} - p_{\xi ,mk,t}^{{{\text{LEM}} ,l}} }}{2} - p_{\xi ,km,t}^{{\text{LEM}}} } \right)^{2}\Delta t^{2} } \right)\pi_{\xi ,km,t}^{l + 1} \\ & = \pi_{\xi ,km,t}^{l} - \rho_{\xi } \frac{{p_{\xi ,km,t}^{{{\text{LEM}} ,l + 1}} + p_{\xi ,mk,t}^{{{\text{LEM}} ,l + 1}} }}{2}\Delta t \\ \end{aligned}$$
(4.54)
  1. 3.

    The iteration termination condition of ADMM algorithm.

    $$\left\{ {\begin{array}{*{20}l} {\left\| {{\varvec{r}}^{l + 1} } \right\|_{2} \le \varepsilon^{{{\text{pri}}}} } \hfill \\ {\left\| {{\varvec{s}}^{l + 1} } \right\|_{2} \le \varepsilon^{{{\text{dual}}}} } \hfill \\ \end{array} } \right.$$
    (4.55)

Appendix 4.2

See Fig. 4.11 and Table 4.5.

Table 4.5 Prosumer’s electricity purchase/selling price from/to distribution system
Full size table
Fig. 4.11
figure 11

Mean arrival rate of workloads

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Appendix 4.3

See Figs. 4.12, 4.13, 4.14 and 4.15.

Fig. 4.12
figure 12

Convergence curve of ADMM algorithm

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Fig. 4.13
figure 13

Operation optimization scheme of PV and FL at node 6

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Fig. 4.14
figure 14

Operation optimization scheme of PV, ESS, and FL at node 8

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Fig. 4.15
figure 15

LEM trading power and price of DC during 10:00–11:00

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Appendix 4.4

See Fig. 4.16.

Fig. 4.16
figure 16

Profit changes of conventional prosumers with the proposed bi-level optimization

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Song, M., Gao, C., Yan, M., Yao, Y., Chen, T. (2025). Optimal Trading Strategy of Data-Center Prosumer in Multiple Local Energy Markets. In: Local Energy Markets. Springer, Singapore. https://doi.org/10.1007/978-981-97-9750-9_4

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