A Neurodynamic Algorithm for Energy Scheduling Game in Microgrid Distribution Networks

In this paper, a competitive energy scheduling strategy game of N-microgrids (MGs) inside a distributed network is considered. Each microgrid (MG) aims to maximize its profit under the noncooperative game frame. The strategy-making of each MG depends on its equipment constraints, the aggregate energy supplies of all MGs, and the energy balance of supplies and demands. To solve above discussed problem, a noncooperative game with linear coupled constraints and a distributed neurodynamic algorithm are proposed to seek the generalized Nash equilibrium (GNE). Besides, the correctness and convergence of the proposed algorithm are analyzed in detail. The effectiveness and feasibility of the proposed method are also illustrated via the simulation example.

This work is supported by Natural Science Foundation of China (Grant Nos: 61773320), Fundamental Research Funds for the Central Universities (Grant No. XDJK2020TY003), and also supported by the Natural Science Foundation Project of Chongqing CSTC (Grant No. cstc2018jcyjAX0583).

Authors and Affiliations

1. Chongqing Key Laboratory of Nonlinear Circuits and Intelligent Information Processing, College of Electronic and Information Engineering, Southwest University, Chongqing, 400715, China

   Shifan Wen & Xing He

2. School of Economic Information and Engineering, Southwestern University of Finance and Economics, Chengdu, 611130, China

   Shifan Wen

Corresponding author

Correspondence to Xing He.

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Appendices

1.1 Appendix A: Proof of Lemma 3

Proof

Since the Assumption 1 and the conditions of \(\alpha \), \({\tilde{\psi }}_1\) are satisfied, and \(\kappa _i\left( t \right) \) is strictly continuous, \(\sum _{i=1}^N{{\dot{\kappa }}_i\left( t \right) }=0\) is satisfied for all \(t\geqslant 0\) such that \(\sum _{i=1}^N{{\dot{\kappa }}_i\left( 0 \right) }=0\), then, there is \(\sum _{i=1}^N{\omega _i\left( t \right) }=\sum _{i=1}^N{M^TP_i}\left( t \right) \). Furthermore, it is easy to obtain

Let \(\varsigma \left( t \right) \triangleq \max \sum _{\left( i,j \right) \in {\mathbb {E}}\left( t \right) }{\left| \omega _i-\omega _j \right| }\ \) and \(\omega \left( t \right) =\frac{1}{2}\Vert \omega \left( t \right) -\frac{1}{N}{{\mathbf {1}}}{{\mathbf {1}}}^T\omega \left( t \right) \Vert ^2\). Then, for all \(t\geqslant 0\), there is [32, 33]

where \(\varsigma \left( t \right) \geqslant 0\) is completely continuous and \(\left( \alpha -\left( N-1 \right) {\tilde{\psi }}_1 \right) \) \( \int _0^{+\infty }{\varsigma \left( t \right) }\leqslant \omega \left( 0 \right) <+\infty ,\ \varsigma \left( t \right) \rightarrow 0\) as \(t\rightarrow +\infty \). Then, for \(t\geqslant {\hat{t}}\) and \({\hat{t}}\) is sufficient large, there is \(\omega \left( t \right) \leqslant \varsigma \left( t \right) \) and \({\dot{\omega }}\left( t \right) \leqslant \left( \alpha -\left( N-1 \right) {\tilde{\psi }}_1 \right) \omega \left( t \right) \), the prove is completed. \(\square \)

1.2 Appendix B: Proof of Lemma 5

If there is \(\varXi ^+\times \varPsi ^+\ne \varnothing \), then, for any \(\left( P^+,{\tilde{\lambda }}^+ \right) \in \varXi ^+\times \varPsi ^+\), \(\left\{ t_{\iota } \right\} _{\iota =1}^{+\infty }\) exists such that \(\lim _{\iota \rightarrow +\infty }P\left( t_{\iota } \right) =P^+\) and \(\lim _{\iota \rightarrow +\infty }{\tilde{\lambda }}\left( t_{\iota } \right) ={\tilde{\lambda }}^+\) [29, 34]. Taking the the limit of \(t_{\iota }\) into Eqs. (22) and (27), we can get \(\left( P^+,{\tilde{\lambda }}^+ \right) \) is the solution of Eq. (28) according to the Lemma 2, and the prove is completed.

1.3 Appendix C: Proof of Theorem 1

Sufficiency. If \(\left( {\tilde{\lambda }}^*,P^* \right) \in \varXi ^*\times \varPsi ^*\), then, \(P^*\in \varPi \) according to the definition of projection in the Sect. 2.2. We can also get \(P^*\in \varPi ^*\cap {\mathcal {K}}^*\) since \({\mathbb {M}}^{\text {T}}P^*-D=0\). From Lemma 1, \(P^*\in {\textit{SOL}}\left( \varPi ,\ \nabla _PF+\frac{\varrho }{N}M{\tilde{\lambda }}^* \right) \), i.e., \(P^*\) is the GNE.

Necessity, If the \(P^*\) is the GNE, it seems easy to claim that there exist \({\tilde{\lambda }}^*\) and

with the following conditions satisfied [35, 36]

1. (1)

   \(\forall i\in {\mathbb {N}}\), \(F_i\) is twice continuously differentiable;

2. (2)

   \(\nabla _PF\) is strictly monotone;

3. (3)

   \(\varPi \) is compact and convex;

4. (4)

   \(0\in {\textit{rint}}\left( \varPi -{\mathcal {K}} \right) \).

There exist a vector \(\theta \in R^{2N}\) such that

then, from (32b) and (32c), \(\theta \in \varGamma _{\varPi }\left( P^* \right) \cap \varGamma _{{\mathcal {K}}}\left( P^* \right) =\varGamma _{\varPi \cap {\mathcal {K}}}\left( P^* \right) \). According the definition of the tangent tone, there exist \(P_{\epsilon }\in \varPi \cap {\mathcal {K}}\) and \(t_{\epsilon }>0\) such that \(P_{\epsilon }\rightarrow P^*,\ t_{\epsilon }\rightarrow \text {0, }\,and\ \lim _{\epsilon \rightarrow \infty } \frac{P_{\epsilon }-P^*}{\ t_{\epsilon }}=\theta \). Therefor

which conflicts with (32a), the prove is completed, the \(P^*\) is the unique GNE.

1.4 Appendix D: Proof of Theorem 2

For all \(t>0\), there is \({\dot{P}}\in \varGamma _{\varPi }\left( P \right) ,\, P\left( t \right) \in \varPi \), and we also get

where

for some \({\hat{\alpha }}>0\) and defining \(\phi \left( t \right) =\left( \phi _{1}^{T}\left( t \right) ,\ldots ,\phi _{N}^{T}\left( t \right) \right) ^T\), then, \(\phi \left( t \right) \) reduces exponentially based on the Lemma 2.

Let \(\chi \left( t \right) =\left[ \begin{array}{c} P\left( t \right) \\ {\tilde{\lambda }}\left( t \right) \\ \end{array} \right] \), \(\nabla _P{\tilde{F}}\left( \chi \right) =\left[ \begin{array}{c} \nabla _PF+\frac{\varrho }{N}M{\tilde{\lambda }}\\ -\frac{\varrho }{N}\left( {\mathbb {M}}^{\text {T}}P-D \right) \\ \end{array} \right] \), \({\mathscr {A}}=\varPi \times R\), \({\tilde{\nu }}\left( \chi \right) ={\textit{Pre}}_{{\mathscr {A}}}\left( \chi -\nabla _P{\tilde{F}}\left( \chi \right) \right) \) and \(\chi ^*\in \varXi ^*\times \varPsi ^*\). Then, the Lyapunov function can be formulated as:

It is easy to get \(V\left( t \right) \geqslant 0\). Moreover, the gradient of \(V\left( t \right) \) is as follows

where \({\tilde{\phi }}\left( t \right) =\left[ \begin{matrix} \phi \left( t \right) ,&{} 0\\ \end{matrix} \right] ^T\nabla _{\chi }V\). \(\varPi \) is bounded and according Lemma 4, \(\phi \left( t \right) \) converges exponentially, then, \(\Vert {\tilde{\lambda }}\left( t \right) \Vert \geqslant {\tilde{\alpha }}_1+{\tilde{\alpha }}_2t\) and \(\Vert {\tilde{\phi }}\left( t \right) \Vert \leqslant \left( {\tilde{\alpha }}_3+t{\tilde{\alpha }}_4 \right) e^{-{\tilde{\alpha }}_5t}\) for some constants \({\tilde{\alpha }}_1,\ {\tilde{\alpha }}_2,\ {\tilde{\alpha }}_3,\ {\tilde{\alpha }}_4,\ {\tilde{\alpha }}_5>0\). Hence,

As \(t\rightarrow +\infty \), \(\lim _{t\rightarrow +\infty } P\left( t \right) =P^*\) and \(\lim \sup _{t\rightarrow +\infty } V\left( t \right) <+\infty \) such that \({\tilde{\lambda }}\left( t \right) \) is bounded, consequently, \(\varXi ^+\times \varPsi ^+\ne \varnothing \). From Eq. (27), there is \(\lim _{t\rightarrow +\infty } \dot{{\tilde{\lambda }}}\left( t \right) ={\mathbb {M}}^{\text {T}}P^*-D=0\). Because of the trajectory of (22) is completely continuous and the right side of the (22) about \(P\left( t \right) \) is consistently continuous in t. For \(P\left( t \right) \) is convergent, \(\lim _{t\rightarrow +\infty } {\dot{P}}\left( t \right) =0\) according the Barbalat lemma, the prove is completed.

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