M. Shi, X. Chen, J. Zhou, Y. Chen, J. Wen and H. He, “Distributed Optimal Control of Energy Storages in a DC Microgrid With Communication Delay,” in IEEE Transactions on Smart Grid, vol. 11, no. 3, pp. 2033-2042, May 2020, doi: 10.1109/TSG.2019.2946173.

文章目錄

• 1. Introduction
• 2. Distributed Optimal Problem in DC Microgrid
• 3. PI Consensus Algorithm based Distributed Optimal Control
• • 3.1 PI Consensus Algorithm
  • 3.2 Distributed Optimal Control
  • 3.3 Stability Analysis
• 4. Performance of Distributed Optimal Control with Time Delay
• • 4.1 Steady-state Solution under Time Delay
• 6. Simulation Results

1. Introduction

renewable energy sources (RESs)
energy storages (ESs)

2. Distributed Optimal Problem in DC Microgrid

Maximum Power Point Tracking (MPPT)

$$v_{busi}=v_{ref}?R_{di}{i}_{ESi}\text{\hspace{1em}(1)}$$

$v_{busi}$：終點總線電壓
$i_{ESi}$：第 $i$ 個 ES 輸出電流
$R_{di}$：下垂系數
$v_{ref}$：總線參考電壓

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在多個 ES 之間的功率分配與下垂系數嚴格成反比，即
$$R_{d1}{i}_{ES1}=R_{d2}{i}_{ES2}=?=R_{dN}{i}_{ESN}\text{\hspace{1em}(2)}$$

總線電壓的內部偏差最小，即
$$\min f(v_{bus})=\sum _{i=1}^N\frac{1}{2}(v_{busi}?v_{ref}{)}^2\text{\hspace{1em}(3)}$$

3. PI Consensus Algorithm based Distributed Optimal Control

3.1 PI Consensus Algorithm

常用的基于平均一致性算法的傳統電壓觀測器是一個比例類型（又稱 P-type）
$${\dot{x}}_i(t)=\dot{y}(t)+\gamma (\dot{y}(t)?x_i(t))+\sum _{j\in N_i}{a}_{ij}(x_j(t)?x_i(t))\text{\hspace{1em}(4)}$$

$x_i$：控制變量
$y_i$：控制輸入
$N_i$：鄰居節點
$\gamma \ge 0$：全局估計參數

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$$\begin{array}{rl}\dot{x}(t)=& ?\gamma x_i(t)+\sum _{j\in N_i}{a}_{ij}(x_j(t)?x_i(t))\\ & ?\sum _{j\in N_i}{b}_{ij}(w_j(t)?w_i(t))+\gamma y_i(t)\\ {\dot{w}}_i(t)=& \sum _{j\in N_i}{b}_{ij}(x_j(t)?x_i(t))\end{array}\text{\hspace{1em}(5)}$$

$w_i(t)$：觀測器內部變量

3.2 Distributed Optimal Control

ES 的伏安特性滿足
$$v_{busi}=v_{ref}+u_i?R_{di}{i}_{ESi}\text{\hspace{1em}(6)}$$

$u_i$：是次級控制的輸出信號

由下垂控制引起的電壓降可以表示為
$$v_{di}=R_{di}{i}_{ESi}\text{\hspace{1em}(7)}$$

$$\min f(u):=\sum _{i=1}^N\frac{1}{2}(u_i?v_{di}{)}^2\text{\hspace{1em}(8)}$$

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控制變量 $u_i$ 設計為
$$\begin{array}{rl}{\dot{u}}_i& =\sum _{j\in N_i}{a}_{ij}(v_{dj}?v_{di})?\sum _{j\in N_i}({\zeta }_j?{\zeta }_i)?\gamma {?}_i(u)\\ {\dot{\zeta }}_i& =\sum _{j\in N_i}{b}_{ij}(v_{dj}?v_{di})\end{array}\text{\hspace{1em}(9)}$$

$a_{ij}$：權重系數
$b_{ij}$：權重系數
$\gamma$：權重系數
${\zeta }_i$：內部變量，定義為 $v_{di}$ 和鄰居差值的積分

$$\begin{array}{rl}{\dot{u}}_i& =\sum _{j\in N_i}{a}_{ij}(v_{dj}?v_{di}+v_{leader}?v_{di})?\sum _{j\in N_i}({\zeta }_j?{\zeta }_i)?\gamma {?}_i(u)\\ {\dot{\zeta }}_i& =\sum _{j\in N_i}{b}_{ij}(v_{dj}?v_{di})\end{array}\text{\hspace{1em}(9)}$$

3.3 Stability Analysis

4. Performance of Distributed Optimal Control with Time Delay

4.1 Steady-state Solution under Time Delay

$$\begin{array}{rl}{\dot{u}}_i(t)& =\sum _{j\in N_i}{a}_{ij}(v_{dj}(t?\tau )?v_{di}(t))?\sum _{j\in N_i}({\zeta }_j(t?\tau )?{\zeta }_i(t))?\gamma {?}_i(u_i(t))\\ {\dot{\zeta }}_i& =\sum _{j\in N_i}{b}_{ij}(v_{dj}(t?\tau )?v_{di}(t))\end{array}\text{\hspace{1em}(23)}$$

6. Simulation Results

根據仿真節點數量，可以得到通信關系的Laplacian Matrix

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