SCIENTIA SINICA Informationis, Volume 48 , Issue 10 : 1395-1408(2018) https://doi.org/10.1360/N112018-00020

Event-triggered-based consensus approach for economic dispatch problem in a microgrid

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  • ReceivedApr 1, 2018
  • AcceptedApr 17, 2018
  • PublishedOct 12, 2018


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  • Figure 1

    Topology 1

  • Figure 2

    (Color online) General consensus algorithm. (a) $\lambda_i$; (b) $P_{G_i}$

  • Figure 3

    (Color online) Event-triggered algorithm. (a) $\lambda$ and event-trigged instants; (b) state error $e(t)$

  • Figure 4

    (Color online) $\mu$. (a) $\mu$ and event-trigged instants; (b) state error $e(t)$

  • Figure 5

    (Color online) $\phi$. (a) $\phi$ and event-trigged instants; (b) state error $e(t)$

  • Figure 6

    (Color online) The evolution before and after capacity constraints are imposed. (a) $\lambda$; (b) $P_{G_i}$

  • Figure 7

    Topology 2

  • Figure 8

    (Color online) Event-triggered algorithm with another given topology. (a) $\lambda$ and event-trigged instants;protect łinebreak (b) $P_{G_i}$

  • Table 1   Generator parameters
    Agent $a_i$ (/MW$^2$) $b_i$ (/MW) $P_{{G_i}{\rm~max}}$ (MW)
    1 0.04 2.0 70
    2 0.035 1.3 80
    3 0.02 2.8 50
    4 0.03 3.0 90
    5 0.05 3.5 80

    Algorithm 1 算法流程

    Require:${\lambda}(0)$, $P_{L_{ik}}$, $P_{G_i}(0)$;

    Output:$\dot{\xi}_i(t)=\hat{\lambda}_i(t)$, $~\,~\frac{1}{2a_i}\dot{\lambda}_i(t)=\frac{1}{2}\sum\nolimits_{i=1}^{n}a_{ij}(\hat{\lambda}_j(t)~-\hat{\lambda}_i(t))+\frac{1}{2}\dot{P}_{G_i}(t)$,



    如果 $P_{G_i}^*~<0$, 那么$P_{G_i}^*=0$; 如果 $P_{G_i}^*>P_{{G_i}{\rm~max}}$, 那么 $P_{G_i}^*=P_{{G_i}{\rm~max}}$. 从而确定 $\Theta$, $\bar{\Theta}$;

    $\mu_i(0)~\Leftarrow~\left\{\begin{aligned} &\frac{\lambda^*}{2a_i}-\frac{b_i}{2a_i}-\bar{P}_{G_i},~~~~i\in~\bar{\Theta},\\ &0,~~~~~~~~i\in~\Theta; \end{aligned}\right.$

    $\phi_i(0)~\Leftarrow~\left\{\begin{aligned} &\frac{1}{2a_i},~~~i\in~\Theta,\\ &0,~~~~i\in~\bar{\Theta}. \end{aligned}\right.$

    Require:$\mu_i(0)$, $\phi_i(0)$;

    Output:$\dot{\mu}_i(t)=\sum\nolimits_{i=1}^{n}a_{ij}(\hat{\mu}_j(t)-\hat{\mu}_i(t))$; $~\,~\dot{\phi}_i(t)=\sum\nolimits_{i=1}^{n}a_{ij}(\hat{\phi}_j(t)-\hat{\phi}_i(t))$;


    $\bar{P}_{G_i}~\Leftarrow~\left\{\begin{aligned} &~\frac~{\bar{\lambda}_i(t)-b_i}{2a_i},~~~0<~\frac~{\lambda^*_i(t)-b_i}{2a_i}<~P_{{G_i}{\rm~max}};\\ &~P_{{G_i}{\rm~max}},~~~\,~\frac~{\lambda^*_i(t)-b_i}{2a_i}\geq~P_{{G_i}{\rm~max}};\\ &~0,~~~~~\,~\frac~{\lambda^*_i(t)-b_i}{2a_i}\leq0. \end{aligned}\right.$结果: $\lambda^*$, $P_{G_i}^*$ 或者 $\bar{\lambda}$, $\bar{P}_{G_i}$.