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Operations of a microgrid with renewable energy integration and line switching

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Abstract

With the development of new technologies and their integration to the conventional power grid, the smart grid with the capacity of satisfying power demand by large amount of renewable energy is emerging. Microgrid, a small-scale power system with clearly defined electrical boundaries and ability of self-supply, especially by distributed renewable energy, plays a big role in this process. In this paper, we study the operations of a microgrid with solar photovoltaic generators, energy storage system, and power exchanges with main power grid. More specifically, a mixed integer programming model is formulated for decision-making, such as scheduling of generators within the microgrid, islanding operations through line switching and power trades between microgrid and the main grid, charging and discharging operations of storage system, and also line switching within the microgrid, by robust optimization for capturing the uncertainties of solar power generation. To solve the robust optimization formulation, we formulate our model in order to apply the column-and-constraint generation algorithm, and perform numerical experiments on several test cases to validate the proposed model and algorithm.

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Acknowledgements

J.L. Ruiz Duarte is supported by the Mexican National Council of Science and Technology (CONACYT) and the Mexican Department of Energy (SENER) for his PhD program. N. Fan is supported by University of Arizona Faculty Seed Grant (2016–2017).

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Authors and Affiliations

  1. Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, 85721, USA

    José Luis Ruiz Duarte & Neng Fan

Authors
  1. José Luis Ruiz Duarte
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  2. Neng Fan
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Corresponding author

Correspondence to Neng Fan.

Appendices

Appendix A

The third level for the optimization model is shown complete as follows

$$\begin{aligned} \max _{(\mathbf p ,\mathbf f ,\mathbf o ,\theta , \mathbf{pm },\mathbf y ,\mathbf r ,\mathbf q )}~&\sum _{t\in \mathscr {T}}\Big (C^{+F}_{m,t} p^{+F}_{m,t} + C^{+N}_{m,t} p^{+N}_{m,t} - (C^{-F}_{m,t} p^{-F}_{m,t} + C^{-N}_{m,t} p^{-N}_{m,t}) \\&-\sum _{g\in \mathscr {G}}(C_g^p p_{g,t} + c_{g,t}^u + c_{g,t}^d) \\&-\sum _{s\in \mathscr {S}}(C_s^+ r_{s,t}^+ + C_s^- r_{s,t}^-) - \sum _{i\in \mathscr {I}}C^{sh}_{i,t} q_{i,t}\Big )\\ s.t.~&-c^u_{g,t}\le -C^u_g\big (\hat{x}_{g,t}-\hat{x}_{g,t-1}\big ) ,~~\forall g\in \mathscr {G},t\in \{2,\dots ,T\}\\&-c^d_{g,t}\le -C^d_g\big (\hat{x}_{g,t-1}-\hat{x}_{g,t}\big ) ,~~\forall g\in \mathscr {G},t\in \{2,\dots ,T\}\\&-c^u_{g,1}\le -C^u_g\big (\hat{x}_{g,1}-\overline{x}_{g,0}\big ) ,~~\forall g\in \mathscr {G}\\&-c^d_{g,1}\le -C^d_g\big (\overline{x}_{g,0}-\hat{x}_{g,1}\big ) ,~~\forall g\in \mathscr {G}\\&p_{g,t}-p_{g,t-1}\le R^u_g\hat{x}_{g,t-1}+\tilde{R}^u_g\big (\hat{x}_{g,t}-\hat{x}_{g,t-1}\big )+P_g^{max}\big (1-\hat{x}_{g,t}\big ), \\&~~\forall g\in \mathscr {G},t\in \{2,\dots ,T\}\\&p_{g,t-1}-p_{g,t}\le R^d_g\hat{x}_{g,t}+\tilde{R}^d_g\big (\hat{x}_{g,t-1}-\hat{x}_{g,t}\big )+P_g^{max}\big (1-\hat{x}_{g,t-1}\big ), \\&~~\forall g\in \mathscr {G},t\in \{2,\dots ,T\}\\&p_{g,1}\le \overline{p}_{g,0}+R^u_g\overline{x}_{g,0}+\tilde{R}^u_g\big (\hat{x}_{g,1}-\overline{x}_{g,0}\big )+P_g^{max}\big (1-\hat{x}_{g,1}\big ),~~\forall g\in \mathscr {G}\\&-p_{g,1}\le -\overline{p}_{g,0}+R^d_g\hat{x}_{g,1}+\tilde{R}^d_g\big (\overline{x}_{g,0}-\hat{x}_{g,1}\big )+P_g^{max}\big (1-\overline{x}_{g,0}\big ),~~\forall g\in \mathscr {G}\\ \end{aligned}$$
$$\begin{aligned}&-p_{g,t}\le -\hat{x}_{g,t}P_g^{min} ,~~\forall g\in \mathscr {G},t\in \mathscr {T}\\&p_{g,t}\le \hat{x}_{g,t}P_g^{max} ,~~\forall g\in \mathscr {G},t\in \mathscr {T}\\&-f_{e,t} \le \hat{z}_{e,t}F_e ,~~\forall e\in \mathscr {E},t\in \mathscr {T}\\&f_{e,t} \le \hat{z}_{e,t}F_e ,~~\forall e\in \mathscr {E},t\in \mathscr {T}\\&-f_{e,t}+B_e\theta _{i_e}-B_e\theta _{j_e}\le (1-\hat{z}_{e,t})M_e ,~~\forall e \notin \mathscr {E}',t\in \mathscr {T} \\&f_{e,t}-B_e\theta _{i_e}+B_e\theta _{j_e}\le (1-\hat{z}_{e,t})M_e ,~~\forall e \notin \mathscr {E}',t\in \mathscr {T} \\&(p_{m,t}^{-N}+p_{m,t}^{-F})-(p_{m,t}^{+N}+p_{m,t}^{+F})-\sum _{e\in \mathscr {E}'}f_{e,t}\le 0 ,~~\forall t\in \mathscr {T}\\&-(p_{m,t}^{-N}+p_{m,t}^{-F})+(p_{m,t}^{+N}+p_{m,t}^{+F})+\sum _{e\in \mathscr {E}'}f_{e,t}\le 0 ,~~\forall t\in \mathscr {T}\\&-p_{m,t}^{-N}-p_{m,t}^{-F}\le \hat{m}_t\sum _{e\in \mathscr {E}'}F_e ,~~\forall t\in \mathscr {T}\\&p_{m,t}^{-N}+p_{m,t}^{-F}\le \hat{m}_t\sum _{e\in \mathscr {E}'}F_e ,~~\forall t\in \mathscr {T}\\&-p_{m,t}^{+N}-p_{m,t}^{+F}\le \hat{m}_t\sum _{e\in \mathscr {E}'}F_e ,~~\forall t\in \mathscr {T}\\&p_{m,t}^{+N}+p_{m,t}^{+F}\le \hat{m}_t\sum _{e\in \mathscr {E}'}F_e ,~~\forall t\in \mathscr {T}\\&p_{m,t}^{-F}\le \overline{Pt}^{-F}_t ,~~\forall t\in \mathscr {T}\\&p_{m,t}^{+F}\le \overline{Pt}^{+F}_t ,~~\forall t\in \mathscr {T}\\&-y_{s,t}\le -S_s^{min} ,~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&y_{s,t}\le S_s^{max} ,~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&y_{s,1}-r_{s,1}^+\eta _s^++r_{s,1}^-/\eta _s^-\le S_{s,0} ,~~\forall s\in \mathscr {S}\\&-y_{s,1}+r_{s,1}^+\eta _s^+-r_{s,1}^-/\eta _s^-\le -S_{s,0} ,~~\forall s\in \mathscr {S}\\&y_{s,T}\le S_{s,0} ,~~\forall s\in \mathscr {S}\\&-y_{s,T}\le -S_{s,0} ,~~\forall s\in \mathscr {S}\\&y_{s,t}-y_{s,t-1}-r_{s,t}^+\eta _s^++r_{s,t}^-/\eta _s^-\le 0 ,~~\forall s\in \mathscr {S},t\in \{2,\dots ,T\}\\&-y_{s,t}+y_{s,t-1}+r_{s,t}^+\eta _s^+-r_{s,t}^-/\eta _s^-\le 0 ,~~\forall s\in \mathscr {S},t\in \{2,\dots ,T\}\\&r_{s,t}^+ \le R_s^+ ,~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&r_{s,t}^- \le R_s^- ,~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&\sum _{e:j_e=i}f_{e}+\sum _{g:i_g=i}p_{g,t}+\sum _{s:i_s=i}r_{s,t}^-+q_{i,t}-\sum _{e:i_e=i}f_{e}-\sum _{s:i_s=i}r_{s,t}^+\\&\le D_{i,t}-\sum _{pv:i_{pv}=i}\hat{PV}_{pv,t} ,~~\forall i\in \mathscr {I},t\in \mathscr {T}\\&-\sum _{e:j_e=i}f_{e}-\sum _{g:i_g=i}p_{g,t}-\sum _{s:i_s=i}r_{s,t}^--q_{i,t}+\sum _{e:i_e=i}f_{e}+\sum _{s:i_s=i}r_{s,t}^+\\&\le -D_{i,t}+\sum _{pv:i_{pv}=i}\hat{PV}_{pv,t} ,~~\forall i\in \mathscr {I},t\in \mathscr {T}\\&q_{i,t}\le D_{i,t} ,~~\forall i\in \mathscr {I},t\in \mathscr {T}\\&p_{g,t},c^u_{g,t},c^d_{g,t}\ge 0 ,~~\forall g\in \mathscr {G},t\in \mathscr {T}\\&p_{m,t}^{-N},p_{m,t}^{-F},p_{m,t}^{+N},p_{m,t}^{+F}\ge 0 ,~~\forall t\in \mathscr {T}\\&y_{s,t},r_{s,t}^+,r_{s,t}^-\ge 0 ,~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&q_{i,t}\ge 0 ,~~\forall i\in \mathscr {I},t\in \mathscr {T} \end{aligned}$$

Appendix B

The dual problem for the third part of the tri-level problem can be written as

$$\begin{aligned} \min _{\alpha ,\beta ,\gamma ,\delta ,\zeta ,\iota ,\kappa ,\lambda ,\mu ,\nu ,\xi ,\pi ,\tau ,\rho }~&\sum _{g}\big (\alpha _{g,1}\big (-C^u_g\big (\hat{x}_{g,1}-\overline{x}_{g,0}\big )\big )+\alpha '_{g,1}(-C^d_g\big (\overline{x}_{g,0}-\hat{x}_{g,1}\big )\big )\big )\\&+\sum _{g}\sum _{t=2}^{T}\big (\alpha _{g,t}\big (-C^u_g\big (\hat{x}_{g,t}-\hat{x}_{g,t-1}\big )\big )\\&+\alpha '_{g,t}\big (-C^d_g\big (\hat{x}_{g,t-1}-\hat{x}_{g,t}\big )\big )\big )\\&+\sum _{g}\big (\beta _{g,1}\big (\overline{p}_{g,0}+R^u_g\overline{x}_{g,0}+\tilde{R}^u_g\big (\hat{x}_{g,1}-\overline{x}_{g,0}\big )+P_g^{max}\big (1-\hat{x}_{g,1}\big )\big )\\&+\beta '_{g,1}\big (-\overline{p}_{g,0}+R^d_g\hat{x}_{g,1}+\tilde{R}^d_g\big (\overline{x}_{g,0}-\hat{x}_{g,1}\big )+P_g^{max}\big (1-\overline{x}_{g,0}\big )\big )\big )\\&+\sum _{g}\sum _{t=2}^{T}\big (\beta _{g,t}\big (R^u_g\hat{x}_{g,t-1}+\tilde{R}^u_g\big (\hat{x}_{g,t}-\hat{x}_{g,t-1}\big )+P_g^{max}\big (1-\hat{x}_{g,t}\big )\big )\\&+\beta '_{g,t}\big (R^d_g\hat{x}_{g,t}+\tilde{R}^d_g\big (\hat{x}_{g,t-1}-\hat{x}_{g,t}\big )+P_g^{max}\big (1-\hat{x}_{g,t-1}\big )\big )\big )\\&+\sum _{g}\sum _{t}(-\gamma _{g,t}\hat{x}_{g,t}P_g^{min} + \gamma '_{g,t}\hat{x}_{g,t}P_g^{max})\\&+\sum _{e}\sum _{t}(\hat{z}_{e,t}F_e(\delta _{e,t}+\delta '_{e,t}))+\sum _{e\notin \mathscr {E}'}\sum _{t}(1-\hat{z}_{e,t})M_e(\zeta _{e,t}+\zeta '_{e,t})\\&+\sum _{t}(\iota _{t}(0)+\iota '_{t}(0))+\sum _{t}\big (\sum _{e\in \mathscr {E}'}F_e\hat{m}_t(\kappa _{t}+\lambda _{t}+\kappa '_{t}+\lambda '_{t})\big )\\&+\sum _{t}(\mu _t\overline{Pt}^{-F}_t+\mu '_t\overline{Pt}^{+F}_t)+\sum _{s}\sum _{t}(-\nu _{s,t}S_s^{min}+\nu '_{s,t}S_s^{max})\\&+\sum _{s}S_{s,0}(\xi _{s,1}+\xi _{s,T+1}-\xi '_{s,1}-\xi '_{s,T+1})\\&+\sum _{s}\sum _{t=2}^{T}(\xi _{s,t}-\xi '_{s,t})(0)+\sum _{s}\sum _{t}(\pi _{s,t}R_s^++\pi '_{s,t}R_s^-)\\&+\sum _{i}\sum _{t}(\tau _{i,t}D_{i,t})+\sum _{i}\sum _{t}(\rho _{i,t}-\rho '_{i,t})(D_{i,t}-\sum _{pv:i_{pv}=i}\hat{PV}_{pv,t})\\ s.t.~&-\alpha _{g,t}\ge -1 ,~~\forall g\in \mathscr {G},t\in \mathscr {T}\\&-\alpha '_{g,t}\ge -1 ,~~\forall g\in \mathscr {G},t\in \mathscr {T}\\&\beta _{g,t}-\beta '_{g,t}-\beta _{g,t+1}+\beta '_{g,t+1}-\gamma _{g,t}+\gamma '_{g,t}+\rho _{i_g=i,t}-\rho '_{i_g=i,t}\\&\ge -C_g^p ,~~\forall g\in \mathscr {G}, t \in \{ 1,\dots ,T-1\} \\&\beta _{g,T}-\beta '_{g,T}-\gamma _{g,T}+\gamma '_{g,T}+\rho _{i_g=i,T}-\rho '_{i_g=i,T}\ge -C_g^p ,~~\forall g\in \mathscr {G}\\&-\delta _{e,t}+\delta '_{e,t}-\zeta _{e,t}+\zeta '_{e,t}+\rho _{j_e=i,t}-\rho '_{j_e=i,t}\\&-\rho _{i_e=i,t}+\rho '_{i_e=i,t}=0 ,~~\forall e\notin \mathscr {E}',t\in \mathscr {T}\\&-\delta _{e,t}+\delta '_{e,t}-\iota _{t}+\iota '_{t}+\rho _{j_e=i,t}-\rho '_{j_e=i,t}\\&-\rho _{i_e=i,t}+\rho '_{i_e=i,t}=0 ,~~\forall e\in \mathscr {E}',t\in \mathscr {T}\\ \end{aligned}$$
$$\begin{aligned}&B_e\zeta _{e,t}-B_e\zeta '_{e,t}= 0 ,~~\forall e\notin \mathscr {E}',t\in \mathscr {T}\\&-B_e\zeta _{e,t}+B_e\zeta '_{e,t}= 0 ,~~\forall e\notin \mathscr {E}',t\in \mathscr {T}\\&\iota _t-\iota '_t-\kappa _t+\kappa '_t+\mu _t\ge -C^{-F}_{m,t} ,~~\forall t\in \mathscr {T}\\&\iota _t-\iota '_t-\kappa _t+\kappa '_t\ge -C^{-N}_{m,t} ,~~\forall t\in \mathscr {T}\\&-\iota _t+\iota '_t-\lambda _t+\lambda '_t+\mu '_t\ge C^{+F}_{m,t} ,~~\forall t\in \mathscr {T}\\&-\iota _t+\iota '_t-\lambda _t+\lambda '_t\ge C^{+N}_{m,t} ,~~\forall t\in \mathscr {T}\\&-\nu _{s,t}+\nu '_{s,t}+\xi _{s,t}-\xi '_{s,t}-\xi _{s,t+1}+\xi '_{s,t+1}\ge 0 ,\\&~~\forall s\in \mathscr {S}, t\in \{1\dots ,T-1\}\\&-\nu _{s,T}+\nu '_{s,T}+\xi _{s,T}-\xi '_{s,T}+\xi _{s,T+1}-\xi '_{s,T+1}\ge 0 ,~~\forall s\in \mathscr {S}\\&-\eta ^+_s\xi _{s,t}+\eta ^+_s\xi '_{s,t}+\pi _{s,t}-\rho _{i_s,t}+\rho '_{i_s,t}\ge -C_s^+ ,\\&~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&\xi _{s,t}/\eta ^-_s-\xi '_{s,t}/\eta ^-_s+\pi '_{s,t}+\rho _{i_s,t}-\rho '_{i_s,t}\ge -C_s^- ,\\&~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&\tau _{i,t}+\rho _{i,t}-\rho '_{i,t}\ge -C^{sh}_{i,t} ,~~\forall i\in \mathscr {I},t\in \mathscr {T}\\&\alpha _{g,t},\alpha '_{g,t},\beta _{g,t},\beta '_{g,t},\gamma _{g,t},\gamma '_{g,t}\ge 0 ,~~\forall g\in \mathscr {G},t\in \mathscr {T}\\&\delta _{e,t},\delta '_{e,t}\ge 0 ,~~\forall e\in \mathscr {E},t\in \mathscr {T}\\&\zeta _{e,t},\zeta '_{e,t}\ge 0 ,~~\forall e\notin \mathscr {E}',t\in \mathscr {T}\\&\iota _t,\iota '_t,\kappa _t,\kappa '_t,\lambda _t,\lambda '_t,\mu _t,\mu '_t\ge 0 ,~~\forall t\in \mathscr {T}\\&\nu _{s,t},\nu '_{s,t},\xi _{s,t},\xi '_{s,t},\pi _{s,t},\pi '_{s,t}\ge 0 ,~~\forall s\in \mathscr {S},t\in \mathscr {T}\\&\tau _{i,t},\rho _{i,t},\rho '_{i,t}\ge 0,\forall ~~i\in \mathscr {I},t\in \mathscr {T} \end{aligned}$$

Appendix C

For reading purposes, some of the subindexes were removed in the following paragraphs. Note that, as \((\rho ^{(\omega )}_i-\rho '^{(\omega )}_i)(D_i-\sum _{pv:i_{pv}=i}PV_{pv})\) is a bilinear term, we need to use some linearization techniques in order to have a mixed-integer linear program. The uncertainty set \(\mathscr {PV}\), expressed in (10), only depending on variables \(a_{pv,t}\in [0,1]\) and \(b_{pv,t}\in [0,1]\), is a convex set. Then, the optimal value for \(PV_{pv,t}\) must be an extreme points of this set, i.e. \(a_{pv,t}\in \{0,1\}\) and \(b_{pv,t}\in \{0,1\}\). Then, the extreme point of \(PV_{pv,t}\) is given by

$$\begin{aligned}&PV_{pv}^* =PV^{mean}_{pv}-a_{pv}\sigma _{pv}+b_{pv}\sigma _{pv} ,\quad \forall pv \end{aligned}$$
(22)
$$\begin{aligned}&a_{pv},b_{pv}\in \{0,1\},\quad \forall pv \end{aligned}$$
(23)

Then, plugging (22) in the above mentioned bilinear term, we get

$$\begin{aligned}&(\rho ^{(\omega )}_i-\rho '^{(\omega )}_i)(D_i-\sum _{pv:i_{pv}=i}PV^{mean}_{pv})\nonumber \\&+\sum _{pv:i_{pv}=i}\sigma _{pv}(\rho ^{(\omega )}_ia_{pv}-\rho ^{(\omega )}_ib_{pv}-\rho '^{(\omega )}_ia_{pv}+\rho '^{(\omega )}_ib_{pv}) \end{aligned}$$
(24)

The terms involving \(\rho ^{(\omega )}_ia_{pv}\), \(\rho ^{(\omega )}_ib_{pv}\), \(\rho '^{(\omega )}_ia_{pv}\), and \(\rho '^{(\omega )}_ib_{pv}\) are still nonlinear. They can be linearized by replacing \(u^{(\omega )}_{pv}=\rho ^{(\omega )}_{i:i_{pv}=i,t}a_{pv}\), \(v^{(\omega )}_{pv}=\rho ^{(\omega )}_{i:i_{pv}=i,t}b_{pv}\), \(u'^{(\omega )}_{pv}=\rho '^{(\omega )}_{i:i_{pv}=i,t}a_{pv}\), and \(v'^{(\omega )}_{pv}=\rho '^{(\omega )}_{i:i_{pv}=i,t}b_{pv}\).

Then, (24) can be replaced by:

$$\begin{aligned} (\rho ^{(\omega )}_i-\rho '^{(\omega )}_i)(D_i-\sum _{pv:i_{pv}=i}PV^{mean}_{pv})+ \sum _{pv:i_{pv}=i}\sigma _{pv}(u^{(\omega )}_{pv}- v^{(\omega )}_{pv}- u'^{(\omega )}_{pv}+ v'^{(\omega )}_{pv}) \end{aligned}$$

and adding the following constraints, that also must be included in the reformulated problem, as part of uncertainty set, such that \(\mathbf U ^{(\omega )},\mathbf{PV }\in \mathscr {PV}\)

$$\begin{aligned}&u^{(\omega )}_{pv,t} \le a_{pv,t}M ,\quad \forall pv,t\in \mathscr {T}\\&u^{(\omega )}_{pv,t} \ge \rho ^{(\omega )}_{i:i_{pv}=i,t}-M(1-a_{pv,t}) ,\quad \forall pv,t\in \mathscr {T}\\&u^{(\omega )}_{pv,t} \le \rho ^{(\omega )}_{i:i_{pv}=i,t} ,\quad \forall pv,t\in \mathscr {T}\\&v^{(\omega )}_{pv,t} \le b_{pv,t}M ,\quad \forall pv,t\in \mathscr {T}\\&v^{(\omega )}_{pv,t} \ge \rho ^{(\omega )}_{i:i_{pv}=i,t}-M(1-b_{pv,t}) ,\quad \forall pv,t\in \mathscr {T}\\&v^{(\omega )}_{pv,t} \le \rho ^{(\omega )}_{i:i_{pv}=i,t} ,\quad \forall pv,t\in \mathscr {T}\\&u'^{(\omega )}_{pv,t} \le a_{pv,t}M ,\quad \forall pv,t\in \mathscr {T}\\&u'^{(\omega )}_{pv,t} \ge \rho '^{(\omega )}_{i:i_{pv}=i,t}-M(1-a_{pv,t}) ,\quad \forall pv,t\in \mathscr {T}\\&u'^{(\omega )}_{pv,t} \le \rho '^{(\omega )}_{i:i_{pv}=i,t} ,\quad \forall pv,t\in \mathscr {T}\\&v'^{(\omega )}_{pv,t} \le b_{pv,t}M ,\quad \forall pv,t\in \mathscr {T}\\&v'^{(\omega )}_{pv,t} \ge \rho '^{(\omega )}_{i:i_{pv}=i,t}-M(1-b_{pv,t}) ,\quad \forall pv,t\in \mathscr {T}\\&v'^{(\omega )}_{pv,t} \le \rho '^{(\omega )}_{i:i_{pv}=i,t} ,\quad \forall pv,t\in \mathscr {T}\\&b_{pv,t},a_{pv,t} \in \{0,1\} ,\quad \forall pv,t\in \mathscr {T}\\&PV_{pv,t}=PV^{mean}_{pv,t}+\sigma _{pv,t}(-a_{pv,t}+b_{pv,t}),\quad \forall pv,t\in \mathscr {T}\\&PV_{pv,t} \ge 0,\quad \forall pv,t\in \mathscr {T}\\&PV_{pv,t} \le \overline{PV_{pv}},\quad \forall pv,t\in \mathscr {T}\\&\sum _{pv}\sum _{t\in \mathscr {T}} (a_{pv,t}+b_{pv,t})\le \varGamma \end{aligned}$$

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Ruiz Duarte, J.L., Fan, N. Operations of a microgrid with renewable energy integration and line switching. Energy Syst 10, 247–272 (2019). https://doi.org/10.1007/s12667-018-0286-8

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