Preventing Overloading Incidents on Smart Grids: A Multiobjective Combinatorial Optimization Approach

Abstract

Cable overloading is one of the most critical disturbances that may occur in smart grids, as it can cause damage to the distribution power lines. Therefore, the circuits are protected by fuses so that, the overload could trip the fuse, opening the circuit, and stopping the flow and heating. However, sustained overloads, even if they are below the safety limits, could also damage the wires. To prevent overload, smart grid operators can switch the fuses on or off to protect the circuits, or remotely curtail the over-producing/over-consuming users. Nevertheless, making the most appropriate decision is a daunting decision-making task, notably due to contractual and technical obligations. In this paper, we define and formulate the overloading prevention problem as a Multiobjective Mixed Integer Quadratically Constrained Program. We also suggest a solution method using a combinatorial optimization approach with a state-of-the-art exact solver. We evaluate this approach for this real-world problem together with Creos Luxembourg S.A., the leading grid operator in Luxembourg, and show that our method can suggest optimal countermeasures to operators facing potential overloading incidents.

Appendix: Nomenclature

The next list describes several symbols that are used within the body of the document

Indices
b	cabinet index, $b \in \begin{Bmatrix}1,\dots ,o\end{Bmatrix}$
f	fuse index, $f \in \begin{Bmatrix}1,\dots ,2n\end{Bmatrix}$
i	cable index, $i \in \begin{Bmatrix}1,\dots ,n\end{Bmatrix}$
j	linear equation index, $j \in \begin{Bmatrix}1,\dots ,leq\end{Bmatrix}$
k	user index, $k \in \begin{Bmatrix}1,\dots ,m\end{Bmatrix}$
Parameters
$\delta$	measurement frequency coefficient; e.g. $\frac{60}{15} = 4$, for 15 min interval
$\lambda$	maximum allowed current load percentage for all cables, e.g. 80%
$aE_k$	active energy for user k, $aE_k = aEC_k - aEP_k$, $aE_k \in \mathbb {R}$
$aEC_k$	active energy consumption for user k, $aEC_k \in \mathbb {R_+}$
$aEP_k$	active energy production for user k, $aEP_k \in \mathbb {R_+}$
$cc_{bf}$	fuse cabinet indicator; 1 if fuse f belongs to the cabinet b, 0 otherwise
$cl_i$	maximum allowed current load in cable i, e.g. 100 A
$cur_k$	amperage of user k, $cur_k = \frac{\sqrt{aE_k^2+rE_k^2}}{\sqrt{3}\cdot 230}$
$I_R$	curtailed amperage for users, e.g. 20 A
$I_{LC}$	maximum allowed amperage for consumers, e.g. 32 A
$I_{LP}$	maximum allowed amperage for producers, e.g. 60 A
leq	number of linear equations, $leq \in \mathbb {N^*}$
m	number of users, $m \in \mathbb {N^*}$
n	number of cables, $n \in \mathbb {N^*}$
o	number of cabinets (including substations), $o \in \mathbb {N^*}$
$Pl_i$	initial active energy for cable i, $Pl_i = \delta \sum _{k = 1}^{m}uc_{ki}RaE_k$
$Ql_i$	initial reactive energy for cable i, $Ql_i = \delta \sum _{k = 1}^{m}uc_{ki}rE_k$
$RaE_k$	real active energy consumption for user k, $RaE_k = aE_k$, if $cur_k < I_{LC}$, (consumer) or $cur_k < I_{LP}$ (producer), and $RaE_k = RGaE_k$ otherwise
$rE_k$	reactive energy for user k, $rE_k = rEC_k - rEP_k$, $rE_k \in \mathbb {R}$
$rEC_k$	reactive energy consumption for user k, $rEC_k \in \mathbb {R_+}$
$rEP_k$	reactive energy production for user k, $rEP_k \in \mathbb {R_+}$
$RGaE_k$	curtailed active energy for user k, $RGaE_k = \sqrt{|230^2\cdot 3\cdot I_R^2 - rE_k^2|}$, $RGaE_k \in \mathbb {R_+}$
$uc_{ki}$	user cable indicator; 1 if user k is connected with cable i, 0 otherwise
$x_{f}^{0}$	initial fuse state; 1 if fuse f is closed, and 0 otherwise; if $f = 2i$, $x_{f}^{0}$ denotes the initial state of the start fuse of cable i, else if $f = 2i+1$, $x_{f}^{0}$ denotes the initial state of the end fuse of cable i

Variables
$A_{jf}$	coefficient matrix element; for equation j and fuse f, $A_{jf} \in \begin{Bmatrix}-1,0,1\end{Bmatrix}$
$dfcab_b$	cabinet visit indicator; 1 if $\sum _{f = 1}^{2n} cc_{bf}|x_f - x^0_f |\ge 1$, 0 otherwise
$l_i$	actual current load percentage, at cable i; $l_i = \max (\frac{100\sqrt{wp^2_{2i}+wq^2_{2i}}}{230cl_i\sqrt{3}}, \frac{100\sqrt{wp^2_{2i+1}+wq^2_{2i+1}}}{230cl_i\sqrt{3}})$
$P_j$	active load vector element; $P_j = Pl_i\cdot r_i$, if equation j is describing the current flow of cable i, and 0 otherwise, $P_j \in \mathbb {R}$
$Q_j$	reactive load vector element; $Q_j = Ql_i\cdot r_i$, if equation j is describing the current flow of cable i, and 0 otherwise, $Q_j \in \mathbb {R}$
$r_i$	reachability cable state; 1 if cable i is powered and 0 otherwise
$wp_f$	actual active energy vector energy element for fuse f; $wp_f \in \mathbb {R}$
$wq_f$	actual reactive energy vector energy element for fuse f; $wq_f \in \mathbb {R}$
$x_{f}$	fuse state; 1 if fuse f is closed, and 0 otherwise; if $f = 2i$, $x_{f}$ denotes the current state of the start fuse of cable i, else if $f = 2i+1$, $x_{f}$ denotes the current state of the end fuse of cable i