# Problem Specifications

In addition to the standard power flow solve_mc_pf, and optimal power flow solve_mc_opf, there are several notable problem specifications included in PowerModelsDistribution.

## Optimal Power Flow (OPF) with On-Load Tap Changers (OLTC)

This problem is identical to mc_opf, except that all transformers are now modelled as on-load tap changers (OLTCs). Each phase has an individual tap ratio, which can be either variable or fixed, as specified in the data model.

### OLTC Objective

objective_mc_min_fuel_cost

### OLTC Variables

variable_mc_oltc_transformer_tap

### OLTC Constraints

constraint_mc_transformer_power(pm, i, fix_taps=false)

## Minimal Load Delta (MLD) Problem Specification

Load shed (continuous) problem. See "Relaxations of AC Maximal Load Delivery for Severe Contingency Analysis" by C. Coffrin et al. (DOI: 10.1109/TPWRS.2018.2876507) for single-phase case.

### MLD Variables

\begin{align} \mbox{variables: } & \nonumber \\ & z^v_i \in \{0,1\}\ \ \forall i \in N \mbox{ - bus voltage on/off variable} \\ & z^g_i \in \{0,1\}\ \ \forall i \in G \mbox{ - generator on/off variable} \\ & z^{b}_i \in \{0,1\}\ \ \forall i \in B\mbox{ - storage on/off variable} \\ & z^d_i \in (0,1)\ \ \forall i \in L \mbox{ - continuous load shedding variable} \\ & z^s_i \in (0,1)\ \ \forall i \in H \mbox{ - continuous shunt shedding variable} \end{align}

### MLD Objective

\begin{align} \mbox{minimize: }\left ( \sum_{\substack{i\in N,c\in C}}{10 \left (1-z^v_i \right )} + \sum_{\substack{i\in L,c\in C}}{10 \omega_{i,c}\left |\Re{\left (S^d_i\right )}\right |\left ( 1-z^d_i \right ) } + \sum_{\substack{i\in H,c\in C}}{\left | \Re{\left (S^s_i \right )}\right | \left (1-z^s_i \right ) } + \sum_{\substack{i\in G,c\in C}}{\Delta^g_i } + \sum_{\substack{i\in B,c\in C}}{\Delta^b_i} \right ) \end{align}

where

\begin{align} \Delta^g_i &>= \left [\Re{\left (S^g_{i}(0) \right )} - \Re{\left (S^g_i \right )} \right ] \\ \Delta^g_i &>= -\left [\Re{\left (S^g_{i}(0) \right )} - \Re{\left (S^g_i \right )} \right ] \\ \Delta^b_i &>= \left [\Re{\left (S^b_{i}(0) \right )} - \Re{\left (S^b_i \right )} \right ] \\ \Delta^b_i &>= -\left [\Re{\left (S^b_{i}(0) \right )} - \Re{\left (S^b_i \right )} \right ] \end{align}

### MLD Constraints

\begin{align} \mbox{subject to: } & \nonumber \\ & z^v_i v^l_{i,c} \leq \left | V_{i,c} \right | \leq z_i^v v^u_{i,c}\ \ \forall i \in N,\forall c \in C \\ & z^g_i S^{gl}_{i,c} \leq S^g_{i,c} \leq z^g_i S^{gu}_{i,c}\ \ \forall i \in G,\forall c \in C \\ & \sum_{\substack{k\in G_i,c\in C}} S^g_{k,c} - \sum_{\substack{k\in L_i,c\in C}} z^d_k S^d_{k,c}- \sum_{\substack{k\in H_i,c\in C}} z^s_k Y^s_{k,c}\left | V_{i,c} \right |^2 \nonumber \\ & = \sum_{\substack{(i,j)\in E_i\cup E_i^R,c\in C}} S_{ij,c}\ \forall i \in N \end{align}