Formulations and Models — amplpower 0.0.0 documentation

# Formulations and Models

This section provides an overview of the formulations and optimization models used in the package.

## Nomenclature

### Indexes

\(n, m\) | Bus indices
\(g\) | Generator index
\(l\) | Transmission line index

### Parameters

\(S_{\text{base}}\) | Base power [MVA]
\(P^D_n\) | Active power demand at bus \(n\) [MW]
\(\theta_{\text{max}}, \theta_{\text{min}}\) | Voltage angle limits [radians]
\(\overline{P}_g, \underline{P}_g\) | Generation limits for \(g\) [MW]
\(c_{2,g}, c_{1,g}, c_{0,g}\) | Cost coefficients for \(g\)
\(x_l\) | Reactance of line \(l\) [p.u.]
\(\overline{P}_l\) | Line capacity for \(l\) [MW]
\(C^F_{l,n}, C^T_{l,n}\) | Incidence matrices for “from” and “to” buses of line \(l\)
\(C^G_{g,n}\) | Incidence matrix for generator \(g\) at bus \(n\)

### Variables

\(P_g\) | Active power generation of \(g\) [MW]
\(\theta_n\) | Voltage angle at bus \(n\) [radians]
\(P^F_l, P^T_l\) | Power flow on line \(l\) (from/to) [MW]

## DC Optimal Power Flow (DC-OPF)

The DC-OPF formulation is a simplified version of the AC-OPF that assumes:

- Voltage magnitudes are fixed at 1.0 p.u.

- Reactive power flows are ignored.

- Small angle differences between buses.

### Objective Function

The objective is to minimize the total generation cost:

\[\text{Minimize: } \sum_{g} \left( c_{2,g} \cdot P_g^2 + c_{1,g} \cdot P_g + c_{0,g} \right)\]

### Constraints

1. Active Power Balance:

   \[\sum_{g} C^G_{g,n} \cdot P_g - P^D_n = \sum_{l} \left( C^F_{l,n} \cdot P^F_l + C^T_{l,n} \cdot P^T_l \right), \quad \forall n\]

2. Line Flow Equations:

   \[P^F_l = \frac{1}{x_l} \cdot \sum_{n} \left( C^F_{l,n} - C^T_{l,n} \right) \cdot \theta_n, \quad \forall l\]
   \[P^T_l = \frac{1}{x_l} \cdot \sum_{n} \left( C^T_{l,n} - C^F_{l,n} \right) \cdot \theta_n, \quad \forall l\]

3. Line Flow Limits:

   \[-\overline{P}_l \leq P^F_l, P^T_l \leq \overline{P}_l, \quad \forall l\]

4. Generator Limits:

   \[\underline{P}_g \leq P_g \leq \overline{P}_g, \quad \forall g\]

5. Voltage Angle Reference:

   \[\theta_{\text{slack}} = 0\]