# Electric Grid System / Optimal Power Flow

Notation

For this class, we use z^* to denote the conjugate of z\in\Complex. We will use \text i or \text j (upright i and j) for the complex number \sqrt {-1} to differentiate from the regular variables i,j.

# Complex Numbers Review

These are links to LibreTexts. (They use \overline z for conjugate)

# Thm. Euler's Equation

e^{\text i\pi} + 1 = 0

# Cor. Euler's Identity

Let m, \theta\in\R

\begin{aligned}
    e^{\text i\theta} &= \cos \theta + \text i \sin\theta
\end{aligned}

If z_1 = m_1e^{\text i\theta_1}, z_2 = m_2e^{\text i\theta_2},

z_1z_2 = m_1m_2e^{\text i(\theta_1 + \theta_2)}

# Physics Setup

In real electrical grids we use alternating current, so voltage v and current c changes with time t.

# Voltage

Let v(t) be the voltage at time t, define:

v(t) = m\cdot \cos(\omega t + \phi)

where m is the magnitude, \omega is the angular frequency, and \phi represents phase shift. All of which are constants in \R.

We can get rid of the cosine function by taking the real part of v(t) in the complex plane.

\begin{aligned}
    v(t) &= m\cdot \cos(\omega t + \phi) \\
         &= \text{Re}(me^{\text i\omega t + \phi})\\
         &= \text{Re}(me^{\text i\phi}e^{\text i\omega t})
\end{aligned}

# Current

For current, define

c(t) = m\cdot\cos(\omega t + \theta)

# Kirchhoff's Laws

For an undirected network G = (N, E), suppose we have |N| = n nodes.

Voltage Law:

\sum_{i} V_i = 0

Current Law:

I_i^g - I_i^d = \sum_{ij\in E} I_{ij}

where I_i^g is the generated current at node i, I_i^d is the demanded current at node i.

By Ohm's Law, current is voltage over resistance:

I_{ij} = Y_{ij}(V_i - V_j)

where Y_{ij} acts as the inverse of the resistance(Z_{ij}) term for the edge ij. (Line admittance wikipedia). This is a known input for the problem.

Y = g + \text ib

for some constant g, b\in \R. (g is conductance, b is susceptance).

AC Power on edge ij is voltage times current's complex conjugate:

S_{ij} = V_iI^*_{ij}\\
S_{ij}\in\Complex

and the power flow on edge ij is:

\begin{aligned}
S_i^g - S_i^d &= \sum_{j\in N} S_{ij} \qquad \forall i\in N\\
S_{ij} &= V_iI^*_{ij}\\
&= V_i\cdot (Y_{ij}(V_i - V_j))^*\\
&= Y_{ij}^*V_iV_i^* - Y_{ij}^*V_iV_j^*\\
&= Y_{ij}^*(|V_i|^2 - V_iV_j^*)
\end{aligned}

The total power transmitted from node i is the sum of S_{ij} over all nodes j that are connected to i.

\begin{aligned}
    S_i &= \sum_{\{j\,\mid\, ij\text{ is connected}\}} S_{ij}\\
    &= \sum_{\{j\,\mid\, ij\text{ is connected}\}}Y_{ij}^*(|V_i|^2 - V_iV_j^*)
\end{aligned}

This is known as the Bus Injection Model.

# Constraints and Objective

# Side Constraints

Generator limits (can only produce power in a range)
```
  S_i^{g, l}\leqslant S_i^g\leqslant S_i^{g, u}
```
for some constants S_i^{g, l}, S_i^{g, u}\in\R
Line thermal limit (can't carry too much power over each edge)
```
  |S_{ij}|\leqslant S_{ij}^u
```
Bus voltage limit (can't produce too much voltage at 1 node)
```
  V_i^l \leqslant |V_i|\leqslant V_i^u
```

Phase angle difference (not too much phase shift)

  |\theta_i - \theta_j|\leqslant \theta_{ij}^\Delta

# Optimal Power Flow

Let S_i = p_i + \text i q_i. The objective is to minimize the total cost:

\min \sum_{i\in N} c_{i,2}p_{i,g}^2 + c_{i,1}p_{i,g} + c_{i,0}