Power Flow Module

This module provides functions for AC, DC and AC/DC power flow analysis.

functions are found in pyflow_acdc.ACDC_PF

Running the Power Flow

For a simple power flow, the function Power_flow() can be used. This function will automatically detect the type of power flow to run (AC, DC or AC/DC) and will run the appropriate power flow.

Power_flow(grid, tol_lim=1e-10, maxIter=100)

Performs power flow calculation.

AC Power Flow

The AC power flow solution is based on [1], to solve the AC power flow Newton Raphson method is used. The equations involving \(P_{slack}\) and \(Q_{slack}\) of each separate AC grid ( \(\Gamma\) = number of AC grids) are not included. In addition, the Q equations of PV nodes are also not included.

The number of unknown variables for \(V_i\) and \(\theta_i\) are (\(|\mathcal{N}_{ac}| - \Gamma - PV\)) and (\(|\mathcal{N}_{ac}| - \Gamma\) ) respectively. These unknown variables are arranged into vectors \(\boldsymbol{\theta}\), \(|V|\), and the composite vector \(\boldsymbol{x}\).

The active power equations at each node i are:

\[P_i = \sum P_{g_i} + \sum \gamma_{rg_i}P_{rg_i} - P_{l_i} + \sum P_{cn_i}\]

\[P_i(x) = \sum_{k=1}^{\mathcal{N}_{ac}} |V_i||V_k|[G_{ik} \cos(\theta_i-\theta_k) + B_{ik} \sin(\theta_i-\theta_k)]\]

The reactive power equations at each node i are:

\[Q_i = \sum Q_{g_i} + \sum Q_{rg_i} -Q_{l_i}+\sum Q_{cn_{i}}\]

\[Q_i(x) = \sum_{k=1}^{\mathcal{N}_{ac}} |V_i||V_k|[G_{ik} \sin(\theta_i-\theta_k) - B_{ik} \cos(\theta_i-\theta_k)]\]

These equations are combined into the function \(\boldsymbol{f(x)} = 0\):

\[\begin{split}\boldsymbol{f(x)} = \begin{bmatrix} P_1(\boldsymbol{x})-P_1 \\ \vdots \\ P_{|\mathcal{N}_{ac}|-\Gamma}(\boldsymbol{x})-P_{|\mathcal{N}_{ac}|-\Gamma} \\ Q_1(\boldsymbol{x})-Q_1 \\ \vdots \\ Q_{|\mathcal{N}_{ac}|-\Gamma-PV}(\boldsymbol{x})-Q_{|\mathcal{N}_{ac}|-\Gamma-PV} \end{bmatrix}\end{split}\]

The Jacobian matrix \(\boldsymbol{J}\) contains the first-order partial derivatives:

\[\begin{split}\boldsymbol{J} = \begin{bmatrix} \boldsymbol{J_{11}} & \boldsymbol{J_{12}} \\ \boldsymbol{J_{21}} & \boldsymbol{J_{22}} \end{bmatrix}\end{split}\]

Where:

\[\begin{split}[i,i] \text{ are the diagonal elements} \\ [i,k] \text{ are the off-diagonal elements}\end{split}\]

\(J_{11}\) (size \(|\mathcal{N}_{ac}|-\Gamma \times |\mathcal{N}_{ac}|-\Gamma\)) contains \(\partial P(x)/\partial \theta\):

\[\begin{split}\boldsymbol{J_{11}}[i,i] &= -Q_i-V_i^2 \cdot B_{ii} \\ \boldsymbol{J_{11}}[i,k] &= V_i \cdot V_k \cdot (G_{ik} \cdot \sin(\theta_i-\theta_k)-B_{ik} \cdot \cos(\theta_i-\theta_k))\end{split}\]

\(J_{12}\) (size \(|\mathcal{N}_{ac}|-\Gamma \times |\mathcal{N}_{ac}|-\Gamma-PV\)) contains \(\partial P(x)/\partial V\):

\[\begin{split}\boldsymbol{J_{12}}[i,i] &= \frac{P_i}{V_i}+ G_{ii} \cdot V_i \\ \boldsymbol{J_{12}}[i,k] &= V_i \cdot (G_{ik} \cdot \cos(\theta_i-\theta_k)+B_{ik} \cdot \sin(\theta_i-\theta_k))\end{split}\]

\(J_{21}\) (size \(|\mathcal{N}_{ac}|-\Gamma-PV \times |\mathcal{N}_{ac}|-\Gamma\)) contains \(\partial Q(x)/\partial \theta\):

\[\begin{split}\boldsymbol{J_{21}}[i,i] &= P_i-V_i^2 \cdot G_{ii} \\ \boldsymbol{J_{21}}[i,k] &= -V_i \cdot V_k \cdot (G_{ik} \cdot \cos(\theta_i-\theta_k)+B_{ik} \cdot \sin(\theta_i-\theta_k))\end{split}\]

\(J_{22}\) (size \(|\mathcal{N}_{ac}|-\Gamma-PV \times |\mathcal{N}_{ac}|-\Gamma-PV\)) contains \(\partial Q(x)/\partial V\):

\[\begin{split}\boldsymbol{J_{22}}[i,i] &= \frac{Q_i}{V_i}-B_{ii} \cdot V_i \\ \boldsymbol{J_{22}}[i,k] &= V_i \cdot (G_{ik} \cdot \sin(\theta_i-\theta_k)-B_{ik} \cdot \cos(\theta_i-\theta_k))\end{split}\]

The Newton-Raphson iteration is then:

\[\begin{split}\begin{bmatrix} \boldsymbol{J_{11}} & \boldsymbol{J_{12}} \\ \boldsymbol{J_{21}} & \boldsymbol{J_{22}} \end{bmatrix} \begin{bmatrix} \boldsymbol{\Delta \theta} \\ \boldsymbol{\Delta |V|} \end{bmatrix} = \begin{bmatrix} \boldsymbol{\Delta P(x)} \\ \boldsymbol{\Delta Q(x)} \end{bmatrix}\end{split}\]

Running the AC Power Flow

AC_PowerFlow(grid, tol_lim=1e-8, maxIter=100)

Performs AC power flow calculation.

Parameter	Type	Description	Default
grid	Grid	Grid to analyze	Required
tol_lim	float	Convergence tolerance	1e-8
maxIter	int	Maximum iterations	100

Example

time,tol = pyf.AC_PowerFlow(grid)

DC Power Flow

It is important to note that ‘DC power flow’ here specifically refers to the flow in DC grids and not to the linearized power flow that is often used as a simplification of AC grids.

The DC power flow solution is based on [2], to solve the DC power flow Newton Raphson method is used.

(1)\[P_{cn_d} - P_{l_d} = U_d \sum_{\substack{f=1; f \neq d}}^{n_{dc}} \left( \left( U_d - U_f \right) \cdot p_e \left(\frac{1}{R_{df}} \right) \right), \left\{ R_{df} \neq 0 \right\}\]

Similar to the AC power flow, the DC power flow is solved by Newton-Raphson method by defining a vector \(\boldsymbol{y}\):

\[\begin{split}\boldsymbol{y} = \begin{bmatrix} U_{1} \\ U_{2} \\ \vdots \\ U_{|\mathcal{N}_{dc}|-s_{DC}} \end{bmatrix}\end{split}\]

\[\begin{split}P_d &= P_{cn_d} - P_{l_d} \\ P_d(y) &= U_d \sum_{\substack{f=1; f \neq d}}^{n_{dc}} \left( \left( U_d - V_f \right) \cdot p_e \left(\frac{1}{R_{df}} \right) \right), \left\{ R_{df} \neq 0 \right\}\end{split}\]

\[\begin{split}\boldsymbol{f(y)} = \begin{bmatrix} P_1(\boldsymbol{y})-P_1 \\ \vdots \\ P_{|\mathcal{N}_{dc}|-s_{DC}}(\boldsymbol{y})-P_{|\mathcal{N}_{dc}|-s_{DC}} \\ \end{bmatrix} = 0\end{split}\]

With \(J_{DC}\), the Jacobian matrix of \(\boldsymbol{f(y)}\) considered as:

\[\begin{split}J_{DC} \cdot \frac{\Delta U_{DC}}{U_{DC}} &= \Delta P_{DC} \\ J_{DC} &= U_{DC}\frac{\delta P_{DC}}{\delta U_{DC}}\end{split}\]

\(J_{DC}\) is a matrix of \(|\mathcal{N}_{dc}|-s_{DC}\) (\(s_{DC}\) is the number of DC slack buses)

\[\begin{split}[d,d] \text{ are the diagonal elements} \\ [d,f] \text{ are the off-diagonal elements}\end{split}\]

\[\begin{split}\boldsymbol{J_{DC}}{[d,d]} &= P_{DC_d}+ U_{d}^2 \cdot \sum^n_{f=1; d\neq f}(\frac{1}{R_{df}} \cdot p_{e}) , \left\{ R_{df} \neq 0 \right\} \\ \boldsymbol{J_{DC}}{[d,f]} &= - p_{e}\cdot \frac{1}{R_{df}} \cdot U_d\cdot V_f , \left\{ R_{df} \neq 0 \right\}\end{split}\]

In contrast to the AC Newton-Raphson, in the DC Newton-Raphson, the power target of the droop nodes changes each iteration.

\[P = -P_{l_d}+P_{conv_0}+ (1-U_d) \kappa\]

Where \(P_{conv_0}\) is the target power in pu of the converter, \(U_i\) is the voltage of the DC bus in pu and \(\kappa\) the droop coefficient in \(P_{pu}/V_{pu}\). For the DC bus that are under droop control the Jacobian is also modified as follows:

\[\boldsymbol{J_{DC}}{[d,d]} = P_{DC_d}+ \kappa \cdot U_d+U_{d}^2 \cdot \sum^n_{f=1; f\neq d}(\frac{1}{R_{df}} \cdot p_{e}) , \left\{ R_{df} \neq 0 \right\}\]

Running the DC Power Flow

DC_PowerFlow(grid, tol_lim=1e-8, maxIter=100)

Performs DC power flow calculation.

Parameter	Type	Description	Default
grid	Grid	Grid to analyze	Required
tol_lim	float	Convergence tolerance	1e-8
maxIter	int	Maximum iterations	100

Example

time,tol = pyf.DC_PowerFlow(grid)

Sequential AC/DC Power Flow

Sequential AC/DC power flow is a method that solves the AC and DC power flows sequentially. It is a three-section process:

1. AC power flow: Solves the AC power flow equations for the AC grid.

2. Converter power flow: Solves the converter power flow equations.

3. DC power flow: Solves the DC power flow equations for the DC grid.

The sequential solver will compare the \(P_{conv}\) of converters in the AC grid until convergence is reached.

Sequential AC/DC Power Flow

Running the Sequential AC/DC Power Flow

ACDC_sequential(grid, tol_lim=1e-8, maxIter=20, change_slack2Droop=False, QLimit=False)

Performs sequential AC/DC power flow calculation.

Parameter	Type	Description	Default
grid	Grid	Grid to analyze	Required
tol_lim	float	Convergence tolerance	1e-8
maxIter	int	Maximum iterations	20
change_slack2Droop	bool	Change slack to droop control	False
QLimit	bool	Enable converter Q limits	False

Example

time,tol,ps_iterations = pyf.ACDC_sequential(grid)

References

[1]

Arthur R. Bergen. “Power systems analysis”. eng. In: Power systems analysis. 2nd ed. Upper Saddle River, N.J: Prentice-Hall, 2000. Chap. 10, pp. 323–352. isbn: 0136919901.

[2]

D. Van Hertem, O. Gomis-Bellmunt, and J. Liang. HVDC Grids: For Offshore and Super- grid of the Future. IEEE Press Series on Power and Energy Systems. Wiley, 2016. isbn: 9781118859155. url: https://books.google.es/books?id=oPP8oAEACAAJ.