System Modelling

This page presents the modelling of the different components of the grid.

The Classes are found in pyflow_acdc.Classes

AC System Modelling

AC node

AC node equivalent circuit with voltage source and impedance

The active and reactive power injections at the AC node are given by [1]:

(1)\[\begin{split}\begin{align} P_{net}^{ac} &= P_{flow}^{ac} \\ P_{net}^{ac} &= \sum P_{g_i} + \sum \gamma_{rg_i}P_{rg_i} - P_{l_i} + \sum P_{cn_i} \\ P_{flow}^{ac} &= \sum_{k \in \mathcal{N}_{ac}} |V_i||V_k| [ G_{ik} \cos(\theta_i-\theta_k) + B_{ik} \sin(\theta_i-\theta_k) ] \qquad \forall i \in \mathcal{N}_{ac} \end{align}\end{split}\]

(2)\[\begin{split}\begin{align} Q_{net}^{ac} &= Q_{flow}^{ac} \\ Q_{net}^{ac} &= \sum Q_{g_i} + \sum Q_{rg_i} -Q_{l_i}+\sum Q_{cn_{i}} \\ Q_{flow}^{ac} &= \sum_{k \in \mathcal{N}_{ac}} |V_i||V_k|[G_{ik} \sin(\theta_i-\theta_k)-B_{ik} \cos(\theta_i-\theta_{k})] \qquad \forall i \in \mathcal{N}_{ac} \end{align}\end{split}\]

(3)\[\begin{split}\begin{align} V_{min} \leq V_{i} \leq V_{max} & \qquad \forall i \in \mathcal{N}_{ac} \\ \theta_{min} \leq \theta_i \leq \theta_{max} & \qquad \forall i \in \mathcal{N}_{ac} \end{align}\end{split}\]

The AC node is modeled using a complex voltage phasor \(V_i = V_i \angle \theta_i\) where:

• \(V_i\) is the voltage magnitude in pu

• \(\theta_i\) is the voltage angle

• \(P_{rg}\) is the active power injection of renewable generation in pu

• \(Q_{rg}\) is the reactive power injection of renewable generation in pu

• \(P_{cn}\) is the active power injection of converter in pu

• \(Q_{cn}\) is the reactive power injection of converter in pu

• \(P_g\) is the active power injection of generator in pu

• \(S_l\) is the complex power demand in pu

Class Reference: pyflow_acdc.Classes.Node_AC

Key Attributes:

Attribute	Type	Description
node_type	str	Node type (‘Slack’ or ‘PQ’ or ‘PV’)
Voltage_0	float	Initial voltage magnitude in p.u.
theta_0	float	Initial voltage angle in degrees
kV_base	float	Base voltage in kV
Power_Gained	float	Active power injection in MW
Reactive_Gained	float	Reactive power injection in MVAr
Power_load	float	Active power demand in MW
Reactive_load	float	Reactive power demand in MVAr
Umin	float	Minimum voltage magnitude in p.u.
Umax	float	Maximum voltage magnitude in p.u.
Gs	float	Shunt conductance in p.u.
Bs	float	Shunt susceptance in p.u.
x_coord	float	x-coordinate preferably in decimal longitude
y_coord	float	y-coordinate preferably in decimal latitude

Example Usage:

import pyflow_acdc as pyf
# Create an AC node
node = pyf.Node_AC('PQ', 1, 0,66, Power_Gained=0, Reactive_Gained=0, Power_load=100, Reactive_load=50, name='Bus1', Umin=0.9, Umax=1.1,Gs=0,Bs=0)

AC branch

AC line π-model

The AC branch is modeled with pi model from [1], [2] :

(4)\[\begin{split}Y_{bus-{branch}} = \begin{bmatrix} Y_{ff} & Y_{ft} \\ Y_{tf} & Y_{tt} \end{bmatrix} = \begin{bmatrix} \frac{Y_s+Y_{sh}}{\mu^2} & -\frac{Y_s}{\mu e^{-j\tau}} \\ -\frac{Y_s}{\mu e^{j\tau}} & Y_{s}+Y_{sh} \end{bmatrix}\end{split}\]

(5)\[\begin{split}\begin{align} Y_{bus_{AC}}[i,i]&+= Y_{ff} + G_{r_i}+jB_{r_i}\\ Y_{bus_{AC}}[i,k]&+= Y_{ft}\\ Y_{bus_{AC}}[k,i]&+= Y_{tf}\\ Y_{bus_{AC}}[k,k]&+= Y_{tt} +G_{r_k}+ jB_{r_k} \end{align}\end{split}\]

(6)\[\begin{split}\begin{align} P_{j,to}^2+Q_{j,to}^2 &\leq S_{j,rating}^2 \qquad \forall j \in \mathcal{B}_{ac}\\ P_{j,from}^2+Q_{j,from}^2 &\leq S_{j,rating}^2 \qquad \forall j \in \mathcal{B}_{ac} \end{align}\end{split}\]

Class Reference: pyflow_acdc.Classes.Line_AC

Key Attributes:

Attribute	Type	Description
fromNode	Node_AC	The starting node of the line
toNode	Node_AC	The ending node of the line
r	(float)	Resistance of the line in pu
x	(float)	Reactance of the line in pu
g	(float)	Conductance of the line in pu
b	(float)	Susceptance of the line in pu
MVA_rating	(float)	MVA rating of the line
Length_km	(float)	Length of the line in km
m	(float)	Number of conductors in the line
shift	(float)	Phase shift of the line in radians
N_cables	(int)	Number of cables in the line
name	(str)	Name of the line
geometry	(str)	Geometry of the line
isTf	(bool)	True if the line is a transformer, False otherwise
S_base	(float)	Base power of the line in MVA
Cable_type	(str)	Type of cable in the line

Example Usage:

import pyflow_acdc as pyf
# Create an AC node
node1 = pyf.Node_AC('PQ', 1, 0,66, Power_Gained=0.5, name='Bus1')
node2 = pyf.Node_AC('Slack', 1, 0,66,name='Bus2')

# In pu
line_1 = pyf.Line_AC(node1, node2, r=0.01, x=0.1, g=0, b=0, MVA_rating=100, N_cables=1, name='Line1')

# Or by cable type in database

line_2 = pyf.Line_AC(node1, node2, S_base=100, Length_km=100, Cable_type='NREL_XLPE_630mm_66kV')

Transmission Expansion Planning Modelling

In transmision expansion planning, each branch subject to modification is modeled independently in their power flow equations. The standard \(\pi\)-model is used to represent branch through its corresponding branch admittance \(Y_{\text{bus-branch}}\).

(7)\[\begin{split}\begin{align} S_{ik}& = P_{ik} + \text{j} Q_{ik} \\ P_{ik} &= |V_i|^2 G_{ff} + |V_i||V_k| \left[ G_{ft} \cos(\theta_i - \theta_k) + B_{ft} \sin(\theta_i - \theta_k) \right] \\ Q_{ik}&=-|V_i|^2 B_{ff}+|V_i||V_k|[G_{ft} \sin(\theta_i-\theta_k)-B_{ft} \cos(\theta_i-\theta_{k}) \end{align}\end{split}\]

AC expandable branch

AC expandable branch model

The AC expandable branch \(h\) is modeled with in admittance matrix model from [4]:

(8)\[\begin{split}\begin{align} S_{h_\text{ik}\text{-eq}} &= n_h \cdot S_{h_\text{ik}} \\ S_{h_\text{ki}\text{-eq}} &= n_h \cdot S_{h_\text{ki}} \end{align}\end{split}\]

(9)\[\begin{split}\begin{align} P_{flow}^{ac} += \sum n_{h} P_{h_\text{ik}} \\ Q_{flow}^{ac} += \sum n_{h} Q_{h_\text{ik}} \end{align}\end{split}\]

Class Reference: pyflow_acdc.Classes.Exp_Line_AC

Inherits from Line_AC with the following additional attributes:

Attribute	Type	Description
base_cost	float	Base cost of the line
life_time	float	Lifetime of the line
np_line_b	float	Base number of lines
np_line_i	float	Initial number of lines
np_line_max	float	Maximum number of lines

Where \(n_b \leq n_i \leq n_{max}\)

AC reconducting branch

AC reconducting branch model

The AC reconducting branch \(u\) is modeled with in admittance matrix model from [4]:

(10)\[\begin{split}\begin{align} S_{u_\text{ik}\text{-eq}} &= (1-\xi_u)\cdot S_{u_\text{ik}} + \xi_u \cdot S_{u_\text{ik}}' \\ S_{u_\text{ki}\text{-eq}} &=(1-\xi_u)\cdot S_{u_\text{ki}} + \xi_u \cdot S_{u_\text{ki}}' \end{align}\end{split}\]

(11)\[\begin{split}\begin{align} P_{flow}^{ac} += \sum P_{u_\text{ik}\text{-eq}} \\ Q_{flow}^{ac} += \sum Q_{u_\text{ik}\text{-eq}} \end{align}\end{split}\]

Class Reference: pyflow_acdc.Classes.Rep_Line_AC

Inherits from Line_AC with the following additional attributes:

Attribute	Type	Description
r_new	float	Resistance of the new conductor in pu
x_new	float	Reactance of the new conductor in pu
g_new	float	Conductance of the new conductor in pu
b_new	float	Susceptance of the new conductor in pu
MVA_rating_new	float	MVA rating of the new conductor
Life_time	float	Lifetime of the conductor
base_cost	float	Base cost of the conductor

AC conductor size selection

AC conductor size selection model

The AC conductor size selection \(a\) is modeled with in admittance matrix model from [4]:

(12)\[\begin{split}\begin{align} S_{a_\text{ik}\text{-eq}} &= \sum_{ \iota \in \mathcal{CT}} \left(\xi_{a,\iota}\cdot S_{a,\iota_\text{ik}} \right) \\ S_{a_\text{ki}\text{-eq}} &=\sum_{ \iota \in \mathcal{CT}} \left( \xi_{a,\iota}\cdot S_{a,\iota_\text{ki}} \right) \end{align}\end{split}\]

(13)\[\begin{split}\begin{align} P_{flow}^{ac} += \sum P_{a_\text{ik}\text{-eq}} \\ Q_{flow}^{ac} += \sum Q_{a_\text{ik}\text{-eq}} \end{align}\end{split}\]

For the branches $a$ an additional equality constraint is introduced to keep one type of branch per connection, such that :

(14)\[\sum_{n \in\mathcal{CT}} \xi_{a, n} = 1 \qquad \forall a \in \mathcal{E}_a\]

(15)\[\begin{split}S_{from_n-eq} = \xi_{a,n} \cdot S_{from_n} \\ S_{to_n-eq} = \xi_{a,n} \cdot S_{to_n}\end{split}\]

Additionally, to limit the number of branch types in the evaluated system, inequality constraints are included such that:

(16)\[\begin{split}\begin{align} &\sum_{a\in\mathcal{E}_a} \xi_{a, n} \leq |\mathcal{E}_a| \cdot \nu_n \qquad \forall n \in \mathcal{CT} \\ &\nu_n \leq \sum_{a\in\mathcal{E}_a} \xi_{a, n} \qquad\qquad\; \forall n \in \mathcal{CT} \\ &\sum_{n\in \mathcal{CT}} \nu_n \leq \mathfrak{N} \end{align}\end{split}\]

Class Reference: pyflow_acdc.Classes.Line_sizing

Inherits from Line_AC with the following additional attributes:

Attribute	Type	Description
fromNode	Node_AC	The starting node of the line
toNode	Node_AC	The ending node of the line
cable_types	list	List of possible cable types for sizing
active_config	int	Index of the currently active cable configuration
Length_km	float	Length of the line in kilometers
S_base	float	Base power in MVA
name	str	Name of the line
geometry	str	Geometry of the line

Important to note that this class only takes in the cable types and not the line parameters. The parameters for these have to be uploaded as yalm files in the Cable_database folder.

DC System Modelling

DC node

DC node equivalent circuit

The AC node is modeled using voltage \(U_d\) where [1]:

(17)\[\begin{split}\begin{align} P_{net}^{dc} &= P_{flow}^{dc} \\ P_{net}^{dc} &= P_{cn_d} + \sum \gamma_{rg_d}P_{rg_d} - P_{l_d} \\ P_{flow}^{dc} &= U_d \sum_{\substack{f \in \mathcal{N}_{dc} \\ f \neq d}} \left( (U_d-U_f) \cdot p_{e} \cdot \left(\frac{1}{R_{df}} \right) \right), \left\{ R_{df} \neq 0 \right\} \qquad \forall d \in \mathcal{N}_{dc} \end{align}\end{split}\]

(18)\[U_{min} \leq U_{d} \leq U_{max} \qquad \forall d \in \mathcal{N}_{dc}\]

• \(U_d\) is the voltage magnitude in pu

• \(P_{rg}\) is the active power injection of renewable generation in pu

• \(P_{cn}\) is the active power injection of converter in pu

• \(P_l\) is the active power demand in pu

Class Reference: pyflow_acdc.Classes.Node_DC

Key Attributes:

Attribute	Type	Description
node_type	str	Node type (‘Slack’ or ‘P’ or ‘Droop’ or ‘PAC’)
Voltage_0	float	Initial voltage magnitude in pu
Power_Gained	float	Active power injection in pu
Power_load	float	Active power demand in pu
kV_base	float	Base voltage in kV
Umin	float	Minimum voltage magnitude in p.u.
Umax	float	Maximum voltage magnitude in p.u.
x_coord	float	x-coordinate, preferably in longitude decimal format
y_coord	float	y-coordinate, preferably in latitude decimal format

Example Usage:

import pyflow_acdc as pyf
# Create an DC node
node = pyf.Node_DC('P', 1, 0,0,525,name='Bus1')

DC line

DC line model

(19)\[\begin{split}p_{e}=\begin{cases} 1, &\text{for asymmetrical monopolar} \\ 2, &\text{for symmetrical monopolar or bipolar} \\ \end{cases}\end{split}\]

(20)\[\begin{split}\begin{align} P_{from,d}=&U_d(U_d-U_f) p_{e} \left(\frac{1}{R_{df}} \right) \\ P_{to,f}=&U_f(U_f-U_d)p_{e} \left(\frac{1}{R_{df}} \right) \\ -P_{e, rating} \leq& P_{to/from} \leq P_{e,rating} \qquad \forall e \in \mathcal{B}_{dc} \end{align}\end{split}\]

Key Attributes:

Attribute	Type	Description
fromNode	Node_DC	The starting node of the line
toNode	Node_DC	The ending node of the line
r	float	Resistance of the line in pu
MW_rating	float	MW rating of the line
km	float	Length of the line in km
polarity	str	Polarity of the line (‘m’ or ‘b’ or ‘sm’)
N_cables	int	Number of parallelcables in the line
Cable_type	str	Type of cable in the line
S_base	float	Base power of the line in MVA

Example Usage:

import pyflow_acdc as pyf
# Create an AC node
node1 = pyf.Node_DC('Slack', 1, 0,0,525,name='Bus1')
node2 = pyf.Node_DC('P', 1, 0,0,525,name='Bus2')

# In pu
line_1 = pyf.Line_DC(node1, node2, r=0.01, MW_rating=100, N_cables=1, name='Line1')

# Or by cable type in database

line_2 = pyf.Line_DC(node1, node2, S_base=100, Length_km=100, Cable_type='NREL_HVDC_2000mm_320kV')

DC line expansion

DC expansion model

The expanded branch object is inside the Line_DC class.

The expanded branch \(e\) is modelled as:

(21)\[\begin{split}\begin{align} P_{from,d}=&U_d(U_d-U_f) p_{e} \left(\frac{n_e}{R_{df}} \right) \\ P_{to,f}=&U_f(U_f-U_d)p_{e} \left(\frac{n_e}{R_{df}} \right) \\ -P_{e, rating} \leq& P_{to/from} \leq P_{e,rating} \qquad \forall e \in \mathcal{E}_{dc} \end{align}\end{split}\]

DCDC converter

DCDC converter model

The DCDC converter is modelled very simply, with the set power injected into the toNode and the power drawn from the fromNode. To include losses there is a simple resistive model. For power flow calculations the power injected in either node is kept constant as the losses are pre calculated with the assumption of \(V_{to} = 1 pu\).

(22)\[\begin{split}P_{from} + P_{to} + P_{loss} = 0 \\ P_{loss} = \left(\frac{P_{to}}{V_{to}}\right)^2 \cdot R\end{split}\]

(23)\[P_{net}^{dc} += \sum P_{DCDC_{from}} + \sum P_{DCDC_{to}}\]

(24)\[\begin{split}\text{DCDC converter}\qquad & \begin{cases} P_{DCDC_{from}}, & \text{if } d \text{ is } \textit{from node} \text{ of } DCDC \\ P_{DCDC_{to}}, & \text{if } d \text{ is } \textit{to node} \text{ of } DCDC \end{cases}\end{split}\]

Class Reference: pyflow_acdc.Classes.DCDC_converter

Key attributes:

Attribute	Type	Description
fromNode	Node_DC	The starting node
toNode	Node_DC	The ending node
R	float	Resistance of the converter
P_set	float	Power set point of the converter
MW_rating	float	Power rating of the converter
name	str	Name of the converter
geometry	str	Geometry of the converter

Example Usage:

import pyflow_acdc as pyf
# Create an AC node
node1 = pyf.Node_DC('Slack', 1, 0,0,525,name='Bus1')
node2 = pyf.Node_DC('P', 1, 0,0,525,name='Bus2')

conv = pyf.add_DCDC_converter(grid,fromNode , toNode ,Pset=0.1, r=0.0001, MW_rating=99999,name='DCDC1')

Inputs can be given in MW or pu. For pu use the r and Pset parameters. For MW use the R_Ohm and P_MW parameters.

ACDC Converter

Asymmetrical monopolar converter configuration

Symmetrical monopolar converter configuration

Bipolar converter configuration

Equivalent converter model

Equivalent converter model is taken from [1].

(25)\[\begin{split}\begin{align} P_s &= -V_i^2 G_{tf}+ V_iV_f[G_{tf} \cos(\theta_i-\theta_f)+B_{tf} \sin(\theta_i-\theta_f)] \\ Q_s &= V_i^2 B_{tf}+ V_iV_f[G_{tf} \sin(\theta_i-\theta_f)-B_{tf} \cos(\theta_i-\theta_f)] \end{align}\end{split}\]

(26)\[\begin{split}\begin{align} P_c &= V_c^2 G_{pr}- V_fV_c[G_{pr} \cos(\theta_f-\theta_c)-B_{pr} \sin(\theta_f-\theta_c)] \\ Q_c &= -V_c^2 B_{pr}+ V_fV_c[G_{pr} \sin(\theta_f-\theta_c)+B_{pr} \cos(\theta_f-\theta_c)] \end{align}\end{split}\]

(27)\[\begin{split}\begin{align} P_{cf}-P_{sf} &= 0 \\ Q_{cf}-Q_{sf}-Q_f &= 0 \end{align}\end{split}\]

(28)\[P_{c_{AC}} + P_{cn_{loss}} + P_{cn_{DC}} = 0\]

(29)\[P_{cn_{loss}} = a\cdot p_{cn} +b \frac{\sqrt{P_{c_{AC}}^2+Q_{c_{AC}}^2}}{V_{c_{AC}}} + \frac{c}{p_{cn}} \left( \frac{P_{c_{AC}}^2+Q_{c_{AC}}^2} {V_{c_{AC}}^2} \right)\]

(30)\[\begin{split}p_{cn} = \begin{cases} 1, & \text{for asymmetrical or symmetrical monopolar} \\ 2, & \text{for bipolar} \end{cases}\end{split}\]

(31)\[\begin{split}\begin{align} P_{s_{AC}}^2+Q_{s_{AC}}^2 &\leq S_{cn_{rating}}^2 \qquad \forall cn \in \mathcal{C}n \\ |P_{cn_{DC}}| &\leq S_{cn_{rating}} \qquad \forall cn \in \mathcal{C}n \end{align}\end{split}\]

Class Reference: pyflow_acdc.Classes.AC_DC_converter

Key Attributes:

Attribute	Type	Description
AC_type	str	Type of AC node (‘Slack’ or ‘PV’ or ‘PQ’)
DC_type	str	Type of DC node (‘Slack’ or ‘P’ or ‘Droop’ or ‘PAC’)
AC_node	Node_AC	AC node connected to the converter
DC_node	Node_DC	DC node connected to the converter
P_AC	float	Active power injection in AC node in pu
Q_AC	float	Reactive power injection in AC node in pu
P_DC	float	Active power injection in DC node in pu
Transformer_resistance	float	Transformer resistance in pu
Transformer_reactance	float	Transformer reactance in pu
Phase_Reactor_R	float	Phase reactor resistance in pu
Phase_Reactor_X	float	Phase reactor reactance in pu
Filter	float	Filter in pu
Droop	float	Droop in pu
kV_base	float	Base voltage in kV
MVA_max	float	Maximum MVA rating of the converter
nConvP	float	Number of parallel converters
polarity	int	Polarity of the converter (1 or -1)
lossa	float	No load loss factor for active power
lossb	float	Linear currentr loss factor
losscrect	float	Switching loss factor for rectifier
losscinv	float	Switching loss factor for inverter
Ucmin	float	Minimum voltage magnitude in pu
Ucmax	float	Maximum voltage magnitude in pu
name	str	Name of the converter

Renewable Source

Renewable source model

For all renewable sources:

(32)\[0 \leq \gamma_{rg} \leq 1 \qquad \forall rg \in \mathcal{RG}\]

For renewables sources connected to AC nodes:

(33)\[\begin{split}\begin{align} Q_{rg}^{min} \leq Q_{rg} \leq Q_{rg}^{max} & \qquad \forall rg \in \mathcal{RG}_{ac} \\ (\gamma_{rg} P_{rg})^2 + Q_{rg}^2 \leq S_{rg,rating}^{2^{max}} & \qquad \forall g \in \mathcal{RG}_{ac} \end{align}\end{split}\]

Class Reference: pyflow_acdc.Classes.RenSource

Key Attributes:

Attribute	Type	Description
node	Node_AC or Node_DC	Node connected to the renewable source
P_MW	float	Active power injection in MW

Type	Figure for folium plotting
Wind
Solar

Generator

Generator model

(34)\[\begin{split}\begin{align} P_{g}^{min} \leq P_{g} \leq P_{g}^{max} & \qquad \forall g \in \mathcal{G}_{ac} \\ Q_{g}^{min} \leq Q_{g} \leq Q_{g}^{max} & \qquad \forall g \in \mathcal{G}_{ac} \\ P_{g}^2+Q_{g}^2 \leq S_{g,rating}^{2} & \qquad \forall g \in \mathcal{G}_{ac} \end{align}\end{split}\]

Generator limits

Class Reference: pyflow_acdc.Classes.Gen_AC

Key Attributes:

Attribute	Type	Description
node	Node_AC or Node_DC	Node connected to the generator
P_MW	float	Active power injection in MW
Q_MVAr	float	Reactive power injection in MVAr
P_MW	float	Active power injection in MW

Type	Figure for folium plotting	Default cost
Hydro
Nuclear
Coal
Solid Biomass
Geothermal
Lignite
Natural Gas
Oil
Waste
Other

Price Zone

Price zone model

Price zone model is taken from [3]. The cost of generation quadratic curve is calculated in Market Coefficients Module.

Class Reference: pyflow_acdc.Classes.Price_Zone

Key Attributes:

References

[1] (1,2,3,4)

B. C. Valerio, V. A. Lacerda, M. Cheah-Mane, P. Gebraad and O. Gomis-Bellmunt, “An optimal power flow tool for AC/DC systems, applied to the analysis of the North Sea Grid for offshore wind integration” in IEEE Transactions on Power Systems, doi: 10.1109/TPWRS.2023.3533889.

[2]

D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady- State Operations, Planning and Analysis Tools for Power Systems Research and Education,” Power Systems, IEEE Transactions on, vol. 26, no. 1, pp. 12–19, Feb. 2011.

[3]

B. C. Valerio, V. A. Lacerda, M. Cheah-Mañe, P. Gebraad, and O. Gomis-Bellmunt, “Optimizing offshore wind integration through multi-terminal DC grids: a market-based OPF framework for the North Sea interconnectors,” IET Conference Proceedings, vol. 2025, no. 6, pp. 150–155, 2025. doi: 10.1049/icp.2025.1198

[4] (1,2,3)

B. C. Valerio, M. Cheah-Mane, V. A. Lacerda, P. Gebraad and O. Gomis-Bellmunt, “Transmission expansion planning for hybrid AC/DC grids using a Mixed-Integer Non-linear Programming approach”