Grid Thévenin equivalent

Background

Modern grids are mind-boggling complex machines often comprised of:

• Millions of users
• Millions of pieces of network equipment (e.g. lines and transformers)
• Thousands of generating systems

It is not feasible for a model of an electrical grid to monitor each piece of equipment (e.g. every toaster in the grid). However, grid operators must model enough equipment for a model to be accurate and useful for a given task. Therefore, the level of detail included in the model must depend on the goal of what you are trying to achieve. Commonly used grid models are:

• Full network models which include all major transmission equipment are used for network-level studies (e.g. investigate interactions between different generating systems).
• Network equivalent models which include details of a particular generating system and then model the remaining grid as a Thévenin equivalent. These models are used for plant-level studies (e.g. investigate the fault ride-through response of a converter).

This tool is designed to help you understand the latter. The most common method for modelling a grid equivalent is using a Thévenin equivalent which is a voltage source V_thev in series with an impedance Z_thev).

The tool is divided into two parts:

1. Z_thev calculation from provided grid data.
2. Calculation of remaining unknowns to complete the Thévenin equivalent.

Tool

Part 1: Calculating Z_thev

Use project-specific information to derive the Thévenin equivalent impedance (Z_thev) in ohms and per unit. The Zthev result can be used for subsequent calculations. This is a critical first step in any power system plant modelling where we need to determine how the external network will be represented. The higher the fault level, the smaller the Zthev will be, and vice versa.

Fault level and short-circuit ratio (SCR) are common measures of grid strength and directly determine the magnitude of Z_thev. The strength of the grid is often characterized by SCR and X/R ratio, however in the power system models it is represented as impedance values (resistance and reactance). A calculation is needed to convert between the different ways of expressing the strength of the external grid.

Project settings

Project name

Project rated active power [MW]

System frequency [Hz]

5060

Connection point voltage [kV]

System base [MVA] (Default 100 MVA)

Calculate Z_thev

Select the type of grid data you have received:

SCR and X/R ratioFault level [MVA] and X/R ratioGrid impedance, |Z| [Ω] and X/R ratio

Grid data:

SCR

X/R ratio

Fault level, SCR and X/R ratio

Fault level = 525.0000 [MVA]

SCR = 3.0000

X/R ratio = 10.0000

Z_thev [Ω]

|Z| = 226.7143 [Ω]

R = 22.5589 [Ω]

X = 225.5891 [Ω]

L = 718.0726 [mH]

Z_thev and Z_base [Ω]

|Z| = 0.190476 [p.u.]

R = 0.018953 [p.u.]

X = 0.189531 [p.u.]

Z_base = 1190.2500 [Ω]

@ 345 kV, 100 MVA

Diagram controls

Title

Show title

Impedance labels

Show impedances

Connection point label

Show connection point text

Fault level details

Show fault level details

Project settings

Show project settings

Mark Zthev calculation complete

Part 2: Calculating V_thev

Once the Z_thev impedance has been determined using the tool above, we can progress to calculating other variables in the grid equivalent representation. The diagram below shows the Grid Thévenin equivalent with a single generator representing your project.

Thevenin equivalent calculator

Solve for:

Vthev∠θVpoc∠θPpoc and Qpoc

|V_thev| [p.u.]

1.0221

|V_poc| [p.u.]

Ppoc [MW]

θ_thev [°]

-0.0000

θ_poc [°]

Qpoc [MVAr]

Diagram controls

Title

Show title

Impedance labels

Show impedances

Connection point label

Show connection point text

Ppoc, Qpoc

Show Ppoc, Qpoc (generator contribution at POC)

Power flow arrows

Show Ppoc, Qpoc flow animations

Vpoc

Show connection point voltage

Vthev

Show Vthev

Project settings

Show project settings

Theory and formulae

Key relations for the Thévenin equivalent and power flow at the POC. Toggle sections and examples to manage detail.

VpocConnection point voltage

VthevThévenin equivalent voltage

ZthevThévenin equivalent impedance

S, P, QConnection point apparent, active and reactive power

IConnection point current

VbaseConnection point base L–L voltage

SbaseSystem base apparent power

SfaultlevelFault level apparent power

PratedProject rated active power

SCRShort circuit ratio

ZbaseBase impedance

X/RReactance-to-resistance ratio

fSystem frequency

Rthev, XthevThévenin equivalent resistance and reactance

LthevThévenin equivalent inductance

Base impedance (Ω):

$$Z_{\mathrm{base}} = \frac{V_{\mathrm{base}}^2}{S_{\mathrm{base}}}$$

Fault level and SCR (when provided or derived):

$$|Z_{\mathrm{thev}}| = \frac{V_{\mathrm{base}}^2}{S_{\mathrm{faultlevel}}}$$

Fault level from SCR and Prated: Fault level = SCR × Prated; SCR = Fault level / Prated.

$$S_{\mathrm{faultlevel}} = \text{SCR} \times P_{\mathrm{plant}},\quad \text{SCR} = \frac{S_{\mathrm{faultlevel}}}{P_{\mathrm{plant}}}$$

Alternatively, |Zthev| may be given directly (|Z| in Ω); then Fault level = Vbase² / |Zthev|.

$$|Z_{\mathrm{thev}}| = |Z|,\quad S_{\mathrm{faultlevel}} = \frac{V_{\mathrm{base}}^2}{|Z_{\mathrm{thev}}|}$$

Resistance and reactance in Ω from |Zthev| and X/R:

$$Rthev = \frac{|Z_{\mathrm{thev}}|}{\sqrt{1 + (X/R)^2}},\quad Xthev = (X/R) \cdot Rthev$$

Convert Rthev and Xthev into per unit on system base:

$$R_{\mathrm{pu}} = \frac{Rthev}{Z_{\mathrm{base}}},\quad X_{\mathrm{pu}} = \frac{Xthev}{Z_{\mathrm{base}}}$$

Equivalent inductance (mH), with Xthev in Ω and f in Hz:

$$Lthev = \frac{Xthev}{2\pi f}$$

Example:

Worked example using current calculator values.

Base impedance (Zbase = Vbase² / Sbase):

$$Z_{\mathrm{base}} = \frac{V_{\mathrm{base}}^2}{S_{\mathrm{base}}} = \frac{(345 kV)^2}{100 MVA} = 1190.25\,\Omega$$

Fault level from SCR and Prated:

$$S_{\mathrm{faultlevel}} = \text{SCR} \times P_{\mathrm{plant}} = 3.0 \times 175 = 525.0\,\text{MVA}$$

|Zthev| = Vbase² / Fault level:

$$|Z_{\mathrm{thev}}| = \frac{V_{\mathrm{base}}^2}{S_{\mathrm{faultlevel}}} = \frac{(345 kV)^2}{525.0 MVA} = 226.71\,\Omega$$

SCR = Fault level / Prated:

$$\text{SCR} = \frac{S_{\mathrm{faultlevel}}}{P_{\mathrm{plant}}} = \frac{525.0 MVA}{175 MW} = 3.00$$

R and X from |Zthev| and X/R = 10:

$$Rthev = \frac{226.71 Ω}{\sqrt{1 + 10^2}} = 22.5589\,\Omega,\quad X = 10 \times Rthev = 225.5891 \,\Omega$$

Convert Rthev and Xthev into per unit on system base:

$$R_{\mathrm{pu}} = \frac{Rthev}{Z_{\mathrm{base}}} = \frac{22.5589 Ω}{1190.25 Ω} = 0.0190\,\text{p.u.},\quad X_{\mathrm{pu}} = \frac{Xthev}{Z_{\mathrm{base}}} = \frac{225.5891 Ω}{1190.25 Ω} = 0.1895\,\text{p.u.}$$

Inductance (f = 50 Hz):

$$L = \frac{1000 \times 225.5891 Ω}{2\pi \times 50 Hz} = 718.07\,\text{mH}$$

Revision history

Version 1 | 10 March 2026

• First release