Exponential Load Model

Authors: Carlos Alegre (eRoots)

Reviewers: Eduardo Prieto Araujo (UPC), Josep Fanals Batllori (eRoots)

Context

Loads are the consumer side of an electrical power system, draining energy from a given bus. Since a load can be a bundle of elements such as shunt elements, regulators, transformers, and many other devices, it is tough to describe accurately how it behaves as a group, and there are many models that try to provide an accurate representation. It is necessary to be able to simulate the behavior of the system under different conditions. The model proposed in this page describes the Exponential Load model [1][2], which is useful for steady-state for forecasting studies mainly.

Model use, assumptions, validity domain and limitations

The exponential load model can be used for steady-state studies. The assumptions made for this model are:

  • The load is modeled as a composite load with exponential relationship between the load and the bus voltage.
  • The exponent value of the function is nearly equal to the slope of the function around the nominal voltage.
  • Although this exponent varies non-linearly as a function of voltage, it can be assumed to have only time dependence for time-series static analysis.
  • This time dependency can be omitted, just assuming a constant value for the exponent. A common combination of exponents is \(a=1\) for active power and \(b=2\) for reactive power.
  • The frequency is constant, and the load is balanced.

The model does not take into account the time-response performance of the load, therefore it is not useful for dynamic studies.

Model description

Parameters

Parameter Description Unit
\(V_0\) Load Nominal voltage \(V\)
\(P_0\) Load Nominal active power \(W\)
\(Q_0\) Load Nominal reactive power \(VAR\)

Variables

Variable Description Unit
\(V\) Phase-to-ground Bus Voltage \(V\)
\(P\) Load Active power \(W\)
\(Q\) Load Reactive power \(VAR\)
\(a(t)\) Exponent for the load active power Unitless
\(b(t)\) Exponent for the load reactive power Unitless

System of equations

$$P(V,t) = P_0 (\frac{V}{V_0})^{a(t)} $$ $$Q(V,t) = Q_0 (\frac{V}{V_0})^{b(t)} $$

Operational principles

The model considers the load as an exponential function of voltage for both active and reactive power.

$$P(V, t) = P_0 (\frac{V}{V_0})^{a(t)} $$ $$Q(V, t) = Q_0 (\frac{V}{V_0})^{b(t)} $$

Where \(a\) and \(b\) are the exponents of the active and reactive power, respectively. These exponents take values that are similar to the slope of the function around the nominal voltage. If they take values of 0, 1 or 2, the load is considered constant power, constant current, or constant impedance, respectively [2] [3]. The exponents can be time-dependent functions for time-series analysis.

Open source implementations

This model has been successfully implemented in :

Software URL Language Open-Source License Last consulted date Comments
Dynawo Link modelica MPL v2.0 29/05/2024 No comment
OpenIPSL Link modelica BSD-3-clause 29/05/2024 No comment
PowerSimulationsDynamics Link julia BSD-3-clause 29/05/2024 No comment

References

[1] IEEE Task Force, “Load Representation for Dynamic Performance Analysis” 1993 IEEE Transactions on Power Systems, Vol. 8, No. 2.

[2] Kundur, Prabha. “Power System Stability and Control” New York, USA, 1994, McGraw-Hill.

[3] Kothari, D. P.; Nagrath, I. J. “Modern Power System Analysis”, 4th ed., New Delhi, India, 2011, Tata McGraw-Hill.

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