Ultimate Guide to Electric Power Engineering: Power System Loads

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1 Load Classification

2 Modeling Applications

3 Load Modeling Concepts and Approaches

4 Load Characteristics and Models

5 Static Load Characteristics:

Exponential Models • Polynomial Models • Combined Exponential and Polynomial Models • Comparison of Exponential and Polynomial Models • Devices Contributing to Modeling Difficulties

6 Load Window Modeling

The physical structure of most power systems consists of generation facilities feeding bulk power into a high-voltage bulk transmission network that in turn serves any number of distribution substations.

A typical distribution substation will serve from 1 to as many as 10 feeder circuits. A typical feeder circuit may serve numerous loads of all types. A light to medium industrial customer may take service from the distribution feeder circuit primary, while a large industrial load complex may take service directly from the bulk transmission system. All other customers, including residential and commercial, are typically served from the secondary of distribution transformers that are in turn connected to a distribution feeder circuit. Fgr. 1 illustrates a representative portion of a typical configuration.

1 Load Classification

The most common classification of electrical loads follows the billing categories used by the utility companies. This classification includes residential, commercial, industrial, and other. Residential customers are domestic users, whereas commercial and industrial customers are obviously business and industrial users. Other customer classifications include municipalities, state and federal government agencies, electric cooperatives, educational institutions, etc.

Although these load classes are commonly used, they are often inadequately defined for certain types of power system studies. For example, some utilities meter apartments as individual residential customers, while others meter the entire apartment complex as a commercial customer. Thus, the common classifications overlap in the sense that characteristics of customers in one class are not unique to that class. For this reason some utilities define further subdivisions of the common classes.

A useful approach to classification of loads is by breaking down the broader classes into individual load components. This process may altogether eliminate the distinction of certain of the broader classes, but it’s a tried and proven technique for many applications. The components of a particular load, be it residential, commercial, or industrial, are individually defined and modeled. These load components as a whole constitute the composite load and can be defined as a "load window."


Generation 15-35 kV Bulk transmission 230 kV and higher Sub-transmission 69-138 kV Large industrial Distribution substation 4-35 kV Primary feeders Light/medium industrial Secondaries Residential/commercial customers

FGR. 1 Representative portion of a typical power system configuration.


2 Modeling Applications

It’s helpful to understand the applications of load modeling before discussing particular load characteristics. The applications are divided into two broad categories: static ("snap-shot" with respect to time) and dynamic (time varying). Static models are based on the steady-state method of representation in power flow networks. Thus, static load models represent load as a function of voltage magnitude.

Dynamic models, on the other hand, involve an alternating solution sequence between a time-domain solution of the differential equations describing electromechanical behavior and a steady-state power flow solution based on the method of phasors. One of the important outcomes from the solution of dynamic models is the time variation of frequency. Therefore, it’s altogether appropriate to include a component in the static load model that represents variation of load with frequency. The lists below include applications outside of Distribution Systems but are included because load modeling at the distribution level is the fundamental starting point.

Static applications: Models that incorporate only the voltage-dependent characteristic include the following.

1. Power flow (PF)

a. Distribution power flow (DPF)

b. Harmonic power flow (HPF)

c. Transmission power flow (TPF)

2. Voltage stability (VS)

Dynamic applications: Models that incorporate both the voltage- and frequency-dependent characteristics include the following.

• Transient stability (TS)

• Dynamic stability (DS)

• Operator training simulators (OTS)

Strictly power-flow based solutions utilize load models that include only voltage dependency characteristics. Both voltage and frequency dependency characteristics can be incorporated in load modeling for those hybrid methods that alternate between a time-domain solution and a power flow solution, such as found in Transient Stability and Dynamic Stability Analysis Programs, and Operator Training Simulators (EPRI User's Manual, 1992; EPRI Final Report EL-5003, 1987; Kundur, 1994).

Load modeling in this section is confined to static representation of voltage and frequency dependencies. The effects of rotational inertia (electromechanical dynamics) for large rotating machines are discussed in Sections 13 and 17 of Power System Stability and Control. Static models are justified on the basis that the transient time response of most composite loads to voltage and frequency changes is fast enough so that a steady-state response is reached very quickly.

3 Load Modeling Concepts and Approaches

There are essentially two approaches to load modeling: component based and measurement based. Load modeling research over the years has included both approaches (EPRI, 1981, 1984, 1985). Of the two, the component-based approach lends itself more readily to model generalization. It’s generally easier to control test procedures and apply wide variations in test voltage and frequency on individual components.

The component-based approach is a "bottom-up" approach in that the different load component types comprising load are identified. Each load component type is tested to determine the relation ship between real and reactive power requirements versus applied voltage and frequency. A load model, typically in polynomial or exponential form, is then developed from the respective test data. The range of validity of each model is directly related to the range over which the component was tested. For convenience, the load model is expressed on a per-unit basis (i.e., normalized with respect to rated power, rated voltage, rated frequency, rated torque if applicable, and base temperature if applicable). A composite load is approximated by combining appropriate load model types in certain proportions based on load survey information. The resulting composition is referred to as a "load window." The measurement approach is a "top-down" approach in that measurements are taken at either a substation level, feeder level, some load aggregation point along a feeder, or at some individual load point. Variation of frequency for this type of measurement is not usually performed unless special test arrangements can be made. Voltage is varied using a suitable means and the measured real and reactive power consumption recorded. Statistical methods are then used to determine load models. A load survey may be necessary to classify the models derived in this manner. The range of validity for this approach is directly related to the realistic range over which the tests can be conducted without dam age to customers' equipment. Both the component and measurement methods were used in the EPRI research projects EL-2036 (1981) and EL-3591 (1984, 1985). The component test method was used to characterize a number of individual load components that were in turn used in simulation studies. The measurement method was applied to an aggregate of actual loads along a portion of a feeder to verify and validate the component method.

4 Load Characteristics and Models

Static load models for a number of typical load components appear in Tables 1 and 2 (EPRI, 1984, 1985). The models for each component category were derived by computing a weighted composite from test results of two or more units per category. These component models express per-unit real power and reactive power as a function of per-unit incremental voltage and/or incremental temperature and/or per-unit incremental torque. The incremental form used and the corresponding definition of variables are outlined below:

delta V = V_act - 1.0 (incremental voltage in per unit)

delta T = T_act - 95°F (incremental temperature for Air Conditioner model), or:

= T_act - 47°F (incremental temperature for Heat Pump model)

delta t = tact – t_rated (incremental motor torque, per unit)

TABLE 1 Static Models of Typical Load Components-AC, Heat Pump, and Appliances

Load Component Static Component Model

TABLE 2 Static Models of Typical Load Components-Transformers and Induction Motors

Load Component Static Component Model

If ambient temperature is known, it can be used in the applicable models. If it’s not known, the temperature difference, delta T, can be set to zero. Likewise, if motor load torque is known, it can be used in the applicable models. If it’s not known, the torque difference, delta t, can be set to zero.

Based on the test results of load components and the developed real and reactive power models as presented in these tables, the following comments on the reactive power models are important.

• The reactive power models vary significantly from manufacturer to manufacturer for the same component. For instance, four load models of single-phase central air-conditioners show a Q/P ratio that varies between 0 and 0.5 at 1.0 p.u. voltage. When the voltage changes, the delta Q/ delta V of each unit is quite different. This situation is also true for all other components, such as refrigerators, freezers, fluorescent lights, etc.

• It has been observed that the reactive power characteristic of fluorescent lights not only varies from manufacturer to manufacturer, from old to new, from long tube to short tube, but also varies from capacitive to inductive depending upon applied voltage and frequency. This variation makes it difficult to obtain a good representation of the reactive power of a composite system and also makes it difficult to estimate the delta Q/ delta V characteristic of a composite system.

• The relationship between reactive power and voltage is more non-linear than the relationship between real power and voltage, making Q more difficult to estimate than P.

• For some of the equipment or appliances, the amount of Q required at the nominal operating voltage is very small; but when the voltage changes, the change in Q with respect to the base Q can be very large.

• Many distribution systems have switchable capacitor banks either at the substations or along feeders. The composite Q characteristic of a distribution feeder is affected by the switching strategy used in these banks.

5 Static Load Characteristics

The component models appearing in Tables 1 and 2 can be combined and synthesized to create other more convenient models. These convenient models fall into two basic forms: exponential and polynomial.

5.1 Exponential Models

The exponential form for both real and reactive power is expressed in Equations 1 and 2 below as a function of voltage and frequency, relative to initial conditions or base values. Note that neither temperature nor torque appear in these forms. Assumptions must be made about temperature and/or torque values when synthesizing from component models to these exponential model forms.

Equations 1 and 2 (or 3 and 5) are valid over the voltage and frequency ranges associated with tests conducted on the individual components from which these exponential models are derived.

These ranges are typically ±10% for voltage and ±2.5% for frequency. The accuracy of these models outside the test range is uncertain. However, one important factor to note is that in the extreme case of voltage approaching zero, both P and Q approach zero.

TABLE 3 Parameters for Voltage and Frequency Dependencies of Static Loads Component/Parameters Resistance space heater Heat pump space heater Heat pump/central AC Room air conditioner Water heater and range Refrigerator and freezer Dish washer Clothes washer Incandescent lighting Clothes dryer Colored television Furnace fan Commercial heat pump Heat pump comm. AC Commercial central AC Commercial room AC Fluorescent lighting Pump, fan, (motors) Electrolysis Arc furnace Small industrial motors Industrial motors large Agricultural H2O pumps Power plant auxiliaries

EPRI-sponsored research resulted in model parameters such as found in Table 3 (EPRI, 1987; Price et al., 1988). Eleven model parameters appear in this table, of which the exponents a and ß and the power factor (pf) relate directly to Equations 3 and 5. The first six parameters relate to general load models, some of which include motors, and the remaining five parameters relate to nonmotor loads- typically resistive type loads. The first is load power factor (pf). Next in order (from left to right) are the exponents for the voltage (av, af) and frequency (ßv, ßf) dependencies associated with real and reactive power, respectively. Nm is the motor-load portion of the load. For example, both a refrigerator and a freezer are 80% motor load. Next in order are the power factor (pfnm) and voltage (avnm, afnm) and frequency (ßvnm, ßfnm) parameters for the nonmotor portion of the load. Since the refrigerator and freezer are 80% motor loads (i.e., Nm = 0.8), the nonmotor portion of the load must be 20%.

5.2 Polynomial Models

A polynomial form is often used in a Transient Stability program. The voltage dependency portion of the model is typically second order. If the nonlinear nature with respect to voltage is significant, the order can be increased. The frequency portion is assumed to be first order. This model is expressed as follows.

5.3 Combined Exponential and Polynomial Models

The two previous kinds of models may be combined to form a synthesized static model that offers greater flexibility in representing various load characteristics (EPRI, 1987; Price et al., 1988). The mathematical expressions for these per-unit models are the following.

The expressions for the reactive components have similar structures. Devices used for reactive power compensation are modeled separately.

The flexibility of the component models given here is sufficient to cover most modeling needs.

Whenever possible, it’s prudent to compare the computer model to measured data for the load.

Table 4 provides typical values for the frequency damping characteristic, D, that appears in Equations 6 through 9, 13, and 14 (EPRI, 1979; Warnock and Kilpatrick, 1986). Note that nearly all of the damping coefficients for reactive power are negative. This means that as frequency declines, more reactive power is required which can cause an exacerbating effect for low-voltage conditions.


TABLE 4 Static Load Frequency Damping


Frequency Parameters

Component Dp Dq

Three-phase central AC 1.09818 -0.663828

Single-phase central AC 0.994208 -0.307989

Window AC 0.702912 -1.89188

Duct heater w/blowers 0.528878 -0.140006

Water heater, electric cooking 0.0 0.0

Clothes dryer 0.0 -0.311885

Refrigerator, ice machine 0.664158 -1.10252

Incandescent lights 0.0 0.0

Florescent lights 0.887964 -1.16844

Induction motor loads 1.6 -0.6


5.4 Comparison of Exponential and Polynomial Models

Both models provide good representation around rated or nominal voltage. The accuracy of the exponential form deteriorates when voltage significantly exceeds its nominal value, particularly with exponents (a) greater than 1.0. The accuracy of the polynomial form deteriorates when the voltage falls significantly below its nominal value when the coefficient ao is non zero. A nonzero ao coefficient represents some portion of the load as constant power. A scheme often used in practice is to use the polynomial form, but switch to the exponential form when the voltage falls below a predetermined value.

5.5 Devices Contributing to Modeling Difficulties


FGR. 2 A typical load window with % composition of load components.

Fluorescent light Incandescent light Air conditioning Refrig. and freezer Hot water heater TV Others Heating Clothes dryer Electric range


TABLE 5 Composition of Six Different Load Window Types

Load Window Type

Load Component

3-phase central AC Window type AC Duct heater with blower Water heater, range top Clothes dryer Refrigerator, ice machine Incandescent lights Fluorescent lights Industrial (induct. motor)


Some load components have time-dependent characteristics that must be considered if a sequence of studies using static models is performed that represents load changing over time. Examples of such a study include Voltage Stability and Transient Stability. The devices that affect load modeling by contributing abrupt changes in load over periods of time are listed below.

Protective relays: Protective relays are notoriously difficult to model. The entire load of a substation can be tripped off line or the load on one of its distribution feeders can be tripped off line as a result of protective relay operations. At the utilization level, motors on air conditioner units and motors in many other residential, commercial, and industrial applications contain thermal and/or over-current relays whose operational behavior is difficult to predict.

Thermostatically controlled loads: Air conditioning units, space heaters, water heaters, refrigerators, and freezers are all controlled by thermostatic devices. The effects of such devices are especially troublesome to model when a distribution load is reenergized after an extended outage (cold-load pickup). The effect of such devices to cold-load pickup characteristics can be significant.

Voltage regulation devices: Voltage regulators, voltage controlled capacitor banks, and automatic LTCs on transformers exhibit time-dependent effects. These devices are present at both the bulk power and distribution system levels.

Discharge lamps (mercury vapor, sodium vapor, and fluorescent lamps): These devices exhibit time dependent characteristics upon restart, after being extinguished by a low-voltage condition-usually about 70%-80% of rated voltage.

6 Load Window Modeling

The static load models found in Tables 1 and 2 can be used to define a composite load referred to as the "load window" mentioned earlier. In this scheme, a distribution substation load or one of its feeder loads is defined in as much detail as desired for the model. Using the load window scheme, any number of load windows can be defined representing various composite loads, each having as many load components as deemed necessary for accurate representation of the load. Fgr. 2 illustrates the load window concept. The width of each sub-window denotes the percentage of each load component to the total composite load.

Construction of a load window requires certain load data be available. For example, load saturation and load diversity data are needed for various classes of customers. These data allow one to (1) identify the appropriate load components to be included in a particular load window, (2) assign their relative percentage of the total load, and (3) specify the diversified total amount of load for that window. If load modeling is being used for Transient Stability or Operator Training Simulator programs, frequency dependency can be added. Let P(V) and Q(V) represent the composite load models for P and Q, respectively, with only voltage dependency (as developed using components taken from Tables 1 and 2). Frequency dependency is easily included as illustrated below.


Table 5 shows six different composite loads for a summer season in the southwestern portion of the United States. This "window" serves as an example to illustrate the modeling process. Note that each column must add to 100%. The entries across from each component load for a given window type represent the percentage of that load making up the composite load.

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