Analysis of electric heating load dependency on voltage

Nokhum Markushevich

This analysis is based on a hypothetical model of electric heaters with the following assumptions:

The heating/cooling processes are described by exponential functions
The load of the heater depends on the voltage in the second power.
The maximum temperature that can be provided by the heater is proportional to the voltage in the second power
The aggregated load of a group of heaters is proportional to the probability of the heaters to be in the ON state.
The energy consumed by the group of heaters is defined for a given time interval (billing cycle)

This is a simplified model of the heating process, but we believe that it can illustrate the conceptual dependencies of the electric heater loads and energy on voltage.

The following conditions have been considered:

Oversized heaters that can provide the target temperature under 95% of the nominal voltage
Undersized heaters that cannot provide the target temperature under sub-nominal voltages
Moderate ambient temperature
Extreme ambient temperatures

The following dependences on voltage are presented in the results:

Load of an individual heater
Aggregated load of a group of heaters
Energy consumption during one ON-OFF cycle
Energy consumption during a given time interval, e.g., during a billing period
Load-to-voltage dependencies of combined heating and other loads in different proportions.

The first two values are not the same because of the changing diversification of the load of the group.

The dependency of a one-cycle energy consumption on voltage is not the same as the dependency of energy during a billing cycle. When the voltage is changed, the ON-time is affected, while the OFF-time is the same. Hence, the total duration of the ON-OFF cycle is changed, and the number of cycles during a given time interval is also changed.

The following equations are used in the analysis:

IF(Templim>Temphigh),

then TimeON =-Tconst x Ln((1-(Temphigh-Templow)/(Templim-Templow)). (1)

If (Templim<=Temphigh),

then TimeON = Tconst x LN((1-(Temphigh-Templow)/(Temphigh+0.01-Templow)).

IF (Templim<=Temphigh),

then TimeOFF = 0, (2)

IF(Templim>Temphigh),

then, TimeOFF= -Tconst-amb x LN(1-(Temphigh-Templow)/(Temphigh-Tempamb))

Timecycle = TimeON + TimeOFF (3)

Ncycles = Timebilling / Timecycle (4)

Loadind ~ Volt2 (5)

Templim = Templim-nom x Volt2 x (1+(Tempamb-Tempnom)/A) (6)

Energycycle = TimeON x Loadind (7)

Energybilling=EnergycyclexNcycles (8)

ProbON = TimeON x Ncycles / Timebilling (9)

Loadaggr~ ProbONxLoadind (10),

where

Templim – temperature in the indoor space that can be reached, if the heater is ON all time

Temphigh – maximum target indoor temperature

Templow - minimum target indoor temperature

TimeON – time during which the heater is ON to raise the temperature from Templow to Temphigh in one heating cycle

TimeOFF - time during which the heater is OFF while the temperature drops from Temphigh to Templow in one heating cycle

Tconst – time constant for the indoor heating process

Tconst-amb - time constant for the cooling process

Tempamb – outdoor temperature

Timecycle – duration of one heating-cooling cycle

Ncycles – number of heating-cooling cycles during the billing period

Timebilling – duration of the billing period

Loadind – load of an individual heater

Volt – voltage applied to the terminal of the heater

Templim-nom - temperature in the indoor space that can be reached, if the heater is ON all time under nominal (reference) conditions

A – adjustment coefficient

Energycycle – energy consumed by a heater during one heating cycle

Energybilling - energy consumed during billing period

ProbON – probability of the heater being ON during the billing cycle

Loadaggr- aggregated load of a group of heaters.

Figure 1 depicts the load of an individual oversized heater, its energy consumed during the time ON, the energy consumed by the heater during a billing cycle, and the aggregated load of a group of such heaters. These dependencies are presented for a moderate ambient temperature. As seen in the figure, while the load of an individual heater is lower under lower voltage, all other parameters are higher.

Figure 2 depicts the same parameters for a colder weather. As seen in this figure, the increase of the energy consumption during the time ON and the aggregate load under low voltages are significantly greater than for the moderate weather. It is due to the longer duration in the ON status.

Figure 1. Load and energy dependencies for an oversized heater, moderate weather.

Figure 2. Load and energy dependencies for an oversized heater, cold weather.

Figure 3 illustrates the difference in the aggregated load for the moderate and cold weather.

Figure 3. Comparison of load dependencies for an oversized heater for a cold and a moderate weather.

Figure 4 depicts the dependencies of the same parameters for undersized heaters under a cold weather. As seen in the figure, under sub-nominal voltages, the energy and the aggregated load reduce with voltage reduction. This is because the heater cannot reach the target indoor temperature and is all the time ON.

Figure 4. Load and energy dependencies for an undersized heater, cold weather

In reality, the area load fed from one power source consists of a combination of the electric heating load and other loads with different load-to-voltage dependencies. Hence, the aggregated load-to-voltage dependency of the area load depends on the proportion of the heating load in the common load and on the dependencies of the other loads.

Figure 5 through Figure 7 illustrate load-to-voltage dependencies of combined load for different proportions of the heating load, for different heaters, and different weather. In this illustrations, the load sensitivity of the non-heating load is positive, namely, 1% reduction of load per 1% reduction of voltage. As seen in the figures, even with a small portion of the heating load, the load sensitivity to voltage can be negative, i.e., the lower the voltage – the higher the load. The figures also show that the aggregated dependencies are different under different ambient conditions.

Figure 5. Dependencies for combined load for oversized heaters. Moderate weather.

Figure 6. Comparison of combined loads for moderate and cold weather. Oversized heaters

Figure 7. Dependencies for combined load for undersized heaters. Cold weather.

Conclusions.

In this paper, the analysis of the load-to-voltage dependencies for the electric heating load is performed based on a hypothetical model of the heating and cooling processes. The analysis serves the purpose of attracting attention to the methodologies of field tests and applications of the load-to-voltage dependencies in power system operations.
As follows from this analysis, the dependencies can be significantly different even for the same season, but under different ambient conditions.
The dependencies can also be different for different areas due to different combinations of heating and other load and due to different sizing of the heaters.
It is possible that under some conditions the load-to-voltage sensitivities can become negative, when higher voltages result in lower loads and lower energy consumption.
The load-to-voltage dependencies for areas with significant electric heating load should be discriminated by areas and ambient conditions to properly apply the conservation voltage control.

References.

Nokhum Markushevich, Aleksandr Berman, and Ron Nielsen, Methodologies for Assessment of Actual Field Results of Distribution Voltage and Var Optimization, presented at IEEE PES 2012 T and D
Nokhum Markushevich, Voltage Reduction Effect in Active Distribution Networks. Available: https://www.scribd.com/document/376755241/Voltage-Reduction-Effect-in-Ac ...