Load Coefficients (Load Varying with Voltages)
It is crucial to comprehend how the electric load reacts to changes in the supply voltage, especially when
voltage reduction is now being utilized more often by utilities as means to reduce electric stress and
demand response. After voltage reduction was implemented at a specific station, various engineering
departments review the load change and compare the impact to their models and expectations. Often, the
results don’t concur with their analysis, mainly due to the lack of modeling proper load coefficients. In
the next paper, I can discuss the impact of voltage reduction on the substation transformers, subtransmission feeders and the transmission system.
There are three types of loads, also known as load coefficients:
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Constant power load
Constant current load
Constant admittance (inverse of the impedance) load
Even though several load flow programs models the electric load as constant power only, the other two
types play a key role as well. Most modern equipment is of a constant power load type. The main
difference between these types is in their response to changes in voltage.
Constant power, 𝑃𝑝 = 𝐼 2 𝑅 ∝ 𝑉0 (𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑣𝑎𝑟𝑦 𝑤𝑖𝑡ℎ 𝑣𝑜𝑙𝑡𝑎𝑔𝑒)
𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑐𝑢𝑟𝑟𝑒𝑛𝑡, 𝑃𝐼 = 𝐼𝑉 ∝ 𝑉 1 (𝑣𝑎𝑟𝑖𝑒𝑠 𝑤𝑖𝑡ℎ 𝑣𝑜𝑙𝑡𝑎𝑔𝑒)
Constant admittance, 𝑃𝑌 =
𝑉2
𝑅
∝ 𝑉 2 (𝑣𝑎𝑟𝑖𝑒𝑠 𝑤𝑖𝑡ℎ 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑠𝑞𝑢𝑎𝑟𝑒).
Pp is the constant power component, PI is the constant current component and PY is the constant
admittance component. The same formulas apply for reactive power as well.
Load coefficients (percentage of each load type mentioned above) were originally calculated via
actual field tests, usually during the summer peak days. Different load pockets/areas/networks
are expected to have different load coefficients, since they have different components, i.e. the
percentage of residential load Vs. commercial loads, industrial loads Vs. non-industrial loads,
etc. Being that an actual test was warranted at the time (summer peak) when it is desirable to
maintain the electric grid and not perform any tests, some of the available calculated load
coefficients are ancient.
For power load flow representation, the real power shows all three types of loads and so does the
reactive power, if the load breakdown is not available, then 100% of the load is applied to the
constant power load (very conservative). The load coefficient formulas, also known as the ZIP
coefficients (Z=impedance, I=current and P=power) are:
𝑉
𝑉𝑜
𝑉 2
𝑉𝑜
𝑉
𝑉𝑜
𝑉 2
𝑉𝑜
𝑃 = 𝑃𝑜 [𝑃𝑝 + 𝑃𝐼 � � + 𝑃𝑌 � � ] and Q= 𝑄𝑜 [𝑄𝑃 + 𝑄𝐼 � � + 𝑄𝑌 � � ]
𝑃𝑜 & 𝑄𝑜 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑟𝑒𝑎𝑙 𝑎𝑛𝑑 𝑟𝑒𝑎𝑐𝑡𝑖𝑣𝑒 𝑝𝑜𝑤𝑒𝑟 𝑎𝑡 𝑉𝑜 .
To illustrate, several load flows were performed for a 138/13 kV substation called “Demo”.
Assume that from historical field tests results, the real power load coefficients for Demo
substation were:
68.7 % 𝑃𝑝 , 23.7% 𝑃𝐼 𝑎𝑛𝑑 7.6% 𝑃𝑌
The reactive power load coefficients percentages were:
10.2 % 𝑄𝑝 , 27.5% 𝑄𝐼 𝑎𝑛𝑑 62.4% 𝑄𝑌
It is clear from the above percentages that the bulk of the real power represents constant power
type, however, only 68.7% and not the entire composition as many think. Also, the bulk of the
reactive power is of a constant admittance type, which varies with the voltage squared. This is an
expected contribution since the reactive power decreases substantially when the voltage is
reduced, being that the reactive power varies with the square of the voltage. Also, being that
power factor is a function of the reactive power and the reactive power decreases as a result of
the reduced voltage, the power factor improves when voltage is reduced. The significant
reduction in reactive power and the improvement in power factor will not be witnessed when the
entire load is modeled as constant power (see explanation below).
During distribution contingencies, especially in the summer, some utilities utilize voltage
reduction in order to reduce the stress on their distribution feeders.
To illustrate further, a PSS/E load flow for substation “DEMO” carrying 100 MW & 40 MVAR, no
capacitors, at 13.75 kV was modeled; the first black rectangle below shows the breakdown of the load
based on the three types of load. As seen, the bulk of the real power 68.7 MW represents the constant
power component of the 100 MW and the bulk of the reactive power 25.2 MVAR represents the constant
admittance component. The red box below shows the breakdown after implementing a 5.3 % voltage
reduction, the total MW dropped by 2% from 100.5 to 98.5 MW and the MVAR dropped by 8% from
40.2 to 37 MVAR. As expected, the constant power load (top line) did not change at all and the biggest
impact was the change to the constant admittance component.
It is also important to study how the load types behave when the voltage increases. The load flow below
shows the impact of raising the voltage by 2% (from 13.753 to 14.049 kV); as expected, the first line
shows the constant power contribution which did not change. The biggest impact is the constant
admittance change.
The load at 13.753 kV was 100.4 +j40.2 [MW + jMVAR]
The load at 13.030 kV was 98.5 +j37 [5.3% kV reduction, 2% MW reduction & 8% MVAR reduction]
The load at 14.049 kV was 101.3 +j41.5 [2% kV Increase, 1% MW increase & 3% MVAR increase]
The % load changes above as a result of varying the station voltage was explained by the formulas in the
beginning of the paper. As far as utilities are concerned, the voltage reduction will provide benefits for
constant admittance and constant current types from load reduction perspective, of course, the electric
stress will be reduced, which is the main reason voltage reduction was implemented in the first place in
order to avoid failures in the distribution feeders; however as seen from the load flow results, the constant
power loads (which is the predominant load type) are not impacted by voltage reduction, i.e. P = VI,
when the voltage decreases the current will increase and when voltage increases the current will decrease
in order to attain a constant power load. It is important to note that as the value of voltage decreases
beyond a certain value/threshold there will be an increased reduction in load due to certain equipment
tripping (going out of service).