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Overloading & Calculations of Transformers

Overloading & Calculations of Transformers

In Article-1 we started to explain introduction of thermal loading and Cooling types of transformers. In this article, we are going to start with based on Article-1.

Life of the transformer is directly related with in the life of insulation Materials. Therefore in this article we are going to analyze formulas, and Calculations.

There are different reasons why transformers become overloaded, or why utilities may choose to overload them beyond their nameplate ratings. One reason is because the load demand has caught up or surpassed the transformer capacity and additional capacity is needed. Due to the complexity and exposure of the power system, no matter how well it is designed, failures are going to occur. It is the primary function of protective equipment to recognize such faults and isolate the faulted element from the rest of the system. This will cause the power flow to find new ways to reach the load demand. Transformers that find themselves on such paths might experience overloads beyond their normal capacity. In fact, depending on economical and reliability reasons, it may be necessary to overload transformers to use 25%, 50%, or more of their transformer life to maintain customers’ loads or to give system operators time to mitigate contingency conditions and prevent a potential blackout. It is possible to intelligently overload transformers to a rating that is still safe to operate by using the IEEE guide for loading of mineral-oil transformers C57.91-1995. The guide outlines the risks, theory, and calculations that make it possible to overload power transformers. Understanding the aging of the insulation and how to calculate the hottest-spot winding temperature are of vital importance in order to know how much a transformer can be safely overloaded.

Aging of Insulation

A power transformer’s solid insulation has two essential characteristics: dielectric strength and mechanical strength. Dielectric strength is maintained until the insulation is exposed to certain elevated temperatures. At that point, the insulation also becomes brittle and loses its mechanical strength. If the elevated temperatures are severe, the insulation will no longer be able to maintain its properties, resulting in insulation failure and ending the useful life of the transformer. Side effects of insulation aging include the formation of water and oxygen. However, with new oil preservation systems, these formations can be minimized, leaving the insulation temperature exposure as the controlling parameter for control personnel.

In the late 1940’s, it was discovered that the aging of insulation is part of a chemical process. Its reactions vary with temperature according to the Arrhenius equation

ƟH: is the winding hottest-spot Temperature, °C

A: is a constant

B: is a constant


The curve indicates that if the winding hottest-spot temperature is allowed to go beyond 110° C, the age of the insulation is accelerated above its normal rate and decelerated for temperatures below 110° C. Transformers with loading factors between 55% and 65% might seldom reached this temperature, resulting is slow insulation aging.

The equation for per unit life curve is shown below:

The 15000 constant value corresponds to the slope of aging and represents most forms of insulation. ΘH is the hottest-spot winding temperature. This per unit life equation is the basis for calculating the aging accelerating factor curve in Figure-2.


The aging accelerating factor is calculated using the following equation:

The aging accelerating factor can be used to calculate the life of a transformer for a predetermined hottest-spot winding temperature. However, to calculate the loss of life of a transformer for a 24-hour cycle, we must use the following equation:

FEQA: Equivalent aging factor for the total time period

N: Total number of intervals

n: Index of time interval

FAAn: Aging accelerating factor at index n

Δtn: Time interval

A transformer’s total loss of life can be derived as a percentage by using the following equation:

According to the IEEE C57.91-1995 guide, the length of insulation’s normal life has been a topic of controversy for decades. Earlier versions of the IEEE C57.91- 1981 settled on an insulation life of 65000 hours for power transformers. The standard now states that this value might be extremely conservative; an insulation life of 180000 hours has now been used for many years.

This section has described transformer insulation’s normal life, and factors that affect its aging. We also established that insulation’s aging is directly connected with the hottest-spot winding temperature. If we are to establish a thermal overload rating, the hottest-spot winding temperature must be known.

The Winding Hottest-Spot Temperature

As it was determined before, an insulation’s mechanical and dielectric properties are deteriorated at temperatures above normal limits. From the previous section, we now know that if the hottest-spot temperature is allowed to go beyond 110° C, the insulation deteriorates at a faster rate than normal. Therefore, the highest temperatures in the transformer aid in calculating insulation integrity.

The winding hottest-spot temperature is given by



ΘA: Ambient temperature

ΔΘTO: Top-oil rise over ambient temperature in ° C

ΔΘH: Winding hottest-spot rise over top-oil Temperature

Ambient Temperature ΘA

To predict an overload for a 24-hour period, discrete 1-hour ambient temperature increments are necessary. The Temperature used should be the maximum daily temperature for the month of interest averaged over several years.

Top-Oil Rise over Ambient Temperature ΔΘTO

As explained in the thermal heat transfer process section, the heat produced by the core, structural parts, and winding of the transformer decreases the specific gravity of the oil, causing it to travel upward toward the top of the tank. During steady state load conditions, the top-oil rise over the ambient may be constant. On the other hand, under transient conditions or when the load increases or decreases, the top-oil rise over the ambient may be continuously changing (2). As a result of such behavior, the top-oil rise over the ambient is given by the following formula:

ΔΘTO,u: Ultimate top-oil rise over ambient temperature in °C

ΔΘTO,i: Initial top-oil rise over ambient temperature in °C

t: Duration of changed load in hours

τ: Thermal time constant for the transformer (accounting for the new load, and accounting for the specific temperature differential between the ultimate oil rise and the initial oil rise)

During a load increase or decrease, a heating transient occurs inside the transformer, causing the temperature to either go up or down. Due to transformer’s large mass, it takes time for the heat to dissipate from an initial value to an ultimate value. Therefore, the initial (ΔΘTO,i) and ultimate (ΔΘTO,u) top-oil rises over the ambient formulas are given below:


ΔΘTO,R: Top-oil rise over ambient at full load (determined during test report)

K: Ratio of load of interest to rated load

R: Ratio of load loss at rated load to no-load loss

n: Empirically derived exponent used to calculate the variations of ΔΘTO with changes in load. Transformers with different cooling modes will have different n values, which approximate effects of change in resistance with changing load.

The time required for the top-oil temperature to reach its ultimate value is a function of the thermal oil time constant τ. The oil time constant for the top-oil rise is given below:


τTO,R: Oil time constant at rated load beginning with the initial top-oil temperature rise at 0° C

PT,R: Total power loss t rated load

C: Thermal capacity of the power transformer (watt-hour per °C)

“The thermal capacity is the heat transfer to a unit quantity of matter in a body which will cause 1° C change in the temperature of the body” (2). Such thermal capacity will depend on the cooling system employed and size of the transformer.

The thermal capacity is calculated as follows:

For ONAN, ONAF cooling modes:

C= 0.06*(weight of core and coils) + 0.04*(weight of tank and fittings) + 1.33*(gallons of oil)

For OFAF cooling modes, either directed or undirected:

C= 0.06*(weight of core and coils) + 0.06*(weight of tank and fittings) + 1.93*(gallons of oil)

Winding Hot-Spot Rise over Top-Oil Temperature ΔΘH

During a continuous or steady state load, the losses cause the temperature of the winding to increase. As the process continues, the heat is transferred to the surrounding oil. This heat transfer process continues until the heat generated by the windings equals the heat taken away by the oil under continuous load. During heat transient conditions, the winding hottestspot rise may change from an initial value to an ultimate value depending on the winding hottest-spot time constant τW. This is given by:


ΔΘH,i: Initial hot-spot rise over top-oil

ΔΘH,u: Ultimate hot-spot rise over top-oil

ΔΘH,R: Winding hot-spot rise over top-oil

ΔΘH/A,R: Winding hot-spot rise over ambient

τ: Winding time constant

m Empirically derived exponent to account for conductor insulation wall thickness, conductor configuration, oil viscosity, oil velocity, cooling mode

ΔΘH/A,R is a value that is determined during manufacturing of the power transformer. The C57.91-1995 also recommends assuming a value of 80 °C for a 65 °C average winding rise and a value of 65 °C for a 55 °C average winding rise on its nameplate, respectively.

Our initial goal was to determine how much overload an oil-immerse transformer can sustain without severely affecting its integrity and knowing that it is still safe to operate. We described in previous sections that the driving factor for insulation credibility is temperature exposure, since at high temperatures insulation can lose its insulation properties. For this reason, it is necessary to establish the hottest-spot winding temperature. If we know the maximum hottest-spot temperature to which the insulation is exposed, we know how much life is lost, allowing us to limit the overload. For practical applications, following the formulas outlined above can be cumbersome, complex, and time-consuming. It will be explained in this paper’s last section that a microprocessor protective relay can perform the algorithm calculations we have outlined, simplifying the job of calculation and application. However, the relay will require the user to input the maximum hottest-spot Temperature where he/she wants the relay to operate. Computer programs can also facilitate the offline analysis and studies that are needed to establish limited ratings. As a result, rating methodologies can be created and refined.

Source: Fundamental Principle of Transformer Thermal Loading and Protection Article.

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