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The Necessity and Challenges of Modeling and Coordinating Microprocessor BasedThermal Overload Functions for Device Protection

Karl Smith, P.E., ABB Inc. Sajal Jain, P.E., ETAP

Abstract— Increased demand and economic constraints oftencause the power system to be operated near its thermal limits andsometimes beyond. One way to mitigate this problem is to use thethermal overload protection functions built in to today’s modernIED (Intelligent Electronic Device) relays that continuously monitorthe thermal load in real time. These functions (ANSI 49) use verysophisticated algorithms that closely replicate the thermal image ofthe protected object (motor, transformer, feeder cable etc.). It isequally important that this function is set and coordinated properlyto avoid mis-operations and equipment damage. Unfortunately, theoperate time curves for these functions are often neglected in relaycoordination software since modeling them is mathematicallychallenging. The steps required to express the curve equation interms of prior load and a trip reference current are lengthy andcomplex. For transformer applications (using two time constants)the operate time must be solved iteratively. Even if a reduced singleequivalent time constant is used to simplify the model there are stilldeviations. In addition to this, engineers often use typical timeconstant values since this data is not always provided from themanufacturer. As a result, time constant setting(s) may needadjustment to improve coordination. The ideal solution would be touse custom curves. However this is time consuming since pointswould have to be entered manually from third party tools. There’salso an increased chance for error. This tutorial based paperdescribes what is required to overcome these modeling challengesfollowed by a detailed discussion on the coordination principlesusing motor and transformer project examples. The result, a singletime constant equation derived from the first order thermal modelthat is easily adapted to specific relay settings (temperature,overload factor etc.). The equation, expressed in per unit of the tripreference current term, can be universally applied to any relay.

Key Terms— Relay coordination software, thermal overloadfunction, first order thermal model, measured current, prior loadcurrent, reference current, maximum temperature, overload factor,weighting factor, time constant, trip time equation, over-currentcoordination.

I. INTRODUCTION

This paper describes a case study for modelling andcoordinating micro-processor based thermal overload curves inrelay coordination software. In particular, for feeders, motors andtransformers. Most modern relays now have their own uniquebrand of the thermal overload (ANSI 49) function. What is lesscommon however, is the ability to define the relay’scorresponding trip time curves in much of the protection designsoftware available today. Without a reliable way to verifysettings and device coordination, mis-operations and equipmentdamage could result. The information in the following sectionsoffers a tutorial based approach for gaining a betterunderstanding of this kind of protection, why it is needed and anovel method for how it can be used efficiently in a project for awide range of relays and software vendors.

II. WHY IS THERMAL OVERLOAD PROTECTION NEEDED

At its simplest, traditional non-directional overcurrentprotection using IDMT (Inverse Definite Minimum Time) canprovide some degree of overload protection. The disadvantageof overcurrent protection however is that it fully resets after theoverload condition subsides even though the protected devicemay still be approaching its thermal limit. In other words thereis no thermal memory. In the event of repetitive overloading thiscould lead to numerous problems such as insulation failure.According to one IEEE study 4.2% of all motors failures are dueto persistent overload [1]. Although setting a longer reset delaytime can help it is limited by its maximum setting (typical valueis 60 sec).

Conversely thermal overload protection has a thermalmemory. It continuously monitors the load and calculatestemperature. As a result, protected circuits can operate closer totheir thermal limits allowing increased utilization of the powersystem.

III. PRINCIPLE OF OPERATION IN THE RELAY

Fundamentally, all thermal overload protection works offthe I2t principle. Simply stated, the temperature rise for aprotected device is proportional to the square of the currentflowing through it. Unlike overcurrent protection, thermaloverload settings use thresholds of temperature and thermalcapacity instead of actual current to determine the operation setpoints (alarm and trip).A. Over Temperature

The actual temperature of the protected device is calculatedusing a thermal counter which replicates the thermal image of theprotected device (i.e., feeder, transformer). The relay trips whenthe actual temperature calculated exceeds a maximumtemperature threshold.B. Thermal Capacity Used

The Thermal Capacity Used (TCU) is calculated based onrated full load current of the motor with an overload factor (k)applied. It is expressed as a percentage of maximum thermalcapacity. The relay trips when the TCU reaches 100 percent.

IV. OVER TEMPERATURE

The derivation of the temperature calculation can beexpressed using the general time domain solution for theelectrically equivalent R-C circuit shown in figure 1 [2]. Formost electrical engineers this is the most practical way tounderstand the thermal process.

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Fig 1. Equivalent R-C circuit with voltage source.

The differential equation of the R-C circuit in figure 1 is:

= (1)

− = (2)

From (1) & (2) the equation becomes:

− = (3)

Re-arranging (3):

=1

( − ) =1

( − ) (4)

The equation is now in the same form as the thermal differentialequation. If voltage of capacitor (C) at t=0 is v0, then the generalsolution becomes:

= + ( − ) 1− (5)

Substituting the thermal overload relay equivalent parametersfrom table 1 into (5) the time step equation becomes [2], [3]:

= + ( − ) ∙ 1−∆

(6)

θn calculatedpresenttemperatureθn-1 calculatedtemperatureatprevioustimestepθfinal calculatedfinaltemperaturewithactualcurrentΔttimestepbetweencalculationsofactualtemperatureτsetthermalTimeconstantofprotecteddevice

Where:

= ∙ (7)

IthelargestphasecurrentIrefsetCurrentreferenceTrefsetTemperaturerise

The actual temperature is then calculated by adding ambienttemperature (set or measured via RTD) to the calculated present

temperature (θn). In the form shown, the thermal counterequations describe how temperature is calculated for the thermaloverload protection (49F) function in a feeder relay.

TABLE 1RELAY /THERMAL EQUIVALENCY FOR R-C CIRCUIT PARAMETERS

R-Ccircuit Thermal Relay Description

θ θ Presenttemperature

RC ℎ τ Timeconstant

ℎθ Final

Temperature

t t ∆t time

There also is a calculation for relay operation time (present timeto trip). This calculation is run whenever the final temperaturecalculated exceeds the set trip (θtrip) temperature.

= − ∙ ln− −

(8)

For transformer applications (49T protection function), thetemperature rise calculation equation (9) requires two timeconstants to describe warming. One for the windings (short timeconstant, 1) and the other for oil (long time constant, 2). Incases where only a single time constant is supplied by themanufacturer, an equivalent time constant can be used [3].

∆ = ∗ ∙ ∙ 1 −∆

+⋯

… (1 − ) ∗ ∙ ∙ 1−∆

(9)

Δθ calculatedtemperaturerise(°C)intransformerImeasuredphasecurrentwithhighestvalueIrefsetCurrentreference(ratedcurrentoftransformer)TrefsetTemperaturerise(temperaturerisesettingwiththesteady-statecurrentIref)psetWeightingfactorp(weightingfactorfortheshorttimeconstant)ΔttimestepbetweencalculationofactualTemperatureτ1setShorttimeconstantofprotecteddeviceτ2setLongtimeconstantofprotecteddevice

Adding ambient temperature to (9) the actual temperature of thetransformer is:

= + (10)

Θ temperatureintransformerΔθ calculatedtemperatureriseintransformerθamb set ambient (environmental) temperature

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Fig 2. Simulation illustrating temperature rise to trip for transformer thermal overload protection.

To determine trip time, iterative methods are required. Theresults from a simulation tool shown in figure 2 graphicallyillustrate the temperature rise to trip principle for the prior loadcurrent and relay settings indicated. Plots for heating curves, with(warm) and without (cold) prior load are shown.

V. THERMAL CAPACITY USED

For motor applications (49M protection function) the relaycalculates percent thermal capacity used, instead of temperature,to evaluate a trip condition (trip occurs when TCU reaches100%). Also an overload factor, instead of temperature, is usedto define the operation set point. When the measured currentexceeds the overload factor times the rated current (maximumpermissive load), the relay picks up and trips after a delaydetermined by the relay’s time constant setting. The overloadfactor is calculated based on the principle of temperature risebeing related to the square of currents follows:

= ∙ (11)

−− = (12)

Taking the square root of both sides of the equation the overloadfactor can be calculated using temperature:

=−− = (13)

To demonstrate with an example, consider a motor that has aclass F (155°C) insulation with a class B temperature rise of 90°C(130°C – 40°C) [4]. Substituting these values into (13) thecalculated value of the overload factor is:

=155 − 40130 − 40 =

11590 = √1.28 = 1.13

Note that increasing the current by a factor of 1.13 yields anincrease in temperature (above ambient) by a factor of about1.28. The increase in current is not proportional to the increasein temperature (figure 3).

Fig 3. Temperature vs percent settings in motor thermal overloadprotection.

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The overload factor is generally set at 8-10 percent above themotor service factor to account for measuring error [5]. Themaximum setting is limited to about 1.2 (typical maximum defaultsetting) since the motor cable is normally sized at 1.25 percent ofthe motor’s full load current rating [5].

Thermal overload protection is also unique in that it depends onnegative sequence current to describe heating of the rotor and thatit that it uses one of three time constants depending on the state ofmotor operation (starting, normal/overload, and stopped). Table 2shows the typical current ranges (relative to rated current) thatdefine each state of motor operation.

TABLE 2CURRENT RANGES FOR MOTOR TIME CONSTANTS

Motor phase current Time constant setting used< 0.12 x Ir Stop> 0.12 x Ir and < 2.5 x Ir Normal> 2.5 x Ir Start

Essentially there are two thermal load equations that describemotor heating:

= × + × …

× 1− × 100%(14)

= × + × …

× 1− × %(15)

IMaximumvalueofmeasuredphasecurrentsIrsetRatedcurrentI2measurednegativesequencecurrentksetvalueofOverloadfactorK2setvalueofNegativesequencefactorpsetvalueofWeightingfactorτsetvalueofTimeconstantnormal/start/stop

The first, θA, to account for hot-spots in the rotor and statorwindings when the motor is started or overloaded. For thiscondition the motor will trip if the overload condition persists longenough (TCU reaches 100%). The second, θB , which always isrunning in the background, to account for the overall thermalcapacity of the motor body. It is not intended for trip. The equationis identical to θA with the exception that a weighting factor (p) isused to define, and limit, the TCU (typically to 50%) when themotor is operating at the maximum permissive load, therebypreventing trip. When the overload condition subsides θA isbrought down linearly to θB at a rate of 1.66% (ie., after motorstarting, or when an overloaded motor is reduced to non-triplevels) [3].

The θB calculation defines the initial (prior) thermal load levelfrom which the θA calculation for overload begins. It can also beused to calculate the initial and subsequent decaying thermal loadduring cooling after the motor is at standstill, that is when themeasured current drops below the stop threshold. Using table 2this would be when measured current drops below 0.12 x Ir.

To calculate thermal load during cooling, the θB equation(which assume zero prior thermal load) must be completed so ittakes prior thermal load into account

= × + × − % …

× 1− × % +(16)

To arrive at the initial or prior thermal load level for the coolingcurve, consider a motor that has been running at a constant loadlong enough (t→∞) such that it reaches a steady thermal level.This omits the exponential curve of equation (15) yielding:

= × + × × % (17)

If the motor is then stopped, the calculated value of θB nowbecomes the initial or prior thermal load value (θP), from whichcooling begins. Expressing in terms of pre-load current (I = IP, I2 =IP2) the equation becomes:

Substituting (18) into (16), assuming zero current (I = I2 = 0)equation (18) is reduced to:

= 0 − % × 1 − × % +

= × (19)

For a motor operating at the maximum permissive load (IP = k xIr), the prior thermal load level for cooling would be equal to theweighting factor (p) setting. The cooling equation then becomes:

= % ×

Figure 4 shows a graphical example of these heating and coolingcharacteristics (not drawn to scale) for a motor which was initiallyoperating at the maximum permissive load (for a long enough timeto reach its maximum steady state thermal level), and then stopped,re-started and stopped again.

= × + × × % (18)

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Fig 4. Thermal behavior for motor running at maximum permissive load (p%), stopped, re-started and stop again.

The thermal simulation in figure 5a show how calculatedthermal capacity varies with respect to time for cold motor start-up (no prior load) [6]. Figure 5b shows a simulation for a motorstopping, re-starting, and then running continuously [6].

Fig 5a. Thermal simulation for cold motor start-up.

Fig 5b. Thermal simulation for motor stopping, re-starting and running.

VI. COLD AND WARM TRIP CURVES

When substation components (ie., feeders, transformers andmotors) are first energized or taken out of service for a sufficientlength of time, they reach equilibrium temperature with theenvironment. This temperature is referred to as the “ambient”temperature. For motors the percent TCU is at zero. The trip timeequations for these conditions describe a “cold” curve since noprior loading has contributed to an increase in temperature orpercent TCU.

If there is prior load (ie., at rated current) then the trip timeequation describes a “warm” curve. The trip time in this case isfaster since starting from a higher temperature or percent TCU

value. For example, feeders or transformers running at rated load(typically the reference current setting) have a specifiedtemperature rise (reference temperature) above ambient. Formotors running at rated load the percent TCU is typicallyapproaching 50%.

To calculate trip time, using the motor for example, the θA

equation (14) must be completed for prior thermal load in terms ofpre-load current:

= × + × − % …

× 1− × 100% +(20)

Solving (20) for time (t) at the tripping level θA = 100%(Appendix 4) and neglecting the negative sequence current, theequation becomes:

= ×− ×− ( ) (21)

For a cold motor, the prior load is assumed to be zero. Theequation then is reduced to:

= ×− ( ) (22)

VII. CHALLENGES MODELING IN COORDINATION SOFTWARE

Whenever the thermal overload (49) function is part of theprotection scheme it is critical to coordinate the trip curves withthe other protective devices in the zone of protection. Although,there are many vendors today that offer protection designsoftware to make relay coordination easier and more efficient,the challenge still remains to integrate the trip curves into therelay coordination software programs.

The largest obstacle to modeling the curves is themathematically complexity. For this reason, the task is often

Motor running atmax permissive load

(k x Ir) for long duration(θp =p%)

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neglected. The same holds true for setting in the relay and testing.Some typical challenges are as follows:

· Trip equation must be expressed in terms ofmeasured load, pre-load and (trip) reference current(typically rated or nominal). Publisheddocumentation does not always show equations inthis form. Derivations, if provided are lengthy andcomplex.

· Trip time equation in its general form cannot beuniversally applied to a specific relay manufacturer(unique settings to define trip current).

· Negative sequence current component andweighting factor (motors).

· Advanced mathematical methods required to solvetrip time iteratively from thermal model using twotime constant equation (transformers).

Fortunately, most of these challenges can be overcome usingthe time solution, expressed in per unit of the trip current, of thefirst order thermal model. From there the trip current can beadapted to the relay’s specific settings independent of the vendor.In some cases a few simplifying assumptions would need tomade. If decided the assumptions cannot be made, then customuser curves can be developed from simulation tools that arereadily available, although less practical.

VIII. FIRST ORDER THERMAL MODEL

Fundamentally all thermal overload protection, andcorresponding trip curves, are derived from the temperature andtime solutions of the first order thermal (Appendix 1) [2]. Thefirst order differential equation resulting from the thermal modelis:

=1 ( )

ℎ − (23)

θ temperature(aboveambient)t timeI measuredRMScurrentRresistanceτTimeconstanthheattransfercoefficientAsurfaceareaofconductor

The temperature solution (Appendix 2) to the first order thermaldifferential equation for constant current is:

= + ℎ − 1− (24)

Where;

q = ℎ ,q = ℎ ,ℎ =

θ0initialtemperature(aboveambient)θfinalfinaltemperature(aboveambient)Innominalcurrent(referencecurrent)θnnominaltemperature(atreferencecurrent)

Solving (24) for time (Appendix 3) the equation becomes:

= ×

⎣⎢⎢⎢⎡

−

− q ⎦⎥⎥⎥⎤

(25)

Where;

=q

(26)

Substituting (26) into (25) and setting I0 to the pre-load current(I0 = IP) the equation now becomes:

= ×−

−(27)

Expressing measured current (I) from (26) in per unit of the tripcurrent the equation is becomes:

= ×−− 1

(28)

For motors, (27) assumes a weighting factor (p) of 100%. Forweighting factors other than 100%, (and still neglecting negativesequence current), the pre-load current term needs to be scaledby the weighting factor (p) in accordance with (21):

= ×− ×− 1

(29)

The equations are now in the general form for modeling inprotection design and relay coordination software. Specific relaysettings, regardless of application or relay can be adapted to thegeneral form of the trip current equation (26). The result, auniversal trip time equation which reduces the modelingcomplexity considerably.

IX. MODELING FOR THE SPECIFIC APPLICATION

The trip time curves for each application (feeder, motor, andtransformer are modeled for both cold and warm curves. Thecold and warm equations for each application are the same withexception that the pre-load variable is removed for the coldcurve. The following sections show how each application wasmodeled and the assumptions that were made.

A. FeederThe trip equation curve for the feeder is the mathematically the

least complex. Figure 6 shows how the warm curve is modeled inprotection design software. The cold curve equation is the samewith exception that the pre-load term (C*C) is removed. For thisparticular vendor some of the variables for inverse definiteminimum time (IDMT) curve modeling were re-used. Fromequation (27) the measured current (I) is now defined in multiples(M) of trip current instead of pickup, and the time constant (τ) isnow used in place of dial setting (TD). The prior load (IP), is nowused in place of the “time multiplier (C) and is entered as a per

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unit value of the trip current. Note how the range for minimumpick-up is set slightly above 1 (minimum multiple is 1.00001) toprevent invalid calculations.

Fig 6. Warm trip time curve model for feeder thermal overload protection.

B. TransformerBecause transformer thermal overload protection uses two

time constants, the operate time solution is complex requiringiterative solutions (i.e., secant method). To simplify the modeling,a single reduced equivalent time constant can be used toapproximate the two time constant curve. From equation (9), theweighting factor would be set to zero (p = 0) and the long timeconstant (τ2) would be set to the value of the single equivalenttime constant. This results in an equation that is identical to thefeeder model with exception of terminology used and the variableselected for ‘Prior Load’ (P). The modeling is shown in figure 7.

To determine the single time constant from the short and longtime constants, conversion tables provided in the relay’s technicaldocumentation can be used [3]. If the time constant(s) are notsupplied by the manufacturer then typical values can be used. Fora distribution transformer a single time constant value of 50minutes (3000 seconds) can be used, and for a power transformer,about 75 minutes (4500 seconds).

Fig 7. Warm trip time curve model for transformer thermal overload

Using the simplified model will always result in somedeviation to the two time constant trip time curve. Thesimulation in figure 8 graphically shows the deviation betweenthe two curves (cold and warm) for a liquid filled distributiontransformer (65ºC rise) using typical time constant values.

Fig 8. Simulation comparing single and two time constant trip time curves for typical distribution transformer

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It can be seen from the graphs that using the single timeconstant approach is less conservative. However, this methodwill improved coordination with downstream protectionprovided there’s adequate margin with the transformer damagecurve. If not, adjusting the single time constant to just fall belowthe damage curve usually results in a close approximation to thetwo time constant trip curve. (See next section on coordinationprinciples)

If using the two time constant equation, then user definedcurves need to be created by manually entering the curve pointlist. This of course is time consuming and increases the chancefor error. Plans to develop two time constant models are nowbeing considered.

C. MotorThe trip time curves for the motor are the same as the feeder

and transformer model (single time constant) with the exceptionthat a weighting factor (C), in percent (used in place of the ‘kmultiplier’), is applied to the prior load term (P*P) to express itin terms prior thermal load. Figure 10 shows how the warm tripcurve is modeled.

Fig 9. Warm trip time curve model for motor thermal overload protection.

X. ADAPTING THE MODEL TO SPECIFIC RELAY SETTINGS

As discussed previously the principle operation for thermaloverload protections works off temperature and thermal capacityused. To model the curves however, the concept of a trip currentwas introduced. It was shown that the trip current , equation (26),can be defined in terms of generalized settings consisting of acurrent reference (expressed in per unit of nominal CT or ratedFLA), temperature reference (temperature rise corresponding toreference current) and trip temperature [7].

Also shown was how temperature rise is related to theoverload factor for motor applications. The key then to definingthe trip time curves for a given thermal overload function is toadapt the general form of the trip current equation to specificsettings.

By substituting relay specific settings into the general form ofthe trip current equation (26), the trip current can be evaluatedand entered in the software modeling page. Used conjunctivelywith equations (27) through (29), the trip time curves can beplotted from the models. The following applications show how

the trip time curves are defined in terms of relay specific settings:A. FeederThe trip current in terms of specific relay settings is defined

as:

=q

=− (30)

Substituting right side of (30) into (27) from previous section(shown below),

= ×−

−

The equation becomes:

= ×

⎣⎢⎢⎢⎡

−

− −⎦⎥⎥⎥⎤

(31)

IrefsetCurrentreference(nominal)τsetTimeconstantImeasuredcurrent(highestphase)Ippre-loadcurrentIrefsetCurrentreference(nominal)TmaxsetMaximumtemperature(levelfortripping)TenvsetEnvtemperatureset(ambient)TrisesetTemperatureraise(aboveambient)B. TransformerWeighting factor (p) is zero (only long time constant (τ2)

setting is used. As discussed in the previous section the value of(τ2) is set to the single reduced time constant.

The trip current in terms of specific relay settings is definedas:

= 100% ∗ −(32)

Substituting right side of (32) into (27) from previous section(shown below),

= ×−

−

The equation becomes:

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= ×

⎣⎢⎢⎢⎢⎢⎢⎢⎡

−

− 100% ∗ −

⎦⎥⎥⎥⎥⎥⎥⎥⎤

(33)

SetLongtimeconstantImeasuredcurrent(highestphase)Ipre-loadcurrentIresetCurrentreference(nominal)TripsetTriptemperature(%ofsetMaximumTemperature)TaxsetMaximumtemperatureTensetEnvtemperature(ambient)TriessetTemperaturerise(aboveambient)

C. MotorThe trip current in terms of specific relay settings is defined

as:= (34)

Substituting the right side of (34) into (29) from previoussection (shown below):

= ×− ×− 1

The equation becomes:

= ×− ×

− ( )(35)

IrefsetCurrentreference(nominal)τsetTimeconstantImeasuredcurrent(highestphase)Ippre-loadcurrentIrefsetCurrentreference(nominal)

XI. PROJECT EXAMPLE - SETTING AND COORDINATINGTHERMAL OVERLOAD CURVES

Like overcurrent protection, thermal overload trip time curvesneed to be coordinated to prevent mis-operation and equipmentdamage due to overload and faults conditions. Many of theguidelines and principles used for overcurrent coordination alsoapply to thermal overload curves. There are however, a few otherthings to consider. Although the primary purpose is to protect foroverload conditions where the load can fluctuate (utilizingthermal memory) it must also protect for faults where the currentis assumed constant. In addition to this prior loading must betaken into account (warm and cold curves). The higher the priorloading, the less margin there is with downstream protection. Formotors the maximum permissible load is the most severe case forchecking coordination. For transformers and feeders theforecasted load can be used. The cold curve is needed to checkcoordination with the damage (transformer) and thermal limit

(motor rotor and stator) curves particularly if typical values or asingle equivalent time constant (transformer) is used.

Thesinglelinediagraminfigure11isforaprojectusingmicro-processor based thermal overload protection forthreeinductionmotors(1000HP,1250HPand1500HP)anda distribution transformer (13.8-4.16 kV/7500KVA) usingtypical impedances.The transformer IED isconnected to a350/5CTonthe13.8kVside.ThemotorIEDs(MTR-A,MTR-BandMTR-C)areconnectedtotheirrespectivefeederCTs(150/5,200/5,and250/5).

Fig 11. Single line diagram for project example

A. Motor CoordinationFor thermal protection of motors, the cold trip time curve (mostsevere case) needs to operate between the warm and cold rotorand stator limits (values generated from the relay coordinationsoftware). The warm trip time curve needs to be checked withdownstream protection. The motor state of operation (starting,normal, and stop) also needs to be known. The otherrequirements for the coordination study follow the same criteriaused for overcurrent coordination. This includes a normalizedcurve plot that takes into account the fault contribution from eachmotor connected to the bus (if there’s more than one) for a faultat the terminals of the largest motor (most severe case sinceclosest to trip curve for main protection). The starting curve isalso required for the coordination study to ensure protection doesnot operate during starting. The coordination guidelines for amedium voltage motor (4.16 kV) are shown in Table 3 [8]. Theplot is shown in figure 12.

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TABLE 3SETTING AND COORDINATION GUIDELINES FOR A MEDIUM VOLTAGE MOTOR

(4.16 KV)

Device Function Recommendations CommentsCT Size 125-150%ofFLA

51 Pickup 115-120%ofFLA

Set belowmotorstatordamagecurve.Setator belowcableampacity

51 TimeDial 2-10secondsabovekneeofmotorcurve

Set belowmotor rotordamagecurve. Setbelow cabledamagecurve

50 Pickup 200%ofLRASet belowcabledamagecurve

Fig 12. Typical coordination plot for a medium voltage motor

Although the 50/51 relay offers some measure of overloadprotection, the intent is for protection against faults. To protectagainst thermal overload, the 49 function in each motor IED willbe used. The function will be set such that its trip time curve willbe between the cold and hot thermal limits of the rotor and stator[1].

For this example two coordination cases will be looked at.The first is with just motor A (MTR-A) during starting/normaloperations (the other motors taken out of service) with a threephase fault (3LG) on the 4.16 kV bus. The second is with all threemotors running in the normal state of operation with motor C(MTR-C) faulted at its terminals. For the warm curve it isassumed all motors have been running at maximum permissibleload (full load amperes with rating factor k applied) long enoughto reach their steady state thermal capacity. Negative sequencecurrents are neglected.

Starting with the first case (MTR-A only), figure 13 showshow the IED settings are entered in the data page of the relaycoordination software. The trip current is equal to the value ofthe overload factor (k = 1.2) and is expressed in per unit of thefull load amperes (FLA). The time constant (typical value) is thesame for starting and normal states of operation ( == 450 seconds). The weighting factor is set to a typical value of50% which determines the prior thermal load of the motor bodyat maximum permissible load. The prior load is expressed in perunit of the trip current. Note that measured current must exceedthe trip current (pickup) for trip timing to begin.

Fig 13. Motor thermal overload settings in relay coordination software datapage (MTR-A)

The curve coordination plot for the first case is shown infigure 14. Neither overcurrent nor thermal overload protectionwill prevent starting (above knee point of starting curve). Amotor starting voltage of 100% is used since this is the mostsevere case (closest margin between motor starting andovercurrent/overload curves). The thermal overload trip curvefor MTR-A falls between the cold and warm stator and rotorlimits and provides backup protection for the motor overcurrentprotection.

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Fig 14. Curve coordination plot showing motor (MTR-A) thermal overload protection example

For the second case (all motors running, MTR-C faulted at motorterminals), a normalized plot is required to account for the faultcontribution from the other motors. The contribution from theother motors results in a higher magnitude of fault current that isseen by the MTR-C IED than by the main relay. The highercurrent will cause the thermal overload protection to operatefaster than the plot indicates unless the curve is shifted to theright with respect to main trip curve. In the relay coordinationsoftware this is accomplished by aligning the curves with totalbus fault current. The fault current each relay sees is calculated

in per unit of the total bus fault (per unit value = 1) and shiftedto the left accordingly. The thermal overload relay which sees ahigher percentage of the total fault current has a higher per unitvalue and therefore is shifted to the left less than the main.Relatively this moves the thermal overload curve closer to themain which is the most severe case. Figure 15 shows thenormalized plot.

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Fig 15. Normalized Coordination plot for motor (MTR-C) example

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B. Transformer CoordinationThe main coordination objective for transformer thermaloverload protection is to ensure the trip time curve is belowdamage curve and above downstream overcurrent protection.The coordination guidelines for a medium voltage transformer(13.8 - 4.16 kV/7500 KVA) are shown in Table 4 [8]. The plot isshown in figure 16.

TABLE 3SETTING AND COORDINATION GUIDELINES FOR A MEDIUM VOLTAGE

Transformer (13.8-4.16 KV)

Device Function Recommendations Comments

CT Size 200% of FLA FLA on baserating

51 Pickup 110-140% of FLA

Set below thetransformerdamage curve.Set at or belowcable ampacity

51 TimeDial

Let-thru current @1.0 second

Set below thetransformerdamage curve.Set at or abovelow voltagemain device

50 Pickup 200% of let-thrucurrent or inrush

Set belowcable damagecurve.Set abovetransformerinrush point

Fig 16. Typical coordination plot for a medium voltage transformer (typical)

For the example, the relay settings from figure 8 will be used.It is assumed that the time constants are unknown so the typicalsingle equivalent time constant (for a distribution transformer)recommended by the relay manufacturer will have to be used asa starting point (3000 seconds) with the weighting factor (p) setequal to zero. It will also be assumed, for the warm curve, thatthe transformer has been operating at 80 percent of full load(prior load) for a long enough time to reach a steady statetemperature that is between ambient (40 ºC) and maximum (105ºC). To simplify the setting calculations the reference current isset to 1.0 per unit of the CT nominal value since it closelymatches the full load amperes for a 65 ºC temperature rise. Usingthe conservative maximum trip temperature of 105 ºCrecommended by the relay technical manual, the trip currentvalue in equation (32) becomes 1.0 in per unit of the nominal CTvalue.

=( 65)

=351.4350 ≈ 1.0

= 100% ∗ −

= 1.0100%100% ∗ 105− 40

65

= 1.0

Figure 17 shows how the settings are entered into the datapage. Note that measured current must exceed the trip current(pickup) for trip timing to begin. In other words the transformercan run at its 65 ºC full load amperes continuously withouttripping.

Fig 17. Transformer thermal overload settings in relay coordination softwaredata page (TR21)

14

To show the deviation (see simulation in lower left corner offigure 18) between the single constant curve (dotted lines) andthe two time constant curves (solid lines) a user defined curve

from figure 8 has been added to the plot (solid line). The curvecoordination plots with (warm curve) and without (cold curve)prior load is shown in figure 18.

Fig 18. Coordination plot showing transformer (TR21) thermal overload protection (typical )

From figure 18 it is observed that the warm trip time curve forthe single time constant transformer thermal overload curve(TR21-OL) modeled in the software overlaps the ‘infrequentfault’ or ‘mechanical’ portion of the damage curve. There are twoexplanations for this, One, that it could be due to the deviationfrom the two time constant model (solid line user curve shownbelow). However because the curve aligns with the ‘frequentfault’ portion of the damage curve (and not below) this could

suggest the single time constant for the typical recommendedvalue is too high, In either case the time constant in the relaycoordination software data page must be lowered slightly belowthe damage curve to achieve proper coordination for faults. Bylowering the single time constant just enough to account for thedeviation with the two time constant curve (600 seconds)coordination is achieved. It is important to note that the timeconstant value is a fixed value determined from equipment

15

properties (nameplate) and is not intended to be adjusted like atime dial in an overcurrent relay. Also note that even after thisadjustment the cold curve only coordinates with the mechanicalportion of the damage curve. However the risk for damage is lowdue to the fact that a double contingency failure of the high sideand main overcurrent backup protection (for frequent faults) isunlikely. The curve coordination plot is shown in figure 19. Due

to the uncertainty of what the time constants really are, there isno basis to pursue further adjustment other than to account forthe deviation between one and two time constant trip time curves.It is unknown whether an overload condition will be risked orwhether the thermal overload protection will be under-utilized.To safeguard against overload a conservative maximumtemperature (temperature rise plus ambient) setting was used.

Fig 19. Coordination plot showing transformer (TR21) thermal overload protection (adjusted )

Generally this size distribution transformer (7500 KVA) isoperated at or less than its full load rating. With only shortoverload durations. Therefore the conservative thermal overload

protection in this example is adequate. Once the full load ratingis exceeded the 600 second time constant is sufficient time for anoperator to respond to an alarm and reduce the load.

16

XII. LESSONS LEARNED AND FUTURE WORK

Modeling thermal overload protection functions in relaycoordination software is an extremely mathematically complexprocess requiring detailed derivations to arrive at the trip timeequations. Having the thermal overload functions alreadymodeled in the software and being able to readily adapt them tospecific relay settings saves considerable time and reduces thechance for error. There is some uncertainty however when datais unavailable (resulting in the use of typical values) or whenassumptions need to be made to simplify the model. Forexample, in motor applications negative sequence current, whichis prevalent in cross the line starting, is neglected in the softwaremodel. Also using a typical value of 50 percent for the motorweighting factor may not always be realistic. For transformerapplications single equivalent time constants are required for thepresent relay coordination software model which result indeviations from the two time constant trip time curves. Often thetime constants are unknown resulting in typical values to be usedwhich are not always suitable. Predicting the degree of overload(how much, how often and how long) for trip reference and priorload current can also be challenging to estimate. Could thesetting approach be too conservative? How much shouldequipment damage be risked to maintain continuity of service?The answer, having good data (which could even be included inthe software libraries if unknown) and building upon an alreadysolid foundation to ensure settings are correct and trip timecurves are coordinated properly.

XIII. REFERENCES

[1] B.Venkataraman, B. Godsey, W. Premerlani, E. Shulman, M. Thakur, R.Midence, Fundamentals of a Motor Thermal Model and its Applications inMotor Protection 58th Annual Conference for Protective Relay Engineers,2005.

[2] Compiled by Petri, EUROPLOT + Thermal differential equation theorydescription MICROENER -PRELIMINARY VERSION-PROTECTHUNGARY 30.10.2009 section 3.3.1, 3.3.3

[3] ABB 1MAC050144-MB 615 series Technical Manual Issued: 06/06/2015Revision E Product version 4.2.1

[4] ABB Online Support for Power and Automation Products Temperature vsper cent settings in thermal overload protection Published 2015-07-20

[5] Craig Wester GE Multilin Presentation[6] ABB1MRS757209 EN Technical Note: Thermal equations of MPTTR –

function August 2010[7] ABB 1MAC004548-AP Rev A October 2015_Modeling Relion® Thermal

Overload Curves in Engineering Software October 2015[8] Thomas P. Smith, P.E., The ABC’s of Overcurrent Coordination, January

2006

XIV. BIOGRAPHY

Karl M. Smith, P.E., received his B.S.E. from the ColoradoSchool of Mines in 1986 and his M.S.E.E. from the Universityof Colorado at Denver in 2000. From 1988 to 1994, he served asa Nuclear Electricians Mate in the United States Navy aboard theU.S.S. Nimitz. He has a diverse substation background whichincludes operations and maintenance, dispatch, testing, paneldesign, project engineering and applications. Karl has worked forseveral utilities and relay manufacturers throughout his careerbefore joining ABB as a Product Application Specialist in 2012.Karl has participated in many relay conferences both as a paperpresenter and through automated testing demonstrations. Karl isa Professional Engineer in the state of Colorado and California.

Sajal Jain, P.E., received his B.Tech. in Electrical Engineeringin 2008 from National Institute of Technology Kurukshetra,India and his MS in Electrical Engineering in 2012 fromUniversity of Southern California, Los Angeles. He started hisprofessional career with NTPC India as Executive Engineer from2008 to 2010, working on project design and consultancy. Since2012, Mr. Jain has been part of ETAP STAR team, working ondesign, development, testing & validation of protection andcoordination features. In addition, he has been instructing ETAPworkshops, managing and modeling various protective devicesfor ETAP libraries. He has been a registered professionalengineer in the state of California since 2014.”

17

Appendix 1

Derivation of thermal differential equation from first order thermal model

The total heat (QT) generated in a conductor for an RMS current (I) flowing through its resistance (R) is equal to the heat impartedto the resistance (QR) plus the heat dissipated to its surrounding (QS):

Q = Q + Q (1)

where,

Q = I (t)Rdt(2)

Q = cmdθ(3)

Q = h A θ dt (4)

c = specific heat capacity m = mass dθ = temperature rise (above ambient) in resistor h = heat transfer coefficient A = surface area of conductorθ = surface temperature (above ambient)

and,

= ℎ (heattimeconstant)

Substituting (2), (3), & (4) into (1) the heat balance equation becomes:

I (t)Rdt = cmdθ + h A θ dt(5)

Re-arranging equation (5):

=1 ( )

ℎ − (6)

18

Appendix 2

Solution to first order thermal differential equation

The first order thermal differential equation from Appendix 1 for constant current is:

=1

ℎ − (1)

The solution to the differential equation is the sum of the steady state and transient components:

= + (2)

To solve the steady state component the differential equation is re-arranged with the θ terms on the left side:

+1

=1∙ ℎ (3)

After enough time θ reaches its steady state value and no longer changes:

= 0(4)

Substituting (4) into (3) and solving for θ yields the steady state component:

= ℎ ( = )(5)

For the transient component the right side of equation (3) is set to zero:

+1

= 0(6)

This is the homogeneous part of the differential equation which can be solved by separation of variables:

= −1

(7)

Integrating both sides of the equation,

= −1

1= −

1

| | = − + (whereCistheintegralconstant)

= ± − +

19

= ± = ± = ± ∙ (8)

and substituting coefficient K for ± yields:

= (9)

To determine for K, substitute (9) into (2),

= + (10)

evaluate θ at time equal zero (initial temperature),

= + ( = )(11)and solve for K:

= − (12)

Substituting (12) into (9) the transient solution is:

= ( − ) ( = )(13)

Substituting (5) & (13) into (2) the solution to the differential equations becomes:

= ℎ + ( − ℎ ) (14)

Re-arranging to express in same form as numerical equation in technical manual:

q = ℎ 1 − + q

q = ℎ 1 − + q − q + q

q = ℎ 1 − + q + −q + q

q = ℎ 1− + q − q − q

20

q = ℎ 1 − + q − q (1− )

q = ℎ 1 − − q (1− ) + q

q = ℎ − q 1− + q (15)

After infinite time (t =¥) the final temperature becomes:

q = ℎ − q (1 − 0) + q

q = ℎ − q + q

Therefore in (15):

ℎ = q (16)

Similarly:

ℎ = q (17)where,

ℎ = (18)

I nominal current (reference current)q nominal temperature at reference current (above ambient)

21

Appendix 3

Calculation for time required to reach specified temperature ( = ) based on load and pre-load measured current

Solving (15) for time:

= − × 1 −q − q

ℎ − q

Expressing natural log argument in terms of common denominator:

= − × ℎ − q

ℎ − q

Applying rule for ‘negative’ natural log (reciprocate argument and remove ‘negative’ sign) yields:

= × ℎ − q

ℎ − q(19)

Substituting (17) into (19) yields:

= × ℎ − ℎ

ℎ − q

Which reduces to:

= ×

⎩⎪⎨

⎪⎧

−

− q

ℎ ⎭⎪⎬

⎪⎫

(20)

Substituting (18) into (20) and re-arranging the time solution is:

= ×

⎩⎪⎨

⎪⎧

−

− q ⎭⎪⎬

⎪⎫

(21)

22

Appendix 4

Solving relay trip time from equation 20 (pg. 5)

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