A Multidisciplinary Model to Couple Power System Dynamics and Building

Dynamics to Enable Building-to-Grid Integration

Yangyang Fu1, Sen Huang2, Yuan Liu2, T. E. McDermott2, Draguna Vrabie2, Wangda Zuo1

1University of Colorado, Boulder, USA2Pacific Northwest National Laboratory, Richland, USA

Abstract

Interactions between power systems and buildings areusually ignored or over-simplified by existing mod-eling and simulation tools. This limits how sys-tem modeling can support Building-to-Grid integra-tion activities. In this paper, we developed a multi-disciplinary model for motor-driven building devicesto consider the interactions. This multidisciplinarymodel considers both mechanical dynamics and elec-trical dynamics of the motor-driven building devices.It characterizes the motor behavior, in response todisturbances from both power systems and buildings.We validated this model by comparing its simulationresults, in terms of the response to a varying voltagesignal, to those from a commonly-used power mod-eling tool. To demonstrate the usage of the devel-oped model, we integrated the developed model intoa simulated building cooling system. We then stud-ied how this simulated system responses to changesin the supply voltage and the thermal load.

Introduction

Power systems and buildings interact in a dynamicand coupled manner. For example, the supply voltagewas found to dramatically affect the instantaneousenergy efficiency of the building systems. (Hood(2004); Bichik et al. (2015); Lee (2014)). The op-eration of building systems, on the other hand, influ-ences the transient performance of the power system,such as the voltage stability (Wu et al. (2006); Heet al. (2012); Li et al. (2017)). Therefore, it is nec-essary to consider those interactions when designingand operating power systems or buildings in order toavoid undesirable side effects. When it comes to theBuilding-to-Grid (B2G) activities, this necessity be-comes even more urgent since more interactions areexpected to be introduced by the B2G activities.

Modeling and simulation is an effective way to in-vestigate the interactions. However, when simulatingpower systems or buildings, the interdependent influ-ences tend to be ignored. One major reason is thatexisting models and simulation tools were developedfrom a single disciplinary perspective and thus havedifficulties in capturing the multidisciplinary interac-tions. For example, in the power system modeling,it is quite common that power factors of the building

system are assumed to be constant (Chassin et al.(2008a)). In the building system modeling, usuallythe influence from power systems is ignored by im-plicitly assuming the supply voltage to be constant(Crawley et al. (2001)).

There were efforts to consider the interactions be-tween buildings and power systems (Chassin et al.(2008b); Bokhari et al. (2014); Clarke (2015)). For ex-ample, a ZIP coefficient model (Bokhari et al. (2014))was proposed to approximate the influences of thevoltage on the building device using a polynomialfunction. However, they over-simplify those interac-tions, and thus may not be able to support lager-scaleapplications. The ZIP coefficient model, for instance,is a static model and ignores the transient dynamics.Thus, it may not be used for the dynamic analysispurposes.

Under specific circumstances, we remark that, ignor-ing or simplifying those interaction may be accept-able. For example, when resistive devices (such as theelectric heater) dominate the power usage in build-ings, we can assume the power factor is constantwithout sacrificing the accuracy too much (Gilbert(1965)). Furthermore, for the static or semi-dynamicanalysis on the power system, the ZIP coefficientmodel can still provide reasonably good approxima-tion on the response from the load side (Hatipogluet al. (2012)). Nevertheless, ignoring or simplifyingthose interactions would generate significant errorsin some applications, especially the B2G activities,such as optimizing the supply voltage to maximizepower saving in a building (Arriffin et al. (2017)). Inthose applications,obviously, ignoring the interactionor only considering the one-directional impacts frompower system to buildings may lead to unrealistic oreven incorrect conclusions.

To support broader applications, we discussed howto consider those interactions in the building sys-tem modeling (Fu et al. (2019)). We also performeda proof-of-concept by developing models for motor-driven building devices. Those models, however, weredesigned primarily to support qualitative analysis andbased on simplified mathematical descriptions of themotor operation. Therefore, they may not be able tobe used directly in a real application.

In this paper, we developed a new multidisciplinarymodel for motor-driven building devices to better rep-resent the interactions between power systems andbuildings. This multidisciplinary model considersboth mechanical dynamics and electrical dynamicsin the motor-driven building devices. It character-izes the motor behavior, in response to disturbancesboth from power systems and buildings. We vali-dated this model by comparing its simulation results,in terms of the response to a varying voltage signal,to those from a commonly-used power modeling tool.To demonstrate the usage of the developed model,we integrated the developed model into a simulatedbuilding cooling system. We then studied how thissimulated system responses to the changes in the sup-ply voltage and the thermal load.

The rest of the paper is organized as below: wefirst describe the studied motor-driven building de-vices, and then elaborate how their dynamic behav-iors are modeled. After that, we validate the modelby comparing its simulation results with that froma commonly-used power analysis tool. We then per-form a case study to demonstrate the usage of thedeveloped model. At last, we discuss the simulationresults and future works.

Motor-driven Building Devices

Typical motor-driven building devices include fan,pump, chiller, etc. They are the major consumersof the electricity in buildings (Webster et al. (2000)).As shown in Figure 1, a typical motor-driven buildingdevice consists of three major components:

• Variable frequency drive (VFD)The VFD is used to adjust the input frequencyand voltage for the motor. It is connected withthe components (such as transformers) in powersystems and transports the electricity power tothe motor. It is noted that VFDs are generallyoptional. In absence of VFDs, the input fre-quency and voltage of the motor depends on thepower system operation directly.

• MotorInduction motors are the commonly used type ofmotors in buildings. A typical induction motorconsists of several sub-components: coil, magnet,stator, and rotor. The coil and stator are con-nected with the electric circuit from the VFDand generates induced magnetic field. The gen-erated magnetic field from the coil and the statorinteract with the rotor, and produces an electro-magnetic torque around the rotor’s axis. Thetorque forces the rotor to rotate at a constant orvarying speed.

• Transitional deviceThe transitional device links the shaft in the ro-tor with that in a mechanical counterpart. Ittransfers the torque from the rotor to the me-

chanical device. A commonly used transitionaldevice is belt.

• Mechanical devices The mechanical deviceconverts the torque into mechanical works. Inbuildings, they usually interact with other sys-tems indirectly via different fluid loops. For ex-ample, for supply air fans, the mechanical devicecirculates the air between the air conditioningsystem and the room, so that the cooling andheating energy can be delivered from the for-mer to the latter. Usually feedback controls areutilized to guarantee that the mechanical sys-tem can deliver a desired amount of mechani-cal works. Those feedback controls monitor thecontrolled variable and send the frequency sig-nal to the VFD in order to modulate the inputfrequency of the motor.

The operation of the motor-driven building devicesare affected by both power systems and other sys-tems in buildings. In that sense, motor-driven build-ing devices act as an interface to connect power sys-tems and buildings. Specifically, the impact of powersystems to the buildings is reflected by the supplyvoltage and the frequency, which affects the amountof electromagnetic torque that motors can generate.The electromagnetic torque determines the amount ofmechanical works that motor-driven building devicescan provide. In addition, the impact of the buildingson the power systems is reflected in the current of thepower flow through the buildings,which influences theactual voltage of the power system. Therefore, mod-eling the motor-driven building devices is the key toconsider the interactions between power systems andthe buildings.

System models

In this section, we elaborate how the motor-drivenbuilding device are modeled at the component level.

• Variable frequency drive (VFD)The VFD model basically has three inputs: inputvoltage,Vin,i, input frequency,fin, and frequencysignal, fsig, and two outputs: output voltage,Vvfd,out,i, and output frequency, fvfd,out. In thisstudy, the VFD is assumed to be ideal, in otherwords,

fvfd,out = fsig (1)

Vvfd,out,i = Vin,i (2)

where the subscript i denotes the phases of thepower system.

• MotorWe considered a three-phase induction motorwith unbalanced supply. It has two major in-puts: the input frequency, fmotor,input, the inputvoltage for each phase, Vmotor,input,i, calculated

Figure 1: Motor-driven Building Devices

by:fmotor,input = fvfd,out (3)

Vmotor,input,i = Vvfd,out,i (4)

The outputs include the apparent power of themotor and the generated electromagnetic torque,τe, obtained by solving a series of differentialequations that represent the electrical dynam-ics and magnetic dynamics of motors (Stankovicet al. (2002)).

• Transitional deviceRegarding the transitional device, the inputs in-clude the τe and the load shaft power, Pshaft

while the outputs include the load speed, ωr andthe load torque, τL. One major parameter forthe transitional device is the load moment inertiaJL. The τL is calculated by solving the followingequations:

τL =Pshaft

ωr(5)

dωr

dt=τe − τLJL

(6)

• Mechanical deviceThe major input of the mechanical devices is themechanical work they provide, w. For the devicesuch as fans or pumps, the mechanical work iscalculated by:

w = ∆pQ (7)

where ∆p and Q are the head and the volumeof the fluid, respectively. The output of the me-chanical device is Pshaft, calculated by

Pshaft =w

ηshaft(8)

where ηshaft are the shaft efficiency. For fan orpumps, ηshaft can be expressed as a quadraticequation as

ηshaft = (b0 + b1(Q

r) + b2(

Q

r)2)r2 (9)

where the normalized speed r can be calculatedbased on the rotation speed ωr and the nominalrotation speed ωr,0, and is shown as

r =ωr

ωr,0(10)

It is noted that all the above formulas, including bothalgebraic equations and deferential equations, shallbe solved simultaneously since some of them are cou-pled. To simplify the implementation process, we useModelica (Fritzson and Engelson (1998)) as the mod-eling tool. Modelica is an equation-based modelinglanguage which allows describing the systems withimplicit equations. Therefore, differential equationscan be directly implemented in Modelica without anymodification. Figure 2 is the diagram of the generatedModelica model for one type of motor-driven device:fan/pump.

Validation

In this section, we validate the developed Modelicamodel against PSCAD (Woodford (2003)), a widelyused power modeling tool. In this validation, we con-sidered a scenario where there are sudden changes inthe supply voltage of a motor-driven device, due toa fault occurs in the power grid. We then modeledthe above scenario with both the developed Modelicamodel and the PSCAD. For the Modelica models, wesimulate it with a solver called DASSL, provided bya commercial software Dymola (Bruck et al. (2002)).The solver DASSL supports variable time steps, and

loaTorExp

w_ref exact=true

M

per

ws_pu

const

k=4*pi/pole

ws_gain

wr_gain

Tl_pu

port_a port_b

f_in

P

Q

VFD

Pump

Motor

Electrical Inputs

Transitional Device

Mechanical Pump

Frequency

Voltage in Φa

Voltage in Φb

Voltage in Φc

Figure 2: the diagram of the Modelica model

the minimum and the maximum time step in theModelica model during the simulation are 0.0154 µsand 0.282 s, respectively. For the PSCAD model, weselect a fixed time step as 50 µs and use the trape-zoidal algorithm for integration.

Figure 3 illustrates the simulation results. We can seethat, at the 5 s, the voltage ramps down to 0.3 p.u.,i.e., decreasing by 70%, in 0.02 s, then stays at 0.3p.u. for 0.3 s. At t=5.32s, the voltage ramps up to1 p.u. in 0.02 s, and finally keeps at 1 p.u. till the 8s. The two models can generate approximately closeresults in general.

The simulated rotor speed, real power, and reactivepower from the two models, in response to the volt-age change, are close in terms of general patterns.There are some relative large differences, however, inthe period from the 0 s to the 3 s, this is becausethe initial values for state variables are different forthe two models. Due to the limitations of the simu-lation environment, we are not able to force the twomodels to have the same initial values. Based on theabove results, we believe that the developed modelprovides reasonable representation on the system dy-namic behaviors, in response to the voltage change,for the studied motor-driven device.

Case Study

To demonstrate how the developed Modelica modelscan help to capture the interactions between powersystems and buildings, we conducted a case study. Inthis case study, we integrated the developed modelwith a simulated building cooling system. We theninvestigated how the model responses to the distur-bances in both power systems and buildings.

This section starts with a brief description on thestudied building cooling system; It then elaboratesthe simulation scenarios and simulation results.

Studied system

We considered a simplified building cooling systemin this study. As shown in Figure 4, this simplifiedsystem contains a water loop and a air loop. Thewater loop consists of an ideal cooling source, a pump,an air handler, and a two-way valve and the air loopcontains an air handler and a load. In the water loop,the ideal cooling source maintains the temperature ofthe leaving water to be 7 oC and the pump circulatescold water between the cooling source and the airhandler. In the air loop, the cold supply air from theair handler then removes the heat from the load witha constant air flow rate. In addition, there are someinteractions between the water loop and the air loop.The water flow rate is modulated by adjusting theopening position of a two-way valve to maintain thecold air leaving the air handler to be around 16 oC.In addition, the frequency of the pump is adjustedto maintain a constant pressure difference in the pipeacross the air handler when the water flow rate varies.

We then modeled the studied system with Mod-elica Buildings library When generating the sys-tem models, we considered two options as well.In the first option (named as “proposed”), thepump is modeled with the developed modelwhile in the second option (“conventional”), thepump is modeled with a module called “Build-ings.Fluid.Movers.SpeedControlled y” from ModelicaBuildings library (Wetter et al. (2014)). “Build-ings.Fluid.Movers.SpeedControlled y” doesn’t con-sider the interactions between the power system andpumps, and thus is used as a reference to better un-derstand the performance of the proposed model. Forboth options, the rest components of system besidesthe pump were modeled with components from theModelica Standard library (Fritzson and Engelson(1998)) and the Modelica Buildings library.

Testing scenarios

In this case study, we considered two scenarios. In thefirst scenario, we studied how disturbances in powersystems affect the studied system by introducing astep change in the supply voltage from the grid andassuming that the thermal load keeps constant. Inthe second scenario, we studied how disturbances inbuildings affect the studied system. Similar to thefirst scenario, we introduced a sudden increase of thethermal load and keep the voltage unchanged.

In both scenarios, the simulations are conducted from0 s to 1600 s, and the signal changes are implementedat the 800s. We use the above simulation time set-tings to make sure the system has sufficient time tobecome steady before and after the signal changes.This can allow us to study both the dynamic and thesteady state responses of the system to the changingsignals.

0.00.20.40.60.81.01.2

Voltage

[pu] Modelica

PSCAD

0.00.20.40.60.81.01.2

Rotor S eed

[ u]

−30−20−10

01020304050

Real P

ower

[MW]

0 1 2 3 4 5 6 7 8Time[s]

0

100

200

300

Reactive Po

wer

[MVAR]

Figure 3: the validation results of the Modelica model

Cooling�Source

Air�Handler

Load

PumpT

ѐp

Pump�control

Supply�temperature�

control

ૠ۱ܗ

۱ܗ

Valve

Figure 4: The studied cooling system

Results

Figure 5 illustrates the simulation result for scenario1. The step change in the voltage from 1.0 p.u. to 0.5p.u. is introduced at 800 s. In response to the sud-den change of the voltage, in the “proposed” option,the real power oscillates dramatically between -10 kWand 15kW in a very short time (less than 0.2 s). Thereal power becomes negative because this rapid volt-age change could force the motor to be in a generatormode and generates some real power during the con-tingency. After that, the real power become relativelysteady at around the 801 s. The steady state valuesof the real power changes slightly before and after thevoltage dip: from 4.15 kW to 4.29 kW . However, inthe “conventional” option, there is no change in thereal power as the voltage is not considered as a inputfor the pump operation. In addition, the “proposed”option shows that the reactive power of the pump ex-

periences an oscillation as well: from around -17.00kV ar to around 3.00 kV ar in 0.2 s. The steady valueof the reactive power also changes slightly, from 2.94kV ar to 2.83 kV ar.

As shown in Figure 6, the primary reason for the os-cillations is that the speed controller of the pump triesto maintain a constant pressure difference. Whenthe voltage decreases, the electromagnetic torque pro-duced by the motor drops. As a result, the motorspeed, driven by the electromagnetic torque, also de-creases, leading to a lower head pressure. To maintaina constant pressure difference in the pipe across theair handle, the speed controller of the pump gener-ates a higher frequency signal to increase the motorspeed. The increased motor speed then generates ahigher head pressure and forces the speed controllerto reduce the frequency signal. The above processrepeats until the frequency signal becomes steady.

The oscillations influence significantly the power con-sumption by the pump . Before the dip, the pumpconsumes 4.15 kW . During the rebalancing process,the pump can consume as much as 13.5 kW , whichincrease the demand by about 2.25 p.u.. However,those oscillations don’t dramatically affect the build-ing operation performance. For example, as shownin, the change of the static pressure is less than 0.03kPa. This is because the mechanical inertial preventsthe load torque changes as the same pattern as theelectromagnetic torque.

Figure 7 illustrates the simulation results for scenario2. At t=800 s, its thermal load increases by 0.5 p.u..As a consequence, more chilled water need to passthrough the air handler, which then decreases thepressure difference. The frequency of the pump isthen increased by the pump controller to maintain aconstant pressure difference.

799.8 800.0 800.2 800.4 800.6 800.8 801.00.00.20.40.60.81.01.2

Voltage

[pu]

799.8 800.0 800.2 800.4 800.6 800.8 801.0−10−505

1015

Rea

l Pow

er[k

W]

ConventionalPropo ed

799.8 800.0 800.2 800.4 800.6 800.8 801.0Time [ ]

−20−15−10−505

Rea

ctive Po

wer

[kVar]

Figure 5: Simulation results of the scenario 1

799.8 800.0 800.2 800.4 800.6 800.8 801.054.054.555.055.556.056.557.0

Frequency

[Hz]

799.8 800.0 800.2 800.4 800.6 800.8 801.0−3−2−10123

Electro

magnetic

Torque

[pu]

799.8 800.0 800.2 800.4 800.6 800.8 801.00.0050.0100.0150.0200.0250.0300.0350.0400.045

Head

[kPa

]

+7.035e1

799.8 800.0 800.2 800.4 800.6 800.8 801.0Time [s]

0.000100.000150.000200.000250.000300.000350.000400.00045

Load Torqu

e[ u]

+6.21e−1

Figure 6: Detailed simulation results of the scenario 1

799.8 800.0 800.2 800.4 800.6 800.8 801.00.40.60.81.01.21.41.61.82.0

Coo

ling

Load

[pu]

799.8 800.0 800.2 800.4 800.6 800.8 801.054.054.555.055.556.056.557.057.558.0

Freq

uncy

[H

z]

799.8 800.0 800.2 800.4 800.6 800.8 801.03456789

1011

Rea

l Pow

er[k

W]

ProposedConventional

799.8 800.0 800.2 800.4 800.6 800.8 801.0Times [s]

2.02.22.42.62.83.03.2

Rea

ctiv

e Po

wer

[kV

ar]

Figure 7: Simulation results of the scenario 2

In the “proposed” option, similar to scenario 2, thepump control contributes mostly to the significantlyoscillations in the real power. We can see that thereal power changes from around 4 kW to around 10kW in just 0.1 s. In addition, there is a significantoscillation in the reactive power as well.

However, in the “conventional” option,we observe no oscillations since “Build-ings.Fluid.Movers.SpeedControlled y” doesn’tconsider the mechanical inertia in the motor andthe transitional device. Therefore, the change of thefrequency signal is much smoother than that in the“proposed” option.

Conclusion

In this paper, we developed a Modelica models toallow considering the interactions between buildingsand the power system. We validated the developedmodel by comparing its simulation outputs to thosefrom a power modeling tool. The validation resultssuggested that the developed models can predict rea-sonable responses of the motor-driven devices to thestudied changing voltage. To demonstrate the usageof the developed model, we conducted a case studywith a simplified building cooling system. The casestudy results suggest that the proposed model pro-vides more realistic representation on how the studiedsystem responses to the disturbances from the supplyvoltage and the thermal load, compared with the ex-

isting models.

In the future study, we will perform a more realis-tic case study by including power systems into thesimulation scope rather than treating it as a bound-ary condition. By doing that, we can better under-stand how the buildings and power grids interact in acoupled manner. In addition, we will also apply theproposed model in the B2G activities to support thedesign or the evaluation of the relevant controls.

Acknowledgement

This work has been supported by the the BuildingTechnologies Office of the U.S. Department of En-ergy’s Office of Energy Efficiency and Renewable En-ergy.

This research is supported by Colorado Energy Re-search Collaboratory (Award No. 22-2018). Thiswork emerged from the IBPSA Project 1, an inter-national project conducted under the umbrella of theInternational Building Performance Simulation Asso-ciation (IBPSA). Project 1 will develop and demon-strate a BIM/GIS and Modelica Framework for build-ing and community energy system design and opera-tion.

ReferencesArriffin, A. M., M. M. Othman, A. A. M. Ka-

maruzaman, I. Musirin, A. Yahya, and M. F. A.Latip (2017). Stochastic approach of voltage opti-

mization to maximize power saving in a building.Indonesian Journal of Electrical Engineering andComputer Science 8 (1), 268–272.

Bichik, A. et al. (2015). Impact of voltage variationon domestic and commercial loads.

Bokhari, A., A. Alkan, R. Dogan, M. Diaz-Aguil,F. de Len, D. Czarkowski, Z. Zabar, L. Birenbaum,A. Noel, and R. E. Uosef (2014, June). Experimen-tal determination of the zip coefficients for modernresidential, commercial, and industrial loads. IEEETransactions on Power Delivery 29 (3), 1372–1381.

Bruck, D., H. Elmqvist, S. E. Mattsson, and H. Ols-son (2002). Dymola for multi-engineering modelingand simulation. In Proceedings of modelica, Volume2002. Citeseer.

Chassin, D. P., K. Schneider, and C. Gerkensmeyer(2008a). Gridlab-d: An open-source power systemsmodeling and simulation environment. In Trans-mission and distribution conference and exposition,2008. t&d. IEEE/PES, pp. 1–5. IEEE.

Chassin, D. P., K. Schneider, and C. Gerkensmeyer(2008b). Gridlab-d: An open-source power systemsmodeling and simulation environment. In Trans-mission and distribution conference and exposition,2008. t&d. IEEE/PES, pp. 1–5. IEEE.

Clarke, J. (2015). A vision for building performancesimulation: a position paper prepared on behalfof the IBPSA Board. Journal of Building Perfor-mance Simulation 8 (2), 39–43.

Crawley, D. B., L. K. Lawrie, F. C. Winkelmann,W. F. Buhl, Y. J. Huang, C. O. Pedersen, R. K.Strand, R. J. Liesen, D. E. Fisher, M. J. Witte,et al. (2001). Energyplus: creating a new-generation building energy simulation program.Energy and buildings 33 (4), 319–331.

Fritzson, P. and V. Engelson (1998). Modelica – aunified object-oriented language for system mod-eling and simulation. In European Conference onObject-Oriented Programming, pp. 67–90. Springer.

Fu, Y., S. Huang, D. Vrabie, and W. Zuo (2019).Coupling power system dynamics and building dy-namics to enable building-to-grid integration. InProceedings of 13th International Modelica Con-ference, OTH Regensburg, Germany, March 4–6,2019. Linkoping University Electronic Press.

Gilbert, E. M. (1965, July 27). Regulated power sup-ply. US Patent 3,197,691.

Hatipoglu, K., I. Fidan, and G. Radman (2012, Sept).Investigating effect of voltage changes on static zipload model in a microgrid environment. In 2012North American Power Symposium (NAPS), pp.1–5.

He, X., R. Zhao, C. Zhu, and H. Yang (2012, Oct).Improving short-term voltage stability problems byvariable-speed air-conditioners. In 2012 15th In-ternational Conference on Electrical Machines andSystems (ICEMS), pp. 1–6.

Hood, G. (2004). The effects of voltage variation onthe power consumption and running cost of domes-tic appliances. In Australasian Universities PowerEngineering Conference (AUPEC).

Lee, K. H. (2014). Optimization of a hybrid electricpower system design for large commercial buildings:an application design guide. Ph. D. thesis, ColoradoSchool of Mines. Arthur Lakes Library.

Li, D., G. Cui, L. Sun, J. Yang, C. Gao, and X. Chen(2017). A control strategy for static voltage sta-bility based on air conditioner load regulation. InSystems and Informatics (ICSAI), 2017 4th Inter-national Conference on, pp. 288–293. IEEE.

Stankovic, A. M., S. R. Sanders, and T. Aydin (2002).Dynamic phasors in modeling and analysis of un-balanced polyphase ac machines. IEEE Transac-tions on Energy Conversion 17 (1), 107–113.

Webster, T., F. Bauman, E. Ring, et al. (2000). Sup-ply fan energy use in pressurized underfloor air dis-tribution systems.

Wetter, M., W. Zuo, T. S. Nouidui, and X. Pang(2014). Modelica buildings library. Journal ofBuilding Performance Simulation 7 (4), 253–270.

Woodford, D. (2003). Pscad users guide. ManitobaHVDC Research Centre.

Wu, B., Y. Zhang, and M. Chen (2006). The effectsof air conditioner load on voltage stability of ur-ban power system. In 6 th WSEAS InternationalConference on Power Systems(PE’06), Volume 6.Citeseer.