IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 3, AUGUST 2004 1685

Monte Carlo Simulation of Residential ElectricityDemand for Forecasting Maximum Demand

on Distribution NetworksDougal H. O. McQueen, Patrick R. Hyland, and Simon J. Watson

Abstract—The prevalent engineering practice (PEP) for max-imum demand estimation in low-voltage (LV) electricity networksis based on an After Diversity Maximum Demand (ADMD) mod-ified by a diversity factor. This method predicts the maximumlikely voltage drop accounting for consumer diversity. However,this approach does not take into account the stochastic na-ture of the demand and is inconsistent with international powerquality standards. We present a Monte Carlo simulation modelof consumer demand taking into account the statistical spreadof demand in each half hour using data sampled from a gammadistribution. The parameters of the gamma distribution are basedon data metered at a number of residential properties fed by onetransformer. The simulated demand is corrected for temperatureand total consumption. The simulated profiles at the residentialproperties are aggregated and the simulated maximum demand iscompared with actual maximum demand at a given transformerand an entire distribution network showing good agreement inboth cases.

Index Terms—Load modeling, Monte Carlo methods, power dis-tribution, simulation.

I. INTRODUCTION

THE common practice within the electricity supply industryfor designing and analyzing low-voltage (LV) distribution

networks still follows the empirical method of Boggis [1]. Thisis based on estimating the maximum demand that is likely tobe incurred by a group of customers within a given time pe-riod, e.g., a year. The methodology uses linear models based onempirical observation. The greatest likely voltage drop is calcu-lated by applying the expected simultaneous maximum demandat each node to an impedance model of the LV network beingconsidered.

When analysis of electricity demand is made from a risk-based perspective where statistical coherence needs to be con-sidered, this method is found wanting. Furthermore, newer in-ternational design standards have been written that require agreater understanding of the nature of the demand and an es-timation of the time duration curve for received voltages.

Manuscript received September 24, 2003. This work was supported in partby Delta Utility Services, Ltd.; Industrial Research, Ltd.; and Orion Networks,Ltd., of New Zealand.

D. H. O. McQueen and S. J. Watson are with the Centre for RenewableEnergy Systems Technology, Department of Electronic and Electrical Engi-neering, Loughborough University, Leicestershire, LE11 3TU, U.K. (e-mail:d.mcqueen@lboro.ac.uk; s.j.watson@lboro.ac.uk).

P. R. Hyland is with Austral Engineering Associates, Ltd., Dunedin, NewZealand (e-mail: p.hyland@aeal.co.nz).

Digital Object Identifier 10.1109/TPWRS.2004.826800

The demand profile for a single house is made up by operationof appliances that are essentially a series of stochastic events.Appliance actuations are largely independent, as are the loadingpatterns placed by neighboring houses. There are some overalldriving factors that will increase the tendency for appliances tobe used coincidently, such as the time of day and the ambienttemperature. However, the result is that residential electricitydemand is largely a stochastic process and therefore exhibitsdiversity.

Infrastructural networks benefit from diversity since the max-imum demand placed by a group of customers is less than thesum of the maximum demands of the individuals; i.e., the in-dividual peak demands are very unlikely to be co-incident. Byunderstanding diversity behavior, network designers are able toreduce capital investment in their networks.

In this paper, we discuss a method of representing residentialdemand on LV networks taking account of diversity and repro-ducing the stochastic nature of the individual profiles, by meansof a Monte Carlo simulation. The method is validated using twoexamples.

II. LOW-VOLTAGE NETWORKS

A. Measuring Maximum Demand

The traditional method used to measure the loading on distri-bution transformers relies on the Maximum Demand Indicator(MDI). The MDI records the maximum demand on each of threephases independently, and the sum of these gives the “maximumloading.” The recorders are read and reset once a year. This re-sults in over-estimation of the true (3-phase) maximum demandas there is diversity among consumers, meaning that it is un-likely phase maxima will be coincident.

Use of the MDI for monitoring a transformer’s maximum de-mand is flawed for two reasons. Firstly, the individual phasemaxima by definition are at the extreme end of the demand dis-tribution. If a typical 20-min integration period is used for sam-pling, this corresponds to a confidence level of 99.996%—wellabove the confidence level that represents “value for money”risk management. Secondly, the MDI as measured representsan event that does not occur if phase currents are not coincident,and thus overestimates the true three-phase maximum demand.

B. Estimating Maximum Demand

When a new transformer is to be placed on a network wherethe loading is unknown, a standardized formulation is used [1].

0885-8950/04$20.00 © 2004 IEEE

1686 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 3, AUGUST 2004

This formulation, which is the basis of most current practice,is based upon the concept of an After Diversity Maximum De-mand (ADMD). The ADMD is the maximum demand, per cus-tomer, as the number of customers connected to the networkapproaches infinity. This is usually derived from the maximumyearly nodal demand on the transmission network divided bythe number of customers served by that network. The ADMD ismultiplied by a diversity factor (DF) that increases the demandper customer as the size of the group decreases. The formulaeused can take many forms but the maximum demand (MD) gen-erally resembles (1), the ADMD is defined by (2) and the diver-sity factor (DF) is given by (3)

(1)

(2)

(3)

wheremaximum demand of the customer group;diversity factor also defined as the ratio ofnoncoincident maximum demand of the cus-tomer group to the coincident group maximumdemand;After Diversity Maximum Demand per cus-tomer;demand of customer at the time of system max-imum demand;number of customers;empirically determined constant.

This method has a number of shortcomings:

• It only provides an estimate of voltage quality at one pointin time. The time duration of the received voltage is notconsidered using this method.

• It calculates the likely maximum demand for a group ofcustomers within a specific period of time, usually oneyear. Coupled with this, demand monitoring typically usesa 30-min sampling rate. This translates to an overly cau-tious statistical confidence level of 99.994%

• The demands are calculated as aggregated sums. There isno means to consider the effects of phase loading unbal-ance and neutral impedance.

A study of the New Zealand electricity industry revealed thatthe majority of distribution companies use ADMD and diver-sity formulae for low voltage network design albeit with widelydifferent assumptions of appropriate ADMD and co-incidencefactor [2].

III. DEMAND MODELING

A. Overview

Solutions of voltage drop problems using statistical modelinghave been formulated using analytical methods employing betafunctions, e.g., [3]–[6]. However, such models become awkwardwhen considering branched networks with a large number ofcustomers. Another common method for demand estimation is

“bottom-up modeling,” whereby the demand for every appli-ance is simulated [7]. Models of these types require knowledgeof appliance demand patterns and are very useful in trying toassess demand side management schemes. However, they aredata intensive and impractical when trying to replicate general-ized household load profiles.

In the past, the use of Monte Carlo simulations in load mod-eling has been restricted to validation of analytical probabilisticmethods, e.g., [5], [8]. This was due to the requirement for alarge number of repeated simulations and the requirement for afast computer. The recent increase in computer power coupledwith decreased cost, have meant that Monte Carlo simulationsfor load flow analysis are now viable and some work has beendone to simulate individual appliance loads [9] and unbalancedthree phase loads [10]. The methodology used here assumes thatelectricity demand is a stochastic process and constructs rep-resentative load profiles for each consumer. The load profilesare then injected at each node of a model representative of thenetwork impedance and the voltage drop solved using a MonteCarlo simulation. This allows the voltage and conductor currentload profiles to be determined and hence the distribution of volt-ages received at each node, yielding a statistically informativedescription of received voltage quality.

The assumption is made that all houses have the same prob-ability of producing any individual loading profile, consistentwith an expected distribution of energy use. This is importantsince it is not known what the energy demand of customers willbe at any particular time in the design stage or even within anexisting network. The model adopted is useful because it sepa-rately specifies energy use and diversity behavior enabling theimpact of changes in energy use and/or patterns to be forecast.

B. Data Used to Build the Model

A scoping study for the methodology based upon a dataset from customers in the Auckland region of New Zealandhighlighted several areas for further investigation such as deter-mining the appropriate sampling interval to use [2]. During thisstudy, the individual electricity consumptions of 100 housessampled at 30-min intervals was recorded over a period of amonth. A further metering program was instigated and resultsfrom this study are used to develop the model used here. Thisprogram measured electricity demand for 21 houses at a 1-minsampling interval for a period of approximately two weeksper house [11]. The sample of houses was carefully chosen toprovide a representative cross-section of domestic electricityconsumption.

C. A Mathematical Model for Residential Electricity Demand

To support the Monte Carlo methodology, a mathematicalmodel must be built for residential electricity demand. Theobjective of developing a mathematical model is so that rep-resentative load profiles may be formed for “any” customer.Load profiling is an important area for such as energy tradingfor supply companies and asset management for distributionand transmission network operators. A number of authors havesought to develop methods to construct representative loadprofiles for domestic, commercial and industrial customers in-cluding statistical modeling of the mean and standard deviationof power curves [13], [14], aggregation of individual loads [15]

MCQUEEN et al.: MONTE CARLO SIMULATION OF RESIDENTIAL ELECTRICITY DEMAND 1687

and fuzzy load models [16]. In this paper, we parameterizeresidential load profiles in terms of overall daily demand andhalf-hourly profiles, using the statistical variations to definegamma distributions.

The mathematical modeling process can be segregated intotwo procedures:

1) The measured data is parameterized to isolate those dy-namics that are responsible for diversity and which areassumed universal, from those that are specific to the pop-ulation being modeled.

2) Simulated load profiles are generated using the informa-tion obtained during the parameterization. This involvesrandom number generation to build the load profiles.

Several different types of models have been investigated in-cluding the use of Markov chains, Fourier series and PoissonRectangular Pulses [11]. However, the model used for this anal-ysis is based on a gamma distribution for the demand withineach time interval. This model has been found to yield good re-sults for the type of problem being addressed here and is com-putationally the most simple. The load model is formed in thefollowing manner.

Parameterization of Measured Data: Firstly, the variabilityof the total daily energy usage is parameterized. The 3 p.m. tem-perature dependence of the daily energy use for eachhouse on a given day is determined. The temperature experi-enced by all houses in the group is assumed to be the same on agiven day. The daily energy use values are determined fromthe metered profile measurements sampled at the requirednumber of intervals throughout the day (between 10 min and30 min)

(4)

where is the total number of time intervals during the day,e.g., there are 48 intervals where a 30-min sampling interval isused.

It is assumed that all houses exhibit the same temperature de-pendence so all data may be normalized to a standard temper-ature. Other authors have fitted polynomial functions to energyuse as a function of temperature [16], however, we find that alinear fit is justified for the sample dataset. A temperature cor-rected daily energy use is thus determined for the referencetemperature

(5)

where is the linear function of daily energy use against tem-perature determined above.

The set of values is fitted using a gamma distributionthat is determined as a function of the mean and

standard deviation of the values. We find there to be astrong linear relationship between and [11] of the form

(6)

where and .The next step is to parameterize the variability at each time

step of the load profile. In order to achieve this, a second set

of gamma distributions determined as a function ofthe normalized means and normalized standard deviations

for each time interval are fitted to the normalized profilemeasurements

(7)

Analysis of the data described in Section III-B shows thereto be a strong correlation between the mean normalized (and di-mensionless) interval load and the corresponding normalizedstandard deviation [11]. The best fit to the data was foundusing a square root relationship

(8)

where and for half-hourly data. Thevalues of and depend on the length of the sampling interval

.Generation of the Simulated Load Profiles: A random daily

energy use for each house is generated by sampling the gammadistribution . The random value is then corrected forthe actual 3 pm temperature on the day being modeled to givethe simulated daily energy use

(9)

For each interval of the load profile, the simulated demandis calculated by randomly sampling the second gamma dis-

tribution and scaling by

(10)

Values of above the fuse limit are rejected.The simulated demands are repeated for all houses and all

days to be modeled. These trials are in turn repeated until theresults have stabilized (typically after 100 repeats).

The generated profiles are then used in conjunction with anappropriate network model described below to determine max-imum demands.

D. Network Model

The network model used in this work considers the pointloads as constant current sources. Newton-Raphson iterationmethods are not employed due to the repeated computationaloverhead with the Monte Carlo simulation technique. The errorfrom this assumption has not been quantified, although the re-sults from the case studies would suggest it has not been signif-icant in these instances.

Conductor impedances have been included as 3 3 matricesthat allow for both phase and neutral conductor resistances andself and mutual inductances. Changes in conductor resistancewith temperature are included whereby the conductor tempera-tures are estimated from the conductor loads, the ambient tem-perature and the rated current carrying capacity at a specific linedesign temperature.

All current is assumed to return via the neutral wire, i.e.,neutral earthing at the customers’ premises and neutral inter-connection between distribution transformers is ignored. Theresidential load power factor is assumed to be 0.98 based onresidential demand analysis [11].

1688 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 3, AUGUST 2004

IV. SIMULATION OF MAXIMUM DEMAND

In order to test the validity and flexibility of the Monte Carlosimulated profiles, two case studies have been undertaken tocompare measured and simulated maximum demand.

A. Arnold St. Transformer

This transformer serves a predominantly residential area of63 houses in Dunedin, New Zealand. Historical MDI recordsfor total transformer demand and individual phases have beenrecorded for the period 1989 to 2001. In addition, a programof load monitoring was carried out for a period of a monthwhere demand was sampled every 10 min and the current ineach phase measured independently. The total load on the trans-former and each of the phases was simulated using the MonteCarlo methodology described in Section III-C and results com-pared with the monitored data. Fig. 1 shows a comparison be-tween the actual and simulated cumulative probability densityfunction (c.p.d.f.) of the individual phase and total transformerdemand. The agreement between the measured and simulatedprofiles is reasonably good. The discrepancies are mainly dueto dynamic factors which are not modeled such as changes inoccupancy over a short period, e.g., holidays, etc., and the factthat street lighting is not included in the model.

The Monte Carlo simulation period was extended to a yearusing representative temperature values for 2000. The simulatedMDI and individual phase maxima were estimated and com-pared with the historic data for 2000. These results are shownin Table I. The agreement between the measured and simulateddata is good with just a 2% error on the MDI. If the ADMDmethodology described in Section II-B is applied then the pre-dicted total maximum demand is 256 kVA which is significantlygreater than the actual phase total recorded in 2000.

B. Dunedin City Distribution Network

The Monte Carlo simulation method is now applied to modelthe loading of transformers in an entire distribution network.Each transformer is linked to the connected customer identifi-cation numbers plus an Australia New Zealand Standard Indus-trial Classification (ANZSIC) code and energy usage statistics.Only transformers with greater than 95% residential customershave been analyzed for this study resulting in a sample of 557transformers. For each transformer, the loading is simulated fora period of ten years. MDI, (true) maximum demands and 99.9thpercentile demands are calculated.

The Dunedin City network used in this analysis has had astatic population over this time period and changes in energyusage patterns are assumed negligible. The scatter plot inFig. 2 shows that the predicted MDIs match the historic valuesvery well. The spread of values can be attributed to: errorsin the energy values from billing data, errors from readingthe MDIs, database entry errors, unbalanced distribution ofcustomers among phases (the simulation assumes balance), andcalibration of the MDIs.

The performance of the Monte Carlo methodology is assessedusing probability density functions (p.d.f.s) formed from theratio of MDI to transformer capacity (Fig. 3). From the shapeof the distribution it can be seen that the transformers in the net-work have been well managed according to a plan under whichtransformers whose MDI exceeds the transformer capacity are

Fig. 1. Cumulative probability density functions for the measured demandon the Arnold St. distribution transformer (solid line) along side the matchingfunctions formed from the Monte Carlo simulated demands (dotted lines).

TABLE IA COMPARISON BETWEEN THE MEASURED AND MONTE CARLO SIMULATED

MAXIMUM DEMAND ON THE ARNOLD ST TRANSFORMER FOR 2000

Fig. 2. Scatter plot of the mean predicted MDIs against the mean historic MDIsfor the transformers in the Dunedin City network over a period of 10 years.

replaced and there are few transformers with very poor demandto capacity ratios. This has resulted in the transformer popula-tion on average utilizing 80% of capacity as measured by MDI.The fit between the measured and predicted distributions showsthe Monte Carlo simulated values match historic values veryclosely. However, the Monte Carlo simulation also allows thethree phase demand at the 99.9th percentile to be calculated.This reveals that the transformer population would, on average,be utilizing only 50% of its capacity.

MCQUEEN et al.: MONTE CARLO SIMULATION OF RESIDENTIAL ELECTRICITY DEMAND 1689

Fig. 3. The p.d.f.s for the ratio of mean historic MDI to transfromer capacity(solid line); the ratio of mean prodicted MDI to capacity (dashed line); andpredicted 99.9th percentile demand to transformer capacity (dotted line).

V. CONCLUSION

A stochastic mathematical model of residential electricity de-mand has been described. The model has been deployed in twocase studies where Monte Carlo load simulation is used withinan LV network to forecast maximum demand showing goodagreement with measured values. The method has the advan-tage that it predicts loading profiles that enable risk analyzes tobe performed such that transformer sizing may be optimized.

Following the case studies, the following points are noted:

• significant effect of phase unbalance on the deliveredvoltages;

• importance of accounting for the self and mutual induc-tances in the network cables;

• atypical situations such as holiday periods must be con-sidered explicitly.

ACKNOWLEDGMENT

D. H. O. McQueen thanks Prof. P. Bodger of the Electricaland Electronic Engineering Department at the University ofCanterbury, New Zealand, and M. McQueen of the Schoolof Information Technology and Electrotechnology, OtagoPolytechnic, Dunedin, New Zealand, for their invaluable helpand advice relating to the research work detailed in this paper.

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Dougal H. O. McQueen received the B.Sc. degree inmathematics in 1996, the Graduate Diploma in 1997,the Post Graduate Diploma in science in 2000, (bothin the field of energy management), and the Mastersdegree in applied science in 2002, all from the Uni-versity of Otago, Dunedin, New Zealand.

He was with Austral Engineering Associates, Ltd.,Dunedin, New Zealand, from 2000 to 2002. In 2002,he was appointed as a Research Associate in theElectronic and Electrical Engineering Department atLoughborough University, Leicestershire, U.K.

Patrick R. Hyland received the masters degree inelectrical engineering from the University of Canter-bury, Christchurch, New Zealand.

He is a Consultant Engineer and a Director ofAustral Engineering Associates, Ltd., Dunedin, NewZealand, and has 23 years’ experience in the elec-tricity industry covering generation, transmission,and distribution in both New Zealand and Australia.

Mr. Hyland is a member of the Electricity Engi-neers Association of New Zealand.

Simon J. Watson received the B.Sc. degree inphysics from Imperial College, London, U.K.,in 1987 and the Ph.D. degree from EdinburghUniversity, Edinburgh, U.K., in 1990.

He worked in the field of renewable energyresearch in conjunction with power systems at theRutherford Appleton Laboratory, Oxfordshire, U.K.,until 1999. He then worked at the green electricitysupply company unit which provides electricitysourced from renewable energy generation todomestic and small commercial customers. In 2001,

he was appointed a Senior Lecturer in the Electronic and Electrical EngineeringDepartment, Loughborough University, Leicestershire, U.K.