Section 6 - Chapter 5 - Chapter 19 - Harmonic Load Identification Using Independent Component Analysis5.1 - Introduction5.2 - Independent Component Analysis - ICA Model5.3 - ICA by Maximization of Nongaussianity5.4 - Harmonic Load Profile Estimation5.5 - Statistical Properties of Loads5.6 - Harmonic Load Estimation5.7 - Case studiesCase 1Case 25.8 - ConclusionAcknowledgment5.9 ReferencesSection 6 - Chapter 5 - Chapter 19 - Harmonic Load Identification Using Independent Component AnalysisEkrem Gursoy, Dagmar Niebur5.1 - IntroductionDue to an increase of power electronic equipment and other harmonic sources, theidentification and estimation of harmonic loads is of concern in electric power transmission anddistribution systems. Conventional harmonic state estimation requires a redundant number ofexpensive harmonic measurements. In this chapter we explore the use of a statistical signalprocessing technique, known as Independent Component Analysis for harmonic sourceidentification and estimation. If the harmonic currents are statistically independent, ICA is ableto estimate the currents using a limited number of harmonic voltage measurements andwithout any knowledge of the system admittances or topology. Results are presented for themodified IEEE 30 bus system.Identification and measurement of harmonic sources has become an important issue inelectric power systems, since increased use of power electronic devices and equipmentssensitive to harmonics, has increased the number of adverse harmonic related events.Harmonic distortion causes financial expenses for customers and electric powercompanies. Companies are required to take necessary action to keep the harmonic distortion atlevels defined by standards, i.e. IEEE Standard 519-1992. Marginal pricing of harmonic injectionsis addressed in to determine the costs of mitigating harmonic distortion. Harmonic levels in thepower system need to be known to solve these issues. However, in a deregulated network, itmay be difficult to obtain sufficient measurements at substations owned by other companies.Harmonic measurements are more sophisticated and costly than ordinarymeasurements because they require synchronization for phase measurements, which isachieved by Global Positioning Systems (GPS). It is not easy and economical to obtain a largenumber of harmonic measurements because of instrumentation installation maintenance andrelated measurement acquisition issues.Harmonic state estimation (HSE) techniques have been developed to asses the harmoniclevels and to identify the harmonic sources in electric power systems -. Using synchronized,partial, asymmetric harmonic measurements, harmonic levels can be estimated by system-wideHSE techniques . An algorithm to estimate the harmonic state of the network partially isdeveloped using limited number of measurements . The number and the location of harmonicmeasurements for HSE are determined from observability analysis . Either for a fully observableor a partially observable network, the number of required harmonic measurements is muchlarger than the number of sources.HSE techniques require detailed and accurate knowledge of network parameters andtopology. Approximation of the system model and poor knowledge of network parameters maylead to large errors in the results. Measurement of harmonic impedances , can be a solution,which again is impractical and expensive for large networks.It is therefore very desirable to estimate the harmonic sources without the knowledge ofnetwork topology and parameters, using only a small number of harmonic measurements.In this chapter we present the estimation of harmonic load profiles of harmonic sourcesin the system using a blind source separation algorithm (BSS) which is commonly referred to asIndependent Component Analysis (ICA). The proposed approach is based on the statisticalproperties of loads. Both the linear loads and nonlinear loads are modeled as random variables. 5.2 - Independent Component Analysis - ICA ModelBlind source separation algorithms estimate the source signals from observed mixtures.The word ‘blind’ emphasizes that the source signals and the way the sources are mixed, i.e. themixing model parameters, are unknown.Independent component analysis is a BSS algorithm, which transforms the observedsignals into mutually statistically independent signals . The ICA algorithm has many technicalapplications including signal processing, brain imaging, telecommunications and audio signalseparation. The linear mixing model of ICA is given as( ) ( ) ( )i i ix t As t n t= +(1)where 1( ) [ ( ),..., ( )]i i N is t s t s t= is the N dimensional vector of unknown source signals,1( ) [ ( ),..., ( )]i i M ix t x t x t= is the M dimensional vector of observed signals, A is an M×N matrix calledmixing matrix and ti is the time or sample index with i=1,2,…,T. In (1), n(ti) is a zero meanGaussian noise vector of dimension M. Assuming no noise, the matrix representation of mixingmodel (1) isX = AS(2)Here X and S are M×T and N×T matrices whose column vectors are observation vectorsx(t1),…, x(tT) and sources s(t1),…, s(tT), A is an M×N full column rank matrix.The objective of ICA is to find the separating matrix W which inverts the mixing processsuch thatY = WX(3)where Y is an estimate of original source matrix S and W is the (pseudo) inverse of theestimate of the matrix A. An estimate of the sources with ICA can be obtained up to apermutation and a scaling factor. Since ICA is based on the statistical properties of signals, thefollowing assumptions for the mixing and demixing models needs to be satisfied: The source signals s(ti) are statistically independent. At most one of the source signals is Gaussian distributed. The number of observations M is greater or equal to the number of sources N(MN).There are different approaches for estimating the ICA model using the statisticalproperties of signals. Some of these methods are: ICA by maximization of nongaussianity, byminimization of mutual information, by maximum likelihood estimation, by tensorial methods , .5.3 - ICA by Maximization of NongaussianityIn this chapter the ICA model is estimated by maximization of nongaussianity. A measureof nongaussianity is