IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 1, FEBRUARY 2015 155
Defining the Unique Signatures of Loads Usingthe Currents’ Physical Components Theory
and Z-TransformN. Calamero, Y. Beck, Member, IEEE, and D. Shmilovitz, Member, IEEE
Abstract—A method to extract unique features from measuredwaveforms of nonlinear loads is presented. These features can beused for the identification of loads connected to the grid. Themethod is based on the currents’ physical components (CPC)electric power transport theory combined with the Z-transform(for implementation using digital signal processing (DSP). A setof admittance-based Z-transform functions that reflect the cur-rent physical components suggested by the CPC is constructed.The resulting transfer functions are shown to reflect the electricphysical significance, and are used for electric load and machineidentification through its electric characteristics. In this paper,the Z-transform analytic expressions are developed and extendedalong with their physical comprehension. Moreover, the strengthof the theory over more traditional spectral analysis is explained.The method presented in this paper is demonstrated and ana-lyzed using real-world measured waveforms (measured by powerquality monitors).
Index Terms—Load identification, power transfer theory,smart grid.
I. INTRODUCTION
T HE GREAT interest in smart grids in recent years hasgiven rise to new research areas and to a renewed inter-
est in the fields of power systems and energy transport theories.Smart grids and increased penetration of renewable powersources [1] require increased power grid monitoring capabili-ties as well as the diagnostics of the various loads connected tothe grid. In residential as well as in commercial and industrialenvironments, there are numerous loads with different imprintson the grid and on microgrids [1]–[3]. It is not only the loads’nature that influences these imprints but also the characteris-tics of the power converters that interface it to the grid [4],[5]. Power monitoring and load identification bear the poten-tial to also improve power quality (complementing PFC andharmonics cancelation methods [6]) There are a few existing
Manuscript received February 27, 2014; revised July 06, 2014 andSeptember 21, 2014; accepted November 16, 2014. Date of publicationDecember 08, 2014; date of current version February 02, 2015. This work wassupported in part by the Israel Smart Grid (ISG) Consortium administered bythe Office of the Chief Scientist of the Israeli Ministry of Industry Trade andLabor. Paper no. TII-14-0243.
N. Calamero is with Israeli Electric Company, Haifa, Israel (e-mail: [email protected]).
Y. Beck is with the Electrical Engineering Faculty, Holon Institute ofTechnolgy, Holon, Israel (e-mail: [email protected]).
D. Shmilovitz is with the Department of Physical Electronics, Faculty ofEngineering, Tel Aviv University, Tel Aviv, Israel (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TII.2014.2378711
methods for load identification in power grids. There are sev-eral studies on load identification: some studies focus on faultevents and power quality events identification, partially withmachine learning [7]–[10]. Other studies on load identificationwere performed using nonintrusive load monitoring and iden-tified electric loads by learning the energy consumption andpattern identification of the load profile [11]–[14]. There havebeen several studies of power quality events identification usingwavelets and fuzzy logic [15]–[17] as well as of load identi-fication by measuring the harmonic impedances and the cur-rent and voltage harmonic parameters [18], [19]. Furthermore,there has been work on the system identification of specificelectric machines using active filters and data summarization[20]–[23]. Recent technological advances have augmented theperformance of power quality monitors and some smart metersbeyond just monitoring power quality events [24]–[26]. In thispaper, a new generic method for electric load identificationis presented. The method is used for identifying loads duringtheir regular operation as well as during periodical faults. Themethod is based on the currents’ physical components (CPC)theory as well as an enhancement via the Z-transform. The CPCtheory is an advanced electric power transport theory that wasinitially presented in 1988 [27]–[30]. In this paper, the theoryis expanded by first using the Z-transform for system identifi-cation and also by using digital signal processing (DSP) toolsfor enhancing the applicability of the theory. The Z-transformhas been widely used in other disciplines, e.g., in radar algo-rithms [31], [32] and speech recognition algorithms [33], but ithas not been projected into electric power. Other works use theZ-transform in the context of electric machinery and electro-magnetic phenomena identification [34], [35], control theory,and filter design [36].
Combining the CPC theory and Z-transform has severalmerits that make the presented theory successful for loadidentification in comparison to other theories.
1) CPC reflects the CPC, which are components that barea physical meaning. Although the validity of these com-ponents and their physical meaning were questionedbecause they are not entirely separable [30], it can beshown that the decomposition maps can be used uniquelyto characterize the waveforms. Therefore, the transferfunctions produced by the CPC theory either have aphysical meaning or are mathematically equivalent to theoriginal waveforms, making them useful for the identifi-cation of the characteristic signatures of electric machinesand other loads.
156 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 1, FEBRUARY 2015
2) The Z-transform displays the overall spectral picture.Therefore, a transfer function that presents the entirespectrum at each point Z has an improved likelihood toobserve fine system characteristics.
3) The merging of the DSP enables the application of pow-erful tools to the CPC theory in particular and to powertheories in general.
4) The orthonormality of the current physical componentsbase maintains the benefits of the Fourier base. Thecurrent physical components maintain the same two rules.a) The total load current is the sum of the physical currents.b) The sum of squares of the currents is equal to the total
current square.These rules imply that the current components are inde-pendent. As a result, the theory proposed herein hasadvantages over the conventional spectral treatment oreven the conventional total admittance Z-transform.
5) The merging of Z-transform and CPC enhances meaning-ful grid indicators such as load unbalance and backwardcurrent (which will not be detected by Z-transform only).Moreover, it may enhance the unique signature in the casethe signature is weak and contained only in one of theproposed components.
6) The theorems that are presented and developed in thispaper and the use of the DSP tools enable the deductionof unique load features.
The massive use of power electronics in modern loadsrequires us to address their impact [38]. Nonlinear loads gen-erate distorted currents, which in turn insert voltage distortions.Nonetheless, these distortions are periodical, i.e., they are har-monic distortions [39]. One objective of this work is to expressthe CPC power theory in terms of the Z-transform. Then, build-ing on this representation facilitates the accomplishment ofthe main objective: identification of the load power properties.Moreover, we developed a generic theory and procedure for theunique identification of these harmonic loads.
In Section II, the CPC theory is briefly presented. InSection III, the enhancement of the CPC theory by the addi-tion and the new formulation via the Z-transform is described.In Section IV, the insights for load identification are explained.Finally, in Section V, the experimental results are shown for theimplementation of the theory.
The computations proposed herein for load identification arepossible for implementation as a module in the metering datamanagement (MDM) central software, or in an additional mod-ule let it be firmware of hardware + firmware inside the metersthemselves.
II. CPC OUTLINE
The CPC theory is a well-known power theory; see for acomprehensive presentation of this theory.
A general way to express the voltage and current waveformsin their Fourier series form [in a linear time invariant (LTI)harmonic system] is given in (1) and (2), respectively,
v(t) = Uo +√2 Re
{∑n∈N
Vne−jnω0t
}(1)
i(t) = Io +√2 Re
{∑n∈N
Ine−jnω0t
}
= YoUo +√2 Re
{∑n∈N
Yn · Vne−jnω0t
}⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
(2)
where Vn and In are the voltage and current phasors. N are allthe harmonic indices in which there are current components inthe frequency domain, and Yn is the admittance for the nth har-monic, which can be expressed as a real Gn part and imaginaryjBn part as follows:
Yn = Gn + jBn = In/Un. (3)
Equation (3) suggests a practical method of extraction ofadmittances while recording voltage and current waveforms ofan unknown load.
The theory is valid for LTI systems as well as for har-monic generating loads (HGL) and quasi-periodic signals witha slowly varying envelope in comparison to the network period.It is important to note that for HGL (namely nonlinear systems),there might be various harmonics existing in the current; how-ever, they are nonexistent in the voltage. In the CPC theory, thetotal current is decomposed of several physical components thatare independent and orthogonal
i(t) = ia(t) + ir(t) + is(t) + iu(t) + iB(t) (4)
where ia is the active current, ir is the reactive current, is isthe scattered (active) current, iu is the unbalanced current (inthree-phase systems), and iB is the backward load generatedcurrent.
The assembly of the current by its physical componentsyields a possibility to enhance the identification of differentloads by means of these currents, their respective values andtheir various transfer functions, as will be demonstrated next.
III. ENHANCEMENT OF THE CPC THEORY USING
THE Z-TRANSFORM
First, the current expression (2) is written by the Z-transform(neglecting the dc component I0) as follows:
I(Z) =∑n∈N
InZ−n. (5)
This expression includes all the harmonics simultaneously.Each harmonic has its own weight at the Z polynomial.
Now, by applying Z as Z � e−jnω0t, expressions (1) and (2)can be written as the actual Z-transform of the digitally sampledsignals as
V(Z) = Uo +√2 Re
{∑n∈M
VnZ−n
}(6)
I(Z) = Io +√2 Re
{∑n∈N
InZ−n
}
= YoUo +√2 Re
{∑n∈N
Yn · VnZ−n
}⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
. (7)
CALAMERO et al.: DEFINING THE UNIQUE SIGNATURES OF LOADS USING THE CPC THEORY AND Z-TRANSFORM 157
Fig. 1. Visualization of a load as a system with an input voltage and outputcurrent.
In the case when the load is LTI, then M in (6) equals Nin (7).
Next, let us consider the problem of any electric load as aproblem of the transfer function, where the voltage is the inputsignal, while the current is considered as the systems’ output,as seen in Fig. 1.
The transfer function of a general load can be written as
Y(Z) � I(Z)
V(Z)=
Io +√2 Re
{ ∑n∈N
InZ−n
}
Uo +√2 Re
{ ∑n∈M
VnZ−n
} � AN (Z)
DM (Z)
(8)
where AN (Z) is the numerator polynomial of rank N (depend-ing on the current harmonic content) and DM (Z) is thedenominator polynomial of M rank (depending on the volt-age harmonic content). In the Z-transform terminology, theCPC leads to a rational polynomial. The Z-transform pre-serves several of the CPC original properties as in the Fourierrepresentation. The current orthonormality and decompositionability are maintained. The following discrete Fourier trans-form (DFT) relations hold for CPC as well as for most powertransport theories:
S2 =∑j
P 2j +
∑k
Q2k
i =∑j
ij +∑k
ik
‖i‖2 =∑j
‖ij‖2 +∑k
‖ik‖2
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(9)
where j relates to all the active harmonic components of thepower and current and k relates to all nonactive harmoniccomponents of the power and currents. At the Z domain, thefollowing CPC presentation of the current can be written as
i(Z) = ia(Z) + is(Z) + ir(Z) + iu(Z) + iB(Z) (10)
and because the components are orthogonal, the followingexpression applies:
‖i(Z)‖2 = ‖ia(Z)‖2
+ ‖is(Z)‖2 + ‖ir(Z)‖2
+ ‖iu(Z)‖2
+ ‖iB(Z)‖2. (11)
Dividing the current expression by the total voltage yields
Fig. 2. Visualization of the total current that is decomposable to the CPC at theoriginal CPC.
Fig. 3. Visualization of the physical components as separable load’s currents.
Expressions (11) and (12) imply the orthogonality of the var-ious components of the admittance transfer functions, namely,
‖Y(Z)‖2 = ‖Ya(Z)‖2 + ‖Ys(Z)‖2 + ‖Yr(Z)‖2
+ ‖Yu(Z)‖2 + ‖YB(Z)‖2(13)
where
Yk(Z) � ik(Z)
V(Z)(14)
and k represents each of the physical components in the CPCtheory. Expression (13) is not mentioned in the original CPCtheory and is introduced as an extension here. It is impor-tant to understand that only the current as in (4) is physicallymaintained, as indicated by Fig. 2. The admittance in (12) isnot physically maintained. It can be viewed as a normaliza-tion of the currents by the total voltage V(Z) or as seen inFig. 3, as parallel-connected branches, which are connected to acommon voltage vTotal. The various transfer functions (admit-tances) in (12) are independent of one another and enable theseparation of a multiproblem into separate admittance physicalcomponents. Moreover, these admittances (transfer functions)enable separate handling of the various transfer characteris-tics per simultaneous phenomena. It is very important to notethat the CPC by means of DSP (CPC–DSP) is not simply aZ-transform of the CPC for the transfer function Y(Z) butrather is an extension.
To summarize the information above, it can be observed thatin the original theory of the CPC, the current Z-transform of(7) is preserving the CPC, but the transfer function at the CPCis maintaining single harmonic relations: Yn = In/Vn(n ∈ N).Equation (8) suggests that the total current is related to the totalvoltage V(Z) as an input, and (14) implies that the current phys-ical component ik(Z) is related to the total voltage V(Z), V(Z)as an input. For the conventional CPC, the Z-transform is pre-served at the current physical components level ik(Z). It differsfrom conventional CPC because the total voltage V(Z) is con-sidered. For the total current, (8) is valid if the load is viewed as
158 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 1, FEBRUARY 2015
a transfer function from the input voltage to the current, as theCPC implies.
IV. ANALYTICAL EXTRACTION OF INFORMATION FOR
ELECTRIC LOAD CLASSIFICATION
A. General
The work presented in [33] and [35] may explain in moredetail the theory of how the zero-pole pairs affect the spec-tra. However, complementary additions are presented in thissection. A series of equations and notifications are presentedin the next sections, and the following is demonstrated.
1) The electrical characteristics of an electric load (ormachine) is comprehended from the transfer functionzeroes and poles.
2) All of the characteristic spectral features of a machineload’s transfer function are calculable from the zero-pole diagram. Analytical expressions for the magnituderesponse, phase response, group response, phase-delay,and transfer function are developed.
The analysis enables an investigation of several electric prop-erties of loads. These results are significant from many aspects,especially in the case of classifying and identifying loads. Theresults are known from other disciplines, such as speech recog-nition [33]. The theorems are presented next in a format suitablefor electric machine identification. The result of the zero-poletheorems is that at the spectra of DSP, features of the load trans-fer functions are: amplitude response, phase response, groupdelay, phase delay—the zeroes and poles usually appear inpairs. The upper bound on the number of zeroes/poles is thesystems sample rate N. The actual count does not exceed thenumber of harmonics at the voltage (denominator) and current(enumerator) waveforms. That count is substantially less thanthe sampling rate. Next, the analysis is shown to be a generalqualitative analysis for demonstrating the theory of uniquelydefining the signature of the loads.
B. Zero-Pole Unit-Like Circle
De Moivre’s theorem states that the n complex roots of 1are located on the unit circle and are equally spaced in termsof the phase angle. In the Z-transform presentation of variousloads, as presented in Fig. 4, the zeroes are marked with an oand represent the roots of the current polynomial. The poles aremarked with an x and represent the roots of the voltage poly-nomial. When applying the Z-transform over a pure resistiveload, the voltage polynomial has the exact same roots as the cur-rent polynomial, and all roots are equally spaced on a unit-likecircle.
It should be mentioned here that the “unit like” circle in thispaper does not refer to the range of convergence (ROC) butrather to the loci of zeros and poles of the admittance transferfunction.
The circle radius is determined by the admittance of the load,as seen in Fig. 4(a). When inductive or capacitive componentsare introduced, the roots of the current polynomial are phase-shifted from the voltage polynomial and are still on the unit like
Fig. 4. (a) Zero-pole diagram of a resistive load. (b) Zero-pole diagram of aresistive-inductive load. (c) Zero-pole diagram of a nonlinear load in which thecurrent harmonics are different from the voltage harmonics.
circle as shown in Fig. 4(b). This is because of the change inthe reactive admittance at each harmonic. From Fig. 4, it is evi-dent that the matching zero-pole pairs represent the RLC linearbehavior of the load. Note that the number of zeroes (current)and number of poles (voltage) are equal. The escaped zeroes,as in Fig. 4(c), represent the nonlinear behavior (HGL) ofthe load.
In this case, the number of zeroes is larger than the numberof poles because there are additional current harmonics.
When introducing slight nonlinearity to the load, somezeroes (current roots) “escape” away from the unit-like circle,as seen in Fig. 4(c). As the load tends to higher nonlinearity,more zero-pole pairs will escape from one another and from theunit-like circle, and the distance in terms of the radius and phaseangle will increase. This characteristic of the zero-pole map isimportant for the unique identification of various loads. It is notobvious at all, at a first glance, as to why a general polynomial
CALAMERO et al.: DEFINING THE UNIQUE SIGNATURES OF LOADS USING THE CPC THEORY AND Z-TRANSFORM 159
of the denominator has on a regular basis the De-Moivre’s rootsdistribution
V(Z) =
N∑n=1
VnZ−n. (15)
The resolution of the theorem is simple when one observesthat the zero poles are spread according to angle θ, where it islinear with harmonic nω0t.
C. Amplitude-Response Zero-Pole Analysis
The amplitude response spectra can be written in a closedform as follows:
log |Y(Z)| =NI∑n=1
log |Z − Zn| −NV∑n=1
log |Z − Z ′n|
= log
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
NI∏n=1
|Z − Zn|NV∏n=1
∣∣Z − Z ′n
∣∣
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
. (16)
The expression in (16) is well known [33]. The expressionseparating NI and NV is a general one. It emphasizes the factthat there is a different number of poles and zeroes.
D. Analytic Analysis of the Group-Delay
In [33]–[36], the group delay feature is used for speechrecognition and radar pattern analysis. In these papers, the ana-lytical results are limited to up to two pole–zero pairs or areused as an empiric result for the general case. Next, the theoryis presented in the general case.
Let Yk(Z) be a transfer function as in (8). The zero-polerepresentation of this expression is as follows:
Yk(Z) =
NI∏n=1
(Z − Zn)
NV∏n=1
(Z − Z ′n)
(17)
where NI and NV are the current and voltage polynomial orderaccordingly. Then, the expression for the spectra of the groupdelay can be written as
vgroup(Z) = −∂θZ(Z)
∂ω
=
NI∑n=1
−1
1 + ϕ2n(ω)
∂Z
∂ω−
NV∑n′=1
−1
1 + ϕ2n′(ω)
∂Z
∂ω (18)
where
ϕn(ω) �Im{(Z − Zn)(ω)}Re{(Z − Zn)(ω)}
Z = e−jθ = e−jωt
⎫⎪⎬⎪⎭. (19)
Fig. 5. Impulse shape of (Z − 1) /(Z − 2).
The proof for (18) is based on: 1) a closed-formexpression for the derivative of: [tan−1(X)]′ = 1/(1 +X2);[tan−1 (X)]′ = 1/
(1 +X2
); and 2) the tan−1 of a product
of variables is the sum of the tan−1 of each variable, andthe tan−1 of a division of two variables is the subtraction ofthe tan−1 of each variable. Equation (18) states that there is adirect link between the zero-pole diagram and the group-delayspectra, and it may be computed. Furthermore, the physicalimplication is that most zeroes and poles are coupled intoidentical zero-pole pairs in the form: (Z − Zi)/(Z − Zi) ≈ 1.When the zeroes and poles deviate from one another, they gen-erate a local minimum and local maximum, as seen in Fig. 5.The interpretation of this characteristic will be demonstrated inthe experimental results section later.
E. Phase-Delay Zero-Pole Analysis
The phase spectrum is defined as the phase angle of theZ-transform as a function of the frequency, and the expressionis separable from the amplitude as follows:
θ(Z) =
N∑n=0
{tan−1(Z − Zn)− tan−1(Z − Z ′
n)}. (20)
The phase delay is defined as
phase-delay = − θ(Z)ω
Z = ejωt
}. (21)
F. Phase-Response Zero-Pole Analysis
The generalized formulation for the phase response is pre-sented in [33]
Y(Z) =
NI∏n=1
|Z − Zn| ejθ(Z−Zn)
NV∏m=1
|Z − Zm| ejθ(Z−Zm)
=
{NI∏n=1
|Z − Zn|}ej
NI∑
n=1θ(Z−Zn)−
NV∑
m=1θ(Z−Zm)
NV∏m=1
|Z − Zm|.
(22)
160 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 1, FEBRUARY 2015
Fig. 6. Feature extraction block diagram.
Equation (22) can be now separated into the amplitude andphase spectra as follows:
|Y(Z)| =
{NI∏n=1
|Z − Zn|}ej
NI∑
n=1θ(Z−Zn)−
NV∑
m=1θ(Z−Zm)
NV∏n=1
|Z − Z ′n|
(23)
and for the phase angle spectra:
θ(Z) =N∑
n=0
{tan−1 · (Z − Zn)− tan−1 · (Z − Z ′
n)}. (24)
G. Procedure Summary
Next, the entire procedure of extracting unique features tothe measured time domain voltage and current waves by meansof the above-mentioned theory is summarized. The process isshown in Fig. 6, as a block diagram, which illustrates the flow ofdata and the analytical and numerical procedures in each stage.The time domain voltage and current waves v(t) and i(t) arefirst transformed to the frequency domain by the fast fouriertransform (FFT) module and are normalized. Note that FFTis a technical algorithm for fast computation of DFT and itsinverse. Therefore, we applied FFT coefficients. The spectra ofthese waves Vn and In are treated in the next module with theCPC and then proceed to the Z domain by the Z-transform inaccordance with the theory presented in (1)–(7) and (10).
The resulting current components along with the voltageV(Z) yield the admittance transfer function components as in(12) and (13). For each individual admittance transfer func-tion, five features are generated (as seen on the right-handside of Fig. 6). These features are the zero-pole diagram,amplitude, phase, phase delay, and group delay as shown inSections IV-B–E.
Fig. 7. Phase voltage and current waveforms of a computer center. The threeupper waveforms are the phase voltages, and the three lower waveforms are thephase currents.
Using the outcome of CPC and Z-transform suggests awealth of uniquely identifiable features. Looking at (19), (20),(21), (24) shows that phase features of Z-transform are usuallymore informative than amplitude features. The phase featuresbear a linear relation to zero-pole pairs, while the amplitudefeatures, as shown in (16), carry a logarithmic relation to zero-pole pairs, making them less sensitive to unique signatures. Ithas also been shown that the active component spectrum isuniform, therefore carrying little signature information, whileactive scattered spectrum is carrying variable spectrum. Finally,in a case where several components are unique to the load, it isnot mandatory to use all of them.
We have implemented the computations according to dia-gram in Fig. 6 in a remote server at the MDM system.
V. EXPERIMENTAL RESULTS
Next, a series of electric machines are analyzed usingthe above-mentioned theory. The current and voltage wave-forms were measured using SATEC EM720 power qualitymonitors. These devices are certified for power quality mon-itoring and energy metering international standards with a256-samples/cycle sampling rate.
A. First Test Case: Computer Center
The measured waveforms of a computer center are shown inFig. 7. The waveforms are analyzed with a MATLAB simula-tion of CPC–DSP code, which creates the transfer functions ofthe various current physical components. Then, each transferfunction is analyzed by the MATLAB FDA tool, which pro-duces the various Z-transform spectral features. Observing thewaveforms in Fig. 8, it is evident that the voltage is pseudo-linear with additional harmonics added to the pure sine wave.However, the current is pulsating and highly nonlinear becauseof the converters’ characteristics.
We first present the DSP spectral features developed for theGActive-conservative(Z) = Ga(Z) +Gs(Z)).
This is performed because the active component by itself hasinsufficient information (because it is a constant number Ge). Itis easy to show that the sum of Ga and Gs is equivalent to theconservative active power [40]. The zero-pole diagram in thiscase is depicted in Fig. 8, where it can be seen that the poles (x)are equally distant along a circle. The highlighted “escaped”
CALAMERO et al.: DEFINING THE UNIQUE SIGNATURES OF LOADS USING THE CPC THEORY AND Z-TRANSFORM 161
Fig. 8. Zero-pole diagram of the active and scattered transfer function of thecomputer center waveforms.
zeroes appear in pairs and are directly represented by the peaksat the phase spectrum according to the theory above. Moreover,the remaining zeroes (current), which are slightly phase-shiftedrelative to the poles (voltage), represent a linear behavior of theload, which includes reactive components.
The various DSP spectra of the nonlinear load are presentedin Fig. 9.
In each graph, black dashed lines are added to emphasize thelocation of the changes and interest in the various spectra. It isclear from Fig. 9 that in all spectra, the location of the peaksor graph change is at the same location regarding the frequencyaxis. In addition to the above-mentioned theory, there are otherobservations.
1) The group delay presented in Fig. 9(a) for the nonlin-ear load of Fig. 7 is characterized by sharp peaks. Thesharpness of the peaks is directly related to the strongnonlinearity of the load. The further the zero escapes, thesharper the peak is (as shown in Fig. 5). These sharp peakswill be compared to the results of a relatively linear loadlater.
2) Observing Fig. 9(a)–(d) and from the developed the-ory, it is evident that the more interesting and uniquelydefined information of the load is located at the phasespectrum functions. The magnitude according to (24) isalso influenced by the zero-pole pairs, but it is much lessobservable because log(|G(Z)|)log(|G (Z)|} is a highlysmoothing and descaled function. A weak effect in theextreme case barely causes an amplitude response but isnoticeable in the phase spectrum.
B. Second Test Case: Electric AC Motor
Next, we consider the voltage and current waveforms mea-sured by the same equipment (as shown in Fig. 10). The mea-surements were taken in a site containing a large ac motor witha motor drive. Because the motor drive consists of power elec-tronics, nonlinearity is introduced into the motor. The zero-polediagram of this load is presented in Fig. 11.
At first glance, it seems that the zero-pole diagram does notinclude “escaped” zeroes and that the poles highly match thezeroes.
Examining the results further, it is noticeable from Fig. 11(b),which is a zoom-in view of the same diagram in Fig. 11(a), thatthere are weak escapes of zeroes from the unit-like circle at
Fig. 9. (a) Group delay spectra of active + scattered. (b) Phase delayspectra of active + scattered. (c) Magnitude response spectra of active +scattered. (d) Phase response spectra of active + scattered.
the marked locations in Fig. 11(b). This means that this loadincludes a small amount of nonlinearity.
Next in Fig. 12 (as in Fig. 9), the various DSP spectra of thisload are presented for each graph. The black dashed lines arealso added to emphasize the location of the changes and interestin the various spectra.
162 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 1, FEBRUARY 2015
Fig. 10. Voltage and current waveforms of an ac motor that is a linear load withan additional HGL.
Fig. 11. (a) Zero-pole diagram of the motor’s active + scattered transferfunction. (b) Zoom-in view of a zero-pole diagram.
The spectrum in Fig. 12(a), being proportional to the deriva-tive of the spectrum in Fig. 12(b), yields a phase shift similar tothe π/2 phase shift between sine and cosine.
The results of Figs. 9 and 12, as proposed by the the-ory, imply the uniqueness of the waveforms for each load.Moreover, an inspection of the waveform peaks of in Fig. 12(a)reveals that they are thicker and less definite compared to thepeaks presented in Fig. 9(a). These results indicate that the lin-ear load can be distinctively identified from a nonlinear load.Furthermore, we can conclude that the sharper the peaks are,the higher the nonlinearity is.
C. Third Test Case: A Dimmer With Variable Excitation Angle
The first and second test cases demonstrated the differ-ence and uniqueness of the features that are produced for two
Fig. 12. Spectrums of the active and scattered transfer function. (a) Groupdelay. (b) Phase delay. (c) Phase response.
significantly different loads to show the general theorems inaction. The next test case, shown in Fig. 13, demonstrates theunique feature generation of a dimmer circuit with two onlyslightly different excitation angles: 85◦ and 90◦. There is hardlyany noticeable difference in the current waveforms, as shownin Fig. 13(a) and (d). Nevertheless, the analysis of Gr(Z),Ga(Z) +Gs(Z) of the CPC–DSP theory reveals noticeablechanges in the various features. The zero-pole diagram revealsdifferent spacing between the zeroes and poles. Moreover, thephase delay of the reactive transfer function Gr(Z) for 85◦
and 90◦ have different characteristics, as can be seen, e.g., inFig. 13(b) and (e) for the group delay and Fig. 13(c) and (f)for the phase delay. Note the dashed arrows which connectthe same graph change in both graphs of the same feature inthe group delay and the circled area in the phase delay fea-tures of each state of operation. These results show noticeabledifferences for a slight angle change in the operation of thedevice. Same results are achieved for the active + scattered
CALAMERO et al.: DEFINING THE UNIQUE SIGNATURES OF LOADS USING THE CPC THEORY AND Z-TRANSFORM 163
Fig. 13. Dimmer circuit. (a) Current waveform with 85◦ excitation angle and the associated FFT distribution. (b) The group delay of analysis of the reactivetransfer function at 85◦ angle. (c) The phase delay of analysis of the reactive transfer function at 85◦ angle. (d) Current waveform with 90◦ excitation angle andthe associated FFT distribution. (e) The group delay of analysis of the reactive transfer function at 90◦ angle. (f) The phase delay of analysis of the reactive transferfunction at 90◦ angle.
transfer functions Ga(Z) +Gs(Z). Another noticeable differ-ence between the various operation modes of the dimmer can beseen in the phase delay. It also shows that the CPC–DSP offersa wealth of features, and it suffices if only one/some of thesefeatures show a significant load signature.
VI. PRACTICAL IMPLEMENTATION FOR SMART GRID
Next, the motivation and practical implementation of thiswork are discussed.
Power measurements use waveform analysis mainly forpower quality analyses, which require fewer resources than theanalyses described here. Therefore, the implementation of thismethod will most likely require larger computational resourcesin comparison to the resources existing in today’s smart metersand power monitors. Nevertheless, many currently availablepower monitors are designed for modularity with the abilityfor extended capabilities by adding modules for the existingfirmware. Moreover, in several load identification applications,the analysis does not require real-time results, and thereforeoffline analysis on remote servers is quite satisfactory. Wecan present numerous examples by which this costly method(these days) will be worthwhile. There are many very expen-sive processes in which adding a system, such as the methodimplementing this theory as explained in this paper, will beworthwhile in the aspect of preventing maintenance. For exam-ple, one application for smart grids might be the detection of
electric loads by waveform measurements recorded by powerquality monitors and also by detecting irregularities in thesewaveforms. Another application could be the installation of thealgorithm based on this theory for a production line of wafers inthe silicon industry or any expensive production line in whichthis system can predict a malfunction prior to it becomingdestructive. Typically, a pause in such a system means lossof millions of dollars, and therefore this method, applied withadditional software for a waveform monitor that will only cost afew thousands of dollars, would be very attractive. At this stage,this technology might be too expensive for implementation inevery power monitor, but the future of smart grids predicts sub-stantially more measurements in the grid, which will imply thatthis theory and similar theories can be implemented in otherhierarchies of the grid.
The above-mentioned spectral features presented in Figs. 9,12, and 13 for the group- and phase-delay spectra as well asthe zero-pole diagram in Figs. 8, 11, and 13 are shown to beunique for each electric load, as was demonstrated theoreti-cally and through examples. The spectrum is characterized withunique peaks with a specific width and height, which character-ize specific loads and other harmonic phenomena. These newlyproposed signatures are significantly more unique than the cur-rent’s physical components spectra without the DSP enhance-ment because they include the load spectral response function,they include the phase information, and the Z-transform is acontinuous function.
164 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 1, FEBRUARY 2015
VII. DISCUSSION AND CONCLUSION
The theory presented in this paper is used for generating fea-tures which define a load, electric machine, or any phenomenonthat can be represented by its current and voltage waveforms.The method combines the CPC power transport theory with theZ-transform to generate unique features for the measured wave-forms of a specific load. The motivation for the development ofthis theory poses advantages compared to more conventionalmachine identification procedures.
1) The admittances originate from the currents with physicalcomponent significance.
2) The Z-transform presents the entire spectral signature ofthe specific admittance with distinct weighting for eachindividual harmonic.
3) The DSP spectral techniques are used to investigate powerproblems and anomalies.
4) The orthonormality introduces separateness for the cur-rent physical components.
In this paper, analytic expressions are given and developed,which demonstrate that all the spectra are analytically compre-hended through the knowledge of the zero-pole diagram. Touniquely describe a load, machine, or electrical phenomena,five transfer functions are introduced (according to the exten-sion of the CPC). These transfer functions are Ya, Yr, Ys, Yu,and YB . While Ya is constant, it leaves four transfer functionsfor which five DSP features, such as amplitude response, phaseresponse, phase delay, group delay, and zero-pole diagram, canbe produced. This means that there are potentially 20 differentspectra for uniquely identifying the electric load. It should bementioned here that not all features exist for every load, andtherefore, in addition to the qualitative analysis, the quantity offeatures is added to the identification of the load.
The spectrum analysis presented in the paper shows first thatfor linear loads, such as R-, L-, and C-like, yields at the most ashift of the roots of the voltage (by a phase angle). The zeroesthat are the current roots are therefore not distant from theirpair of matching poles. With nonlinear loads, there are escapedzeroes, which are distant from the unit-like circle, which in turncause the sharp peaks of the spectrum. It was shown that the fur-ther the zeros migrate, the sharper the peaks and the higher thederivatives for the correlated group delay spectrum in the phase,which indicates a higher nonlinearity of the load. It should bestressed here that these spectra are saved in a database, andtherefore a difference in the spectra is sufficient for uniquelydefining a load or a phenomenon. Test cases A and B, whichinclude different loads, demonstrate this idea clearly. The var-ious spectrum analyses show that the phase spectrum providesmore profound information than the amplitude spectrum. Theresults are demonstrated here on real measured waveforms,demonstrating the proposed technique for uniquely identifyinga load.
A worthy discussion is concerning the added value of CPCZ-transform over simply spectral identification. The addedvalue can be summarized in the next few points.
1) Comparing to only Z-transform, the proposed merger ofCPC and Z-transform preserve physical properties of cur-rents and power. For example, test case 3 deals withhighly similar loads as shown in Fig. 13. Simple DTFT
transform results in pretty much equal load signatureas shown in Fig. 13(a) and (d). However, the proposedmethod reveals notable difference between the two asshown in Fig. 13(b), (c), (e), and (f).
2) The use of CPC enhances meaningful grid indicators suchas load unbalance and backward current (which would nothave been identified through Z-transform alone).
3) The proposed Z-transform combined with CPC mayenhance the unique signature in case in which the sig-nature is weak and contains only in one of the proposedcomponents.
For example, the CPC active admittance manifests as aconstant variable. Thus, it is advantageous to focus on the scat-tered components solely (without the constant) to enhancesphenomena related to active power.
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N. Calamero was born in Tel Aviv, Israel, on June8, 1967. He received the B.Sc. degree in electri-cal and electronic engineering, and the M.Sc. degreein electrical engineering from the Technion IsraelInstitute of Technology, Haifa, Israel, in 1990 and1998, respectively.
During 1995–2005, he worked with Intel RnD,Haifa, at chip design. During 2005–2008, he workedwith Elspec’s RnD Department, Cesaria, Israel. From2005 until today, he works with the MeteringDevelopment Laboratory, National Metering Unit,
Israel Electric Company, Haifa, as a Systems Engineer and Deputy ChiefMetrology Officer. His research interests include control algorithms for activefilters and grid diagnostics. He is a Member of the Smart Meters StandardCommittee at the Israel Standards Institute, Tel-Aviv, Isreal.
Y. Beck (M’07) was born in Tel Aviv, Israel, onNovember 30, 1969. He received the B.Sc. degree inelectronics and electrical engineering, and the M.Sc.and the Ph.D. degrees in ground currents due tolightning strokes from Tel Aviv University, Tel Aviv,Israel, in 1996, 2001, and 2007, respectively.
Since 1998, he has been with the InterdisciplinaryDepartment, Faculty of Engineering, Tel AvivUniversity. In 2008, he joined the HIT-Holon Instituteof Technology, Holon, Israel, as a Lecturer, and since2010, is acting as the Head of the Energy and Power
Systems Department, Faculty of Engineering. His research interests includesmart grid technologies, lightning discharge phenomena, lightning protectionsystems, power electronics, and photovoltaic systems.
D. Shmilovitz (M’98) received the B.Sc., M.Sc., andPh.D. degrees in electrical engineering from Tel-AvivUniversity, Tel-Aviv, Israel, in 1986, 1993, and 1997,respectively.
During 1986–1990, he worked with R&D for theIsrael Air-Force (IAF), where he developed powerprocessing topologies for avionic systems. During1997–1999, he was a Post-Doctorate Fellow withNew York Polytechnic University, Brooklyn, NY,USA. Since 2000, he has been with the Faculty ofEngineering, Tel-Aviv University, where he heads the
Power Electronics and Power Quality Research Group. He has authored andcoauthored over 100 papers, of which over 40 are in refereed journals. Hisresearch interests include general circuit theory, switched-mode converters,power quality, special applications of power electronics (such as for alterna-tive energy sources) powering of autonomous sensor networks, and implantedmedical devices.
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