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An Overview of Harmonic Analysis and Resonances of Large Wind Power Plant

K.N Md Hasan1, Kalle Rauma2, P. Rodriguez3, J. Ignacio Candela4, Raul S. Muñoz-Aguilar5, Alvaro Luna6

1,3-6Technical University of Catalonia,

C/ Colom 1 08222, Terrassa, Spain khairulnisakmh@yahoo.com1

2Aalto University, School of Electrical Engineering, Espoo, Finland

kalle.rauma@luukku.com2

Abstract—This paper presented an overview of harmonic analysis and resonance in large wind power plant. The harmonic related problem becomes a concern as the growing numbers of nonlinear load and the increased possibility of harmonic resonances occurrence. In WPP, several conditions may contribute to the resonance phenomenon that will amplify the effect of the harmonic frequency. General overview of international standards and grid codes and basic modeling of WPP for harmonic analysis is also presented. Furthermore, an overview of the harmonic analysis methods are reviewed with emphasize given on the harmonic resonance analysis. A simple three-bus system example is evaluated applying three resonance analysis methods.

Index Terms—Harmonic Analysis, Offshore Wind Power Plant, Resonance, and Harmonic modeling.

INTRODUCTION ind energy is one of the identified potential renewable energy resources that has received serious attention

worldwide. Rapid development in wind power generation started in 1990s and now, many countries have installed a vast number of MW-range wind power plants that have been integrated to the main grid. According to World Wind Energy (WWEA) 2009 report, the worldwide capacity has reached 159 GW by the year 2009 with the growth rate of 31.7% from previous year. By 2020, the predicted global capacity of wind generation will be 190 GW [1]. Offshore wind power has also shown a positive growth with a considerable amount of offshore wind turbines installed at sea. Currently, 12 countries have started to venture into OWPPs and a rapid growth within these few years is expected. The top 5 countries looking from the total offshore capacity are United Kingdom, Denmark, Netherlands, Sweden and Germany [2]. As of 30th of June in 2010, total installed offshore wind power capacity is 2,396 MW with 118 offshore wind turbines were fully grid connected totaling to 333 MW during the first half of the year 2010 [2].

Integration of WPP´s into the main grid has raised the concern over the power quality, and one of the main reasons is harmonic distortion. For OWPP which comprises of a great number of large capacities of wind turbines, harmonic analysis and especially harmonic resonance has become necessary. Resonance will cause amplification of voltage distortion due to parallel resonance or result in high harmonic current due to

series resonance at the resonance frequency. The problem of resonance arises because of the WPP containing both inductive source and capacitive source characteristics. Transmission line, such as a long HVAC cable used to connect the OWPP to the main grid imposes different characteristics of internal impedance of the system because it has higher capacitance and raises the possibility of resonance occurrence, as it shifts the natural frequency at rather a lower value [3].

Furthermore, the usage of reactive power compensation equipment and power factor correction devices may also alter the system impedance and might lead to resonance. Since resonance is characterized by the inductive and the capacitive elements in the network, change in system impedance leads to alteration of resonant point that makes the issue more complicated. Different configuration or operating condition in the large WPP will impose different impedance in the system, where impedance of the system is changing with the number of turbines in operations.

The harmonic study and analysis in WPP is usually performed either at planning stages or when there is a requirement to do so. Nowadays wind farm must meet a more stringent limits imposed by the TSO. The analysis is usually assisted by the advance computer modelling. Resonance conditions are located by using frequency domain analysis of impedances and amplification factors, namely the frequency scan. The mitigation of harmonics in WPP involved the implementation of harmonic filter, which is commonly passive type of filter. In this paper, an overview of nowadays OWPP and a number of harmonic analysis carried out will be presented. Then, the available harmonic analysis method is reviewed with the emphasis on the resonance phenomenon. Furthermore, a quick review of the codes and standard on the harmonic requirement is done which is followed by the basic modeling of OWPP for the harmonic analysis.

HARMONICS IN WIND POWER PLANT

Overview of Harmonics in OWPP Harmonic studies in WPP become necessary as the numbers

of nonlinear loads in the system network are increasing and the system response characteristics are also changing due to the usage of cables and capacitors which raise the harmonic resonant issue. A recommended practice is that when a network

W

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consists of 25%-30% or more of nonlinear loads, there is a need to conduct a harmonic analysis study. With the increasing usage of power electronic devices in modern WPP, it is required for the wind farm operator to conduct a harmonic power flow analysis to ensure that the harmonic distortions are within the limits imposed.

In OWPP, the common wind turbine (WT) used are of the variable-speed wind turbine (VSWT). The harmonic spectrum is mainly of the lower order of harmonics. This has become a concern in the occurrence of resonance. A frequency scan result of VSWT-DFIG has a significant amount of low-order harmonics (5th, 7th, 11th and 13th) which dominate the spectrum [4]. In the study by [5] on harmonic emissions of variable speed wind turbine, namely synchronous generator, DFIG and SCIG with full rated converter, it is shown that harmonic spectrum of WTs are extending up to few kilohertz with low-order harmonics are always present and dominate the spectrum.

A number of studies on harmonic in WPPs have been conducted. In a case of 20 x 500 kW VSWTs wind farm in western Greece, significant lengths of submarine lines as well as overhead lines are used in connecting the island to the main grid [6]. It was found that impedance at the HV side is less affected by the variation of loads and the load modeling is not really important. However, for MV side, the variation of harmonic impedance is dominated by parallel resonance (between busbar capacitance and inductive impedance of upstream system) and load modeling is critical. Horns Rev 2 is among the largest OWPPs that use HVAC submarine cable and land cable. Studies by [7-8] show that the higher is the number of WT in operation, the more the impedance of the system is reduced and this is obvious for a large WPP. The equivalent admittance in relation to the number of WTs in operation shows that for a higher number of WTs, additional resonance point appears as in the case study, a lower harmonic order at around 200 Hz is observed. This might be due to the transformer series reactance and the HV cable shunt capacitor. The analysis on the effect of the shunt reactor on the parallel resonance shows that there is a little effect on the whole system. While the high capacitance cable line had resulted in a low frequency parallel resonance.

In a case study in one plant in China, monopole operation of HVDC caused repeated capacitor failure at a tuned harmonic filter at a transmission substation connected to a 525 kV system. The bank absorbed high fourth harmonic current (THD = 169%). The three reactive compensation units installed previously had been designed to avoid resonance at 3rd, 5th and 7th harmonic. Since the monopole operation of HVDC produced fourth harmonic current, it had caused system resonant at that frequency [9].

Resonance Phenomenon Most power system circuit elements are primarily inductive.

When shunt capacitor/s elements are present in the system, usually capacitor bank or capacitance of a long cable, it can cause cyclic energy transfer between the inductive and the capacitive elements at the natural frequency of resonance. With a combination of source reactance and shunt reactance at certain location, impedance seen by the current source becomes very large, which results in either series resonance or parallel resonance [10]. In harmonic studies, the driving point impedance as seen from harmonic source bus is examined to locate the series and parallel resonance frequencies and the resulting impedance.

Parallel resonance occurs when the reactance of inductive elements that is in parallel with the reactance of capacitive elements cancelling each other out. The frequency at this point is called the parallel resonant frequency, which can be expressed as follows:

CLLR

CLf

eqeqeqp

121

41

21

2

2

ππ≈−= (1)

Capacitive elements can be capacitance of capacitor bank or

the shunt capacitance of long cables and inductive elements can be inductance of transformer or series reactors. At parallel resonant frequency, the apparent impedance seen from the harmonic current source becomes very high [10].

Shunt capacitor and the inductance of a transformer or a distribution line may appear as a series LC circuit to a source of harmonic current in certain condition. Series resonance is a condition when a low impedance is seen at resonant frequency which causes high current and high voltage distortion even at a location with no or little harmonic emission [11]. The series combination of the transformer inductance and capacitor bank is very small and only limited by its resistance, hence harmonic current will flow freely in this circuit.

INTERNATIONAL STANDARDS AND CODES FOR HARMONICS In this paper, seven standards and grid codes are reviewed on

the harmonic distortion requirement. Note that the requirements stated are only for the current or voltage distortion. Widely used standard are the IEEE Std 519 and IEC 61000-series and other grid codes reviewed are VDEW, Nordic, ElKraft System and Eltra, EirGrid and The New Zealand Electrical Code. IEEE Std 519 stated a more general requirement of current and voltage harmonics that is applied to individual equipment. Different THDV and TDD are imposed for customers connected to different voltage levels, 69 kV and below, 69 kv –

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161 kV and 161 kV and above [12]. IEC 61000-series addressed the harmonics requirement in

several sections, namely IEC 61000-3-2, IEC 61000-3-4 and IEC 61000-3-6 addressing harmonic limits, IEC 61000-4-7 addressing testing and measurement techniques of harmonics and IEC 61400-21 addressing measurement and assessment of power quality of connected wind turbines. IEC 61400-21 required the current harmonic and interharmonic of WECs must be measured and reported. The harmonic limits for voltage harmonic distortion for medium and high voltage are given for odd and even harmonic order, whereby the harmonic current injections present in the system must not be causing the voltage harmonic to exceed the limit [13]. Nordic grid code is also outline the minimum voltage harmonic limits for each harmonic order (up to 25 and more) which is more stringent than the IEC while EirGrid grid code basically applying the same limits as in IEC 61000- standards [14-15].

The most common grid codes that are being referred to, is the VDEW – the German Electricity Association grid code. The maximum allowed of harmonic current components, Iu,v is depending on the short-circuit power, Ssc at the connecting point, where Iu,v is a product of Ssc and a constant. These constants are given in the table for voltage rating of 10 kV, 20 kV and 30 kV up until the 25th harmonic order [16]. The grid code of ElKraft System and Eltra (Regulation TF 3.2.6) implies almost similar way of VDEW but only covered for 10-20 kV grid. The harmonic current is calculated based on the harmonic voltage limit and dependent on substation load and the short-circuit power [17]. It must be emphasized that the application of the harmonic limits needs to be done correctly as a misinterpretation might lead to unnecessary situation. Furthermore, the measurement of harmonics must be in accordance to the stated standard.

HARMONIC ANALYSIS METHODS AND RESONANCES Harmonic analysis of a wind power plant can be very

complex and tedious to perform since the system network is large and there is a limitation of data acquired. The system network is usually simplified to an equivalent network that represents the total network. The available harmonic analysis techniques are basically divided into three; frequency domain, time domain and the combination of both or the hybrid techniques. Besides the important of determining the current and voltage harmonic level, the resonance conditions are also a big concern. Hence, few available methods of identifying resonance are also reviewed. Furthermore, the basic modeling for harmonic analysis is presented in this section.

Fig.1. Classification of harmonic analysis method

Methods Harmonic analysis methods are evolved in several directions.

One of the directions is ability to accurately model the harmonic sources and the harmonic interaction in the system. In a broader view, the classification of harmonic analysis method as reviewed by authors is shown in Fig. 1. A more detailed description is given for the resonance analysis while the other technique is described in a more generalized way.

Resonance Analysis

Resonance will cause a severe effect in the WPP system if the harmonic injection from its sources ‘matches’ the resonance frequency. The common method used in identifying the point of resonance is the frequency scan. Another two methods which perform the resonance analysis based on eigenvalue-sensitivity are the harmonic resonance modal analysis (HRMA) and the eigenvalue state-space resonance analysis (ESRA) [18-23].

Frequency scan method has been reported in many literatures as the most common method used in harmonic analysis in examining resonant condition [24-25]. It is a direct manipulation of the nodal analysis, where the nodal admittance matrix is evaluated at every specific frequency for every harmonic current.

[ ] [ ] [ ] 1≠⋅= hforVYI hhh (2)

The nodal harmonic impedance is obtained from the inverse of the nodal admittance matrix. The diagonal elements of this matrix corresponding to harmonic self-impedance of respective buses and non-diagonal elements are transfer impedances, related to the effect on the voltage of bus i when harmonic current is injected at bus j. Only one entry of the harmonic current of 1.0 p.u will be injected to evaluate the driving point impedance looking into the system from one injection node. The impedance scan plot reveals the resonance points indicated by the peaks (highest impedance), but it doesn´t indicate the severity level of the harmonic problem. Different configurations of system network result in different characteristic impedance. In WPP, the usage of long cables, present of reactive capacitance and different number of wind turbine in operations are needed to consider. Hence, the frequency scan must be done for each case. Even though this method is an acceptable method, but it only gives the resonance

Resonance Analysis

Frequency Domain

Time Domain Hybrid Techniques

1. Frequency Scan 2. Modal Analysis 3. State-space analysis

1. Current-source 2. Harmonic Power Flow

1. Time variant analysis

1. Norton 2. Transfer function 3. Harm admittance matrix 4. Forward/Backward

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point in the system without any further information. This resonance point is also might contain the ‘coupling effect’ of the adjacent bus, which is not really the ‘true’ resonance.

HRMA method is proposed in [18] and further continuation work is found in [19-21]. In the HRMA method, the modal analysis of the nodal admittance matrix is performed. The harmonic resonance is said to occur when the nodal voltages are very high, which is associated with the admittance matrix approaching singularity. This singularity of Y is due to the very small value of eigenvalue which approaches zero. Through eigenvalue decomposition, a further analysis on the resonance point and the sensitivity can be performed which can reveal the ‘true’ resonance point and the potential point of the most influenced bus in the network. The eigenvalue decomposition of Y is given by:

[ ] [ ] [ ] [ ]EVEV TLY ⋅⋅= λ (3)

Where [Lev] and [Tev] is the left and right eigenvector respectively and [λ] is the diagonal eigenvalue admittance matrix. The modal impedances, ZM is obtain by the inverse of [λ], such as:

[ ] [ ] [ ] [ ] [ ]ITLV EVEV ⋅⋅⋅= −1λ (4)

where [ ] [ ] 1−= EVEV TL . The highest ZM is the point of smallest λ and referred to as the critical mode. The frequency corresponding to this point is called the critical mode frequency. The plot of ZM as a function of frequency can reveal the modal impedance plot and the critical mode, the point of the resonance. The corresponding left and right eigenvectors at the critical mode entry will further assist to analyze the observability and the controllability of bus. The sensitivity analysis is done by multiplying [Tev] by [Lev]. For critical mode at n-bus (λ-1

nn), the n-column of [Tev] is multiplied by n-rows of [Lev]. The entries this 3-by-3 sensitivity matrix are the participation factors (PC) with the diagonal elements are PCs of corresponding buses. Its absolute value give the information on how strongly each of the buses is involved in the resonance at critical mode frequency. Hence, the highest PC can be defined as the bus with highest excitability and observability and is the effective location to inject signals to cancel the harmonics.

State-space representation of power system network and its corresponding eigenvalue analysis is commonly applied in the transient-analysis application. It is extended to harmonic resonance analysis which implies that the eigenvalue of the ‘A’ matrix reveals the harmonic resonance frequency and the eigen-analysis is able to provide more information on how ‘involved’ is the bus [22-23]. In this method, the power system network (such as the WPP) is represented in the state-space representation. The system variables are the voltages and currents, where the usual practice is that the capacitor voltage and inductor current. The state-space equation is given in (5).

CxyBuAxx

=+= (5)

The formation of the state-space itself become more complex as the network involved is very large, such as WPP. Taking Laplace Transform of (5), the impedance of the system can be found as:

BUAsICCXYBUAsIX

⋅−⋅==⋅−=

−

−

1

1

)()( (6)

The system harmonic impedance, Z is given as:

BAsI

AsIZ

BAsIXY

k ⋅−

−=

⋅−= −

)det()det(

)( 1

(7)

Where Y is the harmonic voltage, U is the harmonic current and k is representing the bus-k. From here, the eigenvalues of the system impedance are obtained which is given by its poles that reveal the ‘resonance frequency’ and the zeros are corresponding to self-impedance of the node k. The eigenvalue sensitivity is performed such as described for the HRMA method. However, the eigenvalue and its sensitivity values obtained from this method are not associated to the same meaning. In this method, the frequency of transient response represents the natural frequency, where the resonance will occur when the frequency of excitation approaches the natural frequency –in other words, the resonance frequency. The eigenvalues of the two methods can only be associated to each other when the damping coefficient is nearly zero, which corresponding to the resonance frequency. The A matrix of ESRA method are representing the time-domain characteristic of the network and number of eigenvalue modes equal to the size of A matrix. The eigenvalue has a discrete point, giving point of potential resonance frequency only while the HRMA method giving the modes for all frequencies.

Frequency Domain Technique Frequency domain is commonly used in harmonic analysis. Basically, it is a reformulation of the conventional load flow that includes the nonlinear load. The developments of the methods are going on in the direction of integrating the harmonic interaction (voltage dependence) of the network and also formulation based on fundamental power. Current source method is basically the same as the frequency scan, but the nonlinear loads are represented by a current source of known magnitude and phase. More specifically, nonlinear loads are represented using a summation of currents where each entry in the sum corresponds to a term of known frequency in the Fourier series representation of the load current – called ‘vector’. The corresponding vector harmonic voltage is obtained for the harmonic current injection. The drawback of this method is that the harmonic spectrum of the non-linear load must be pre-determined and are not considering the

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interaction with the network [24]. Simplest iterative scheme based on Gauss concept – use fix point iteration which is called the iterative harmonic analysis. Basically, the current injection method as described before is used but the harmonic current vector are obtained using the ‘adjustment’ of the harmonic voltage. The latest value of the distorted terminal voltage is used to update the ac voltage harmonic for the next iteration and the harmonic current vectors are updated automatically [25]. However, the typical spectra of the harmonic current are still required. The harmonic power flow analysis employs the power flow method where the iterative analysis is done. There are a number of versions of HPF with variation in term of modeling the harmonic sources at different frequency. Instead of putting in the typical value of harmonics, a set of harmonic estimation values is used and the obtained value is used and solved using iterative technique. Mainly here, the different is on the complexity of the harmonic source modeling or representation. The early concept of HPF is introduced by Xia and Heydt for the case of symmetrical power system by using the concept of power flow study. The insertion of the harmonic effect is done to modify the power flow concept and analyze the impact of harmonic element produced by non-linear loads. The HPF requires the information of the three phase ac-dc power flow, multi-harmonic three-phase representation of linear element in power system and the harmonic domain representation of nonlinear load individually.

Time-domain Technique The time domain method has an advantage of representing

the nonlinearities in the system and time-varying situation. The time domain method is more robust as it represents the network and harmonic sources as time varying equivalents and simulation is carried out until all transients have fully decayed away. However, the complexity involved in the time domain method requires a very long computation time. Acceleration techniques are usually used to accelerate the process [26].

Hybrid-Technique Basically, in a hybrid solution, an admittance matrix is

performed according to the standard power delivery system, but the non-linear loads are represented in time domain differential equation. Load model is simulated with initial estimate of network voltage until it reaches steady state. From here, a new harmonic current vector is obtained for each nonlinear load, [Ih]. These new current vectors are then injected following the same step as in the current source method. It is run for all frequencies of interest and this will yield the system voltage vector. The process is carried out until the frequency domain networks are converged, where the entire set of equations is solved iteratively by applying the Newton-Raphson or Gauss method. In the Norton-equivalent method, the nonlinear component is represented as a Norton equivalent instead as

current source, where the Norton admittance represents linearization of component response to variation in terminal voltage harmonics.

For the transfer function method, the representation of the converter as the harmonic sources is done in term of transfer function which accounted for the interactions of ac and dc side of the converter [26]. The dc voltage is calculated by summing up the phase voltage multiplied by its associated transfer function and the ac current is define by dc current multiplied by the ac transfer function:

dcac

dcdc

iNYI

vYNV

φφ

φφφ

=

= ∑ (8)

∅ is 0o, 120o and 240o for phase a, b and c and Y is the transfer function. The harmonic coupling matrix equation is used in solving the interaction for the ac current and the dc voltage. Transfer function concept is useful because it provides the cause-effect relationship information, but the linearization process for simplification is usually carried out which reduces the accuracy of result for low harmonic orders where the solution accuracy decreases with increasing distortion magnitude and frequency [26].

The forward/backward method integrates the backward sweep and forward sweep techniques with the frequency scan formulation and the three phase component models to perform the harmonic analysis. This technique avoids the usage of admittance matrix and solving the standard nodal analysis formula to obtain harmonic voltages that reduces the processing time. The nonlinear loads are treated as harmonic current sources and other shunt devices are considered as injection current in the analysis. The harmonic current vectors are formulated and the backward sweep is used to define/calculate the harmonic current flow through the branch. Branch current and voltage drop and bus voltage caused by harmonic current are obtained by applying the forward/backward technique [27-28].

Basic Modeling for Harmonic Analysis The modeling method for harmonic analysis must be

properly carried out. In WPP, it is important to obtain the real data of each equipment used in order to sufficiently model the components. This modeling includes cables, load models, generators, synchronous machines, MV collector and transformers. In this section, only model of cable, MV collector and transformer is discussed.

Cable Cable and transmission line need to be modeled correctly to

represent the actual transmission and distribution line. Modeling of cables can be complex and tedious since exact representation of the cable characteristics must be taken into

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account. The characteristics of cable conductors are nonlinear frequency dependent because of the proximity and skin effect. It is often important to include the shunt capacitance in the representation for lines and cables when performing studies in which frequencies above the 25th harmonic are important. For WPP, common ‘pi’ model is used to represent the transmission line and for a cable, it is more accurate to use a ‘modified’ pi model [8,29]. The later includes the effect of the ‘long-line effect’ and skin effect correction factors and is shown in Fig. 2.

Fig. 2. Modified pi model

The impedance, Z and shunt admittance, Y are calculated as in (9) and (10). The frequency dependence is a model using frequency polynomial characteristic according to (11)

(9)

(10)

)532.0187.0(1 hRR += (11)

Zc = / = characteristic impedance, γl = √ , γ is the propagation constant and l is cable length.

MV collector For a large WPP that consists of large number of wind

turbine, it is more convenient to model the whole plant as a medium voltage collector model, where several numbers of WTs are combined and modeled together. Several factors to consider are such as different length of the cable connecting each WTs, current injection of each WT and the calculation complexity. The connection between WTs is not identical; it might vary in cable length and type. Hence, the impedance of each cable connecting WTs will be different and the evaluation of this can be complex. The representation of equivalent network for collector system is done as referred to [8]. The number of radial collector points is Y and for each radial, there are X number of wind turbines connected. The current at each radial collector, IY can be written as:

YyII x

x xy ,....2,1,1

== ∑ = (12)

Where y is the respective number of radial collector and Ix is current generated by each WT of X number WT on each radial

network. The equivalent impedance is given by:

∑ =− ⎟⎟⎠

⎞⎜⎜⎝

⎛= x

x xyeqn ZyxZ

1

2

(13)

Where Zx is the cable impedance between xth and (x + 1)th wind turbine in the y-radial network. The equivalent impedance of each radial collector is in parallel to each other and can be further simplified into equivalent impedance, ZRC.

∑∑

= −

=⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

= Y

y yeqnY

yy

yRC Z

X

XZ

1

2

1

(14)

Capacitance of each radial collector is added together as they only vary with voltage and voltage in the collector system can be regarded as uniform. Hence, the capacitance, CRC: ∑ ∑= =

= Y

y

y

x xRC CC1 1

(15)

Where Cx is the cable shunt capacitance between xth and (x + 1)th wind turbine in the y-radial network.

Transformer For transformer model, two types of impedances are usually

taken into account, which are the leakage impedance and the magnetizing impedance. It is important to remember that the resistive component of this impedance is not constant with frequency. It is also important to include the transformer phase-shift effect. The typical model of transformer is given in Fig. 3.

Fig. 3. Transformer model

Wind Turbine Generator

The most of WT the generator types employ the induction machine. The equivalent circuit of induction model commonly used is given in Fig. 4 [11]. The model ignores the magnetizing impedance and taking into account the harmonic frequency dependent and the skin effect.

Fig. 4. Equivalent circuit of induction machine model

The motor impedance at fundamental frequency (h = 1) is:

( )2

tanh1sinh/1cos1sinh

lZ

llZ

Y

lZZ

cc

c

γγγ

γ

π

π

=−=

=

2472

Zmotor = R1 + R2/s + j(X1+X2)

and, at harmonic frequencies, the impedance machine is given by:

()1(

1Xjhs

hhbhaRZ

hBmotor +

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −±±+=

Where s

rsh h

hs

ωωω

±−±

=

EXAMPLE – 3 BUS SYSTEM In this section, an example of a 3-bus system

[21] is analyzed for the resonance condition. Toutline in the previous sections is applied toability of identifying the resonance. The 3-bushown in Fig. 5.

Fig. 5. The example-Three bus system

Table 1 Parameters of three-bus systemThree-bus system parameters

Ls = 0.796 mH Rs= 0.04 Ω LT1 = LRT1 = RT2= 0.835 Ω C1 = C2 = 3.448 µF A simple three-bus system is analyzed forresonance analysis to illustrate the concept. Ntwo shunt capacitances present located at bBus 2 is in between the RT1 and LT2. Tresonance frequencies of the system, fR1 and fR2

HzXXf SCR 30381 ==

HzXXXf LSCR 578)2(2 =+=

Fig. 6. The frequency scan of the three-bus

500 1000 1500 20000

500

1000

1500

2000

2500

Mag

nitu

de (

abs)

Frequency (Hz)

Bus 1

Bus 2

Bus 3

of the induction

)2X+ (16)

m as presented in The three methods o demonstrate the us system used is

m

m

LT2 = 10.61 mH

r the three type Note that there are bus 1 and bus 3. The approximate 2 are as follows:

system

Fig. 7. The modal impedances o

Table 2 The eigenvalues of the

1.0e+004 * -0.0026 + 1.9457i 0 0 0 -0.0026 - 1.9457i 0 0 0 -0.0039 + 0.36

0 0 0 The admittance matrix of the sfrequency scan and HRMA merepresentation is also formed baselements present. The frequency simpedance of the system are shorespectively. The eigenvalue obtamethod is shown in the Table 2.

The results from the three metphenomenon observed at all three method shows that the resonance is buses whereby the bus 2 experienfrequencies. It is difficult to identibus at resonance condition. By analysis, it is seen that the modal ithat one mode experiences resonanMode 1 has a resonance frequency aresonance frequency at 578 Hz, experiences any resonance condeigenvectors and participation factaccordingly for the resonance frequgiven by the largest entries in the eithe bus 1 is the most participating bu3 for the mode 3. Hence, it providepotential location for injecting signaFrom Table 2, the natural frequencyresonance frequency obtained byapproximately the same; fR1 = 3097

CONCLUSIO

Harmonic analysis in WPP is inv

2500 3000 3500

0 500 1000 15000

500

1000

1500

2000

2500

Frequency [Hz]

Mag

nitu

de (

abs)

Modal 1

Modal 2

Modal 3

7

f the three-bus system

state-space methods

0 0

627i 0 -0.0039 - 0.3627i

system is formed for the ethod and the state-space sed on the energy storage scan result and the modal

own in Fig. 6 and Fig. 7, ained from the state-space

thods reveal the resonance buses. The frequency scan experienced by all the three

nces the resonance at both fy the significance of each

using modal impedance impedance plot only shows nce at a specific resonance. at 3040Hz and mode 3 has a

while mode 2 does not dition. The corresponding tors (PC) can be obtained

uency and the highest PC is igenvector. For the mode 1, us and the bus 3 and the bus es the information the most als for cancelling harmonics. y which corresponding to the y state-space methods are Hz and fR2 = 577 Hz.

ONS volving a large and complex

2000 2500 3000 3500

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network. Besides the conformance of the harmonic levels in the network to the requirement, the possible resonance phenomenon is a concern in the large WPP. The harmonic analysis and the resonance analysis able to assist in identifying the occurrence as shown by resonance analysis performed on the three-bus system.

ACKNOWLEDGMENT This research work has been supported by the Spanish

Ministry of Science and Innovation under the projects ENE 2008-06841-C02/ALT and TRA2009-0103.

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