A Selected Harmonics Compensation Method with Distributed Energy Resources

Damoun Ahmadi and Jin Wang Department of Electrical and Computer Engineering

The Ohio State University Columbus, OH 43210

Email: wang@ece.osu.edu

Abstract-- Distributed Energy Resources (DERs) are expected to provide ancillary functions including reactive power generation and harmonics compensation. Meanwhile, multilevel inverters and two level converters with Optimal Pulse Width Modulation (OPWM) will be commonly used for DERs at high power levels. The challenge is to realize harmonic compensation with limited switching angles. This work presents a four equations based method to realize Selected Harmonics Compensation (SHC) and power generation with DERs. Since the power converter is requested to realize both harmonics filtering and power generation; limitations on harmonics compensation at the given power are discussed. Power Hardware-in-the-loop based verifications are shown at the end of the paper. Index Terms--Active Power Filter, Distributed Energy Resources, Power Hardware In the Loop, Selected Harmonics Compensation

I. INTRODUCTION

The development of different types of distributed generation, such as fuel cells, photovoltaics, and wind turbines, have provided great opportunities for the implementations of medium/high power inverters. In these applications, the frequency of the Pulse Width Modulation (PWM) is often limited by switching losses and electromagnetic interferences caused by high dv/dt. Thus, to overcome these problems, Selective Harmonic Elimination (SHE) based Optimal Pulse Width Modulation (OPWM) are often utilized in both two-level inverters and multilevel inverters to reduce the switching frequency and the Total Harmonic Distortion (THD) [1-12]. Pulse Width Modulation (PWM) method was proposed for inverters in 1960s and digitalized in 1970s [13-14]. Soon after the birth of the basic PWM method, in 1964, Turnbull proposed the SHE idea [15]. In this method, harmonic components are described as functions of the switching angles in trigonometric terms. For multilevel inverters, If N is the total number of switching transitions, the Fourier series expansion of the symmetric staircase waveform can be expressed as:

)sin())cos(...)(cos(4)( 1,...5,3,1

tmmmmVtV N

m

dc ωθθπ

ω += ∑∞

=

(1)

where m is the order of the harmonic, and Nθ are the thN switching angle. Based on (1), the following group of equations can be utilized to calculate the N switching angles

and realize the selective harmonic elimination up to thmorder. It should be noted that the value of “m” could be much higher than “N”.

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=++++

=++++=++++

=++++

0)13cos(...)cos()cos()cos(...

0)7cos(...)7cos()7cos()7cos(0)5cos(...)5cos()5cos()5cos(

))cos(...)cos()cos()(cos(4

321

321

321

321

N

N

N

fNdc

mmm

VV

θθθθ

θθθθθθθθ

θθθθπ

(2)

In this equation group, the first equation is used to guarantee the amplitude of the fundamental component ( FV ), and the other equations are utilized to ensure the elimination of selected harmonics. Thus, by calculating the N switching angles, N-1 number of harmonics can be eliminated [16], [17]. Most of the methods used for SHE are eventually based on solving complex groups of equations. Then, for higher number of switching transients, it is quite difficult or time consuming to solve these nonlinear equations with current computation methods. Based on harmonics injection and equal area criteria, the authors have recently proposed the four-equations method for multilevel inverters. In this method, regardless the number of voltage levels, few simple equations is needed for switching angle calculations [18-24]. This method can be also utilized for harmonics compensation and power generation as shown in this paper. The paper is organized in the following way: Section II provides the system topology for harmonics compensation. Section III presents the detailed description of the improved four-equation based method for harmonics compensation. The limitations of the proposed method are studied in section IV. In section V, online harmonics detection method is discussed, and finally, different case studies for real time verification and conclusion are provided in section VI and section VII respectively.

204U.S. Government work not protected by U.S. copyright

II. SELECTED HARMONICS COMPENSATION AND POWER GENERATION

Nowadays, distributed energy resources (DERs) are being considered as promising generation sources to meet the continuously increasing energy demand and to improve reliability of electric power systems [25-29]. To increase the system efficiency and maximize the return of the investments on DERs, ancillary functions such as harmonics and power compensations have also been considered. This means that the distributed recourses can be utilized as Active Power Filters (APF) while supplying active/reactive power to the load and the grid [30-38]. In this case, DERs will be controlled to:

1) Generate desired active and reactive power, for the load and the grid;

2) Compensate selected harmonic currents caused by nonlinear loads.

The system topology is shown in Fig. 1. It should be noted that this grid-tie inverter will often work with high modulation indices. Equation (3) shows the relation between the current from local distributed sources, grid voltage vg, inverter output voltage vm, and loads current:

hmfmhmmsgmhmsgm vvxixivxiivv +=×+×+=++= )()()( 22 (3)

where fv , is the fundamental voltage, hmi and hmv

, are the load harmonics, and the related harmonics voltage in DERs respectively. To generate this voltage, multilevel inverters with staircase waveforms using DERs can be utilized. By using the Fourier series expansion of the staircase waveform in multilevel inverters, the following group of equations can be utilized to calculate the switching angles (

Nθθθ ,...,, 51 ) and realize the selective harmonics:

21 sss iii +=

hms ii +lv

lxgv

hms ii +2mx

mv

1si

Fig. 1. Harmonic and power compensation by multilevel inverters

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=++++

=++++=++++

=++++

mN

N

N

fNdc

Vmmm

VV

VV

)13cos(...)cos()cos()cos(...

)7cos(...)7cos()7cos()7cos()5cos(...)5cos()5cos()5cos(

))cos(...)cos()cos()(cos(4

321

7321

5321

321

θθθθ

θθθθθθθθ

θθθθπ

(4)

In (4), fV is the fundamental component of output voltage in

multilevel inverters and mVVV ,...,, 75 are the voltages that are required to compensate the selected harmonics in the load.

III. THE PROPOSED FOUR-EQUATIONS METHOD FOR HARMONICS COMPENSATION

The four equations based method is modified and proposed for harmonics compensation in APF by DERs, as shown in Fig. 2. This method is an iteration process, and using the equal area criteria. This idea is depicted in Fig. 3 for staircase waveform.

The following equation group forms the foundation of the method. The first equation is to calculate the junction point ( kδ ) of the reference and the voltage level; it can be solved with Newton-Raphson method:

))cos(

)sin()...5sin(arctan(

51

)(

kF

kmsks

k

iidc

k V

mhhV

δ

δδδ

++=

∑=

(5)

Fig. 2. Four equation based method in multilevel inverters for APF

π

Fig. 3. Equal area criteria in four equations based method

205

The second equation is to find the switching angles ( kθ ):

)))cos()(cos())...5cos()5(cos(5

))cos()(cos((*/1

115

11

1

1)(

1)()(

−−

−−

−

==

−−−−

−+−= ∑∑

kkms

kks

kkFk

k

iidck

k

iidckdck

mmmhh

VVVV

δδδδ

δδδδθ (6)

The third equation is to find the harmonics content ( mh ) in the staircase waveform:

)))(cos()(cos(2

1

)(kk

N

k

kdcm mm

mV

h θπθπ

−−=∑=

(7)

The fourth equation is to calculate the new reference waveform ( refV ) for switching angle calculation:

)sin()sin( tmhtVV msFref ωω −= (8)

where msh is the combination of the harmonics in the staircase waveform and the voltage reference for harmonics compensation based on the load current ihm, as shown in Fig. 1. Thus, msh can be calculated as:

)(,...2,1

)( ireferencem

iter

iimms hhh −

=

−= ∑ (9)

With online iterations of above equations, active and reactive power generation can be controlled by regulating FV , and at the same time, limited active power filter function can be realized by including )sin( tmhms ω− in the switching angle calculation.

IV. THE LIMITATIONS OF THE PROPOSED METHOD

Two limitations of the proposed methods are identified and discussed as follows:

A) Trade-off Between Fundamental Component Generation and Harmonics Compensation

To realize power generation at fundamental frequency and harmonics compensation at the same time, maximum output voltage is limited by the dc bus input. Without injecting a third harmonics, the first and foremost harmonic is the fifth harmonics. With zero-degree phase shift, the peak value of fifth harmonics will be added to the peak of fundamental voltage. This is illustrated in Fig. 4. To achieve the desired power generation and harmonics compensation, the peak value of (8) needs to be smaller than the summations of voltage levels of the multilevel inverter:

∑=−

− <−+

+−+= N

iidc

mreferencem

referenceFref Vtmhh

thhtVV

1)(

55

)sin()(

...)5sin()()sin(

ωωω (10)

where N is the number of voltage levels in the multilevel inverter. This tradeoff is also embodied in the available area for APF at a given power. For an APF approach, since all the selected harmonics are odd numbers, each harmonic increases the total area and limits the available area for the fundamental frequency. For a half-cycle sinusoidal signal at fundamental frequency, the total area needed for the signal equals the magnitude of FV2 . However, for harmonics compensation in APF, this area will equal:

mF Vm

VVVVareaTotal 2...112

72

522 1175 +++++= (11)

Therefore, by adding these harmonics, the power generation capability of the system can be decreased when APF function is emphasized. Conversely, if loads THD are increased significantly, APF function will be limited at a given active and reactive power request.

B) Junction Points Calculation

In the proposed method, the reference voltage starts as a pure sinusoidal signal, and is then updated in each sampling cycle with additional harmonics compensation components. Without compensating the harmonics current generated by the load, after few irritations, the harmonics in the voltage reference will decrease dramatically and there will be no problem in calculating the junction points of the reference and the voltage levels. However, when the harmonics compensation for load current is also included, the selected harmonics significantly change the reference signal. Thus, as shown in Fig. 5, there could be multiple junction points between the reference and one single voltage level, which can lead to problems in angle calculation.

Fig. 4. 5th harmonics injection for APF.

206

Fig. 5. Multiple junction point for the same dc level

V. ONLINE HARMONICS DETECTION METHOD

For active power filtering, swift and reliable harmonics detection algorithm is essential. Thus, a simple yet accurate selected harmonics detection method is proposed, based on the Instantaneous Reactive Power (IRP) theory. As illustrated in Fig. 6, current at any frequency, including the fundamental and selected harmonics components, can be individually transformed from a-b-c to d-q axes with (12).

a

b

c

d

q

ai

bi

ci

t5ωt1ω

tmω

0θωθ += tmm

Fig. 6. Different frequencies transformation s from a-b-c to d-q.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+−=⎥

⎦

⎤⎢⎣

⎡

c

b

a

mmm

mmm

qm

dm

iii

ii

)3

2sin()3

2sin(sin

)3

2cos()3

2cos(cos

32

πθπθθ

πθπθθ (12)

where mω is the angular speed at concerned frequency. For selected harmonics detection, if (12) is applied to load current at a defined mm fπω 2= , any components that have frequencies other than mf will not be resulting in any dc value in d-q axes, thus can easily be eliminated by a low-pass filter. Based on this idea, the measured currents will be transformed from a-b-c to d-q axes at the selected harmonics frequencies. The dc result for each transformation equals the harmonic magnitude for that selected frequency. Therefore, magnitude for each selected harmonics can be detected with minimum calculation.

VI. REAL TIME VERIFICATION FOR THE PROPOSED METHOD

To verify the performance of the proposed method, real-time simulation based Hardware-in-the Loop (HIL) test is utilized.

In general, HIL is a cost effective and flexible technique to test a control algorithm of an embedded system, when the real system test is either too expensive or not available to the algorithm developer. In this paper, both Control Hardware-in-the-loop (CHIL) and Power Hardware in the Loop (PHIL) are utilized to evaluate the proposed methods. In CHIL, all the electrical components, including the multilevel inverter is simulated at real-time. The simulations communicate with a real TI DSP controller, which controls the multilevel inverter in the simulation. Thus, the proposed algorithm can be tested at real-time. In the PHIL, the grid and nonlinear loads are simulated at real time; but the multilevel inverter is built with real 1200 V, 200 A rated Intelligent Power Modules (IPM). The control unit of the multilevel inverter is the same DSP board in HIL. The dc voltages of the multilevel inverter are provided with multiple rectifiers’ power through individual isolation transformers. The system diagram of the PHIL and different parts of the setup are shown in Fig. 7 and Fig. 8 respectively. The real-time simulator simulates the grid and nonlinear load at real-time. Simulated results, including load currents, grid voltage and grid current, are converted to analog signal as the feedback and input to the DSP control board of the multilevel inverter. At the same time, the voltage output of the multilevel inverter is sensed and fed to the real-time simulator as the control input of a controlled voltage source. Thus, all the algorithms can be tested as if they are tested in the field but without real high current.

Fig. 7. Power hardware in the loop diagram for active power filter

Fig. 8. Real time simulator and multilevel inverters setup for PHIL

207

A) CHIL based simulation for selected harmonics compensation with high number of dc levels

With CHIL, it is possible to test the algorithms with more voltage levels without building more circuits. For selected harmonics compensation, as an example, an 11-level multilevel inverter is simulated to compensate 0.2 p.u of 5th and 0.14 p.u. of 7th harmonics of a three-phase nonlinear load. The line impedance, from the multilevel inverters’ side ( mX ), and from the load side ( LX ) are chosen 3% and 4% respectively which are typical magnitudes for the power systems. The simulated parameters are listed in Table 1. All results for current and harmonics are shown in per unit. Load current, multilevel inverter’s output voltage and the grid current are shown in Fig. 9 (a), 9(b), and 9(c) respectively. Since it is a real-time simulation, the results are monitored with a digital Oscilloscope. The results show that harmonics from the nonlinear load is compensated effectively. The grid side currents have much less harmonics contents than the load currents. The measured THD of grid current is less than 3%.

Table 1. Real time simulation parameters for HIL

Parameters Grid Voltage

Fundamental Current

Each DC Bus

Magnitude mX LX

Per Unit (P.U)

5.0 1.0 1.0 0.03 0.04

(a)

(b)

(c)

Fig. 9. HIL results for APF; (a) load current (b) multilevel inverters voltage (c) the grid current.

B) PHIL based simulation

The simulation parameter of PHIL based verification is summarized in Table 2. The compensation target is 0.2 p.u of 5th and 0.14 p.u of 7th harmonics for a single phase nonlinear load with a five level inverter:

Table. 2. Real time simulation parameters for PHIL Parameters Grid

Voltage Fundamental

Current Each DC

Bus Magnitude

mX LX

Per Unit (P.U)

2.0 1.0 1.0 0.03 0.04

Case 1: Selected harmonic compensation without active power injection:

In this case, the multilevel inverter is just used for harmonics compensation. As shown in Fig. 10, the THD in the grid current is decreased significantly to less than 3%. Since, there is no need to increase the fundamental voltage for power generation; the result shows that the area under the dc levels can be utilized effectively for the harmonics compensation.

Fig. 10. SHE without power compensation.

Case 2: Selected harmonic compensation with active power injection:

Limited amount of active power generation can be provided together with selected harmonic compensation. When effectiveness of selected harmonic compensation is to limit the THD of the grid current to 5% with the same nonlinear load, the generated active power can be as high as 1 p.u. The results are shown in Fig. 11. At higher active power injection, the grid current THD will not be improved as desired. This is shown in Fig. 12, where the active power injection is increased to be more than 1 p.u.

208

Fig. 11. SHE with power compensation for low power

Fig. 12. SHE for high power generation from the renewable sources

VII. CONCLUSION

In this paper, a simple and practical method is proposed for selected harmonics compensation and power generation for distributed energy resources with multilevel inverter interfaces. Methods for both switching angle calculation and harmonics detection are proposed. The trade-off between harmonic compensation and power generation is discussed based on proposed switching calculation method. HIL and PHIL based simulation/experimental results verify the effectiveness of the proposed method and the discussions on the trade-off between harmonics compensation and power generation.

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