41.2.3.1 Current Reference Generation

There are many possibilities to determine the reference current required to compensate the nonlinear load. Normally, shunt active power filters are used to compensate the displacement power factor and low-frequency current harmonics generated by nonlinear loads. One alternative to determine the current reference required by the VSI is the use of the instantaneous reactive power theory, proposed by Akagi [1], the other one is to obtain current components in d-q or synchronous reference frame [2], and the third one is to force the system line current to follow a perfectly sinusoidal template in phase with the respective phase-to-neutral voltage.

Instantaneous reactive power theory

This concept is very popular and useful for this type of application and basically consists of a variable transformation from the a, b, c reference frame of the instantaneous power, voltage, and current signals to the α, β reference frame. The transformation equations from the a, b, c reference frame to the α, β coordinates can be derived from the phasor diagram shown in Fig. 41.9.

The instantaneous values of voltages and currents in the α, β coordinates can be obtained from the following equations:

$\left[\begin{array}{c}\hfill v_{\alpha }\hfill \\ \hfill v_{\beta }\hfill \end{array}\right]=\left[A\right]\cdot \left[\begin{array}{c}\hfill v_a\hfill \\ \hfill v_b\hfill \\ \hfill v_c\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_{\alpha }\hfill \\ \hfill i_{\beta }\hfill \end{array}\right]=\left[A\right]\cdot \left[\begin{array}{c}\hfill i_a\hfill \\ \hfill i_b\hfill \\ \hfill i_c\hfill \end{array}\right]$

  (41.1)

where A is the transformation matrix, derived from Fig. 41.9 and is equal to

$\left[A\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{lll}1\hfill & -1/2\hfill & -1/2\hfill \\ 0\hfill & \sqrt{3}/2\hfill & -\sqrt{3}/2\hfill \end{array}\right]$

  (41.2)

This transformation is valid if and only if $v_a\left(t\right)+v_b\left(t\right)+v_c\left(t\right)$ is equal to zero and also if the voltages are balanced and sinusoidal. The instantaneous active and reactive power in the α, β coordinates are calculated with the following expressions:

$p\left(t\right)=v_{\alpha }\left(t\right)\cdot i_{\alpha }\left(t\right)+v_{\beta }\left(t\right)\cdot i_{\beta }\left(t\right)$

  (41.3)

$q\left(t\right)=-v_{\alpha }\left(t\right)\cdot i_{\beta }\left(t\right)+v_{\beta }\left(t\right)\cdot i_{\alpha }\left(t\right)$

  (41.4)

It is evident that p(t) becomes equal to the conventional instantaneous real power defined in the a, b, c reference frame. However, in order to define the instantaneous reactive power, Akagi introduces a new instantaneous space vector defined by expression (41.4) or by the vector equation:

$q=v_{\alpha }x{i}_{\beta }+v_{\beta }x{i}_{\alpha }$

  (41.5)

where the vector q is perpendicular to the plane of α, β coordinates, to be faced in compliance with a right-hand rule; v_α is perpendicular to i_β; and v_β is perpendicular to i_α. The physical meaning of the vector q is not “instantaneous power” because of the product of the voltage in one phase and the current in the other phase. On the contrary, v_αi_α and v_βi_β in Eq. (41.3) obviously mean “instantaneous power” because of the product of the voltage in one phase and the current in the same phase. Akagi named the new electric quantity defined in Eq. (41.5) “instantaneous imaginary power,” which is represented by the product of the instantaneous voltage and current in different axes, but cannot be treated as a conventional quantity.

The expression of the currents in the $\alpha -\beta$ plane, as a function of the instantaneous power, is given by the following equation:

$\begin{array}{c}\left[\begin{array}{c}\hfill i_{\alpha }\hfill \\ \hfill i_{\beta }\hfill \end{array}\right]=\frac{1}{v_{\alpha }^2+v_{\beta }^2}\cdot \left\{\left[\begin{array}{cc}\hfill v_{\alpha }\hfill & \hfill v_{\beta }\hfill \\ \hfill v_{\beta }\hfill & \hfill -v_{\alpha }\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill p\hfill \\ \hfill 0\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill v_{\alpha }\hfill & \hfill v_{\beta }\hfill \\ \hfill v_{\beta }\hfill & \hfill -v_{\alpha }\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill 0\hfill \\ \hfill q\hfill \end{array}\right]\right\}\\ \text{≡}\left[\begin{array}{c}\hfill i_{\alpha p}\hfill \\ \hfill i_{\beta p}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill i_{\alpha q}\hfill \\ \hfill i_{\beta q}\hfill \end{array}\right]\end{array}$

  (41.6)

and the different components of the currents in the $\alpha -\beta$ plane are shown in the following expressions:

$i_{\alpha p}=\frac{v_{\alpha }p}{v_{\alpha }^2+v_{\beta }^2}$

  (41.7)

$i_{\alpha q}=\frac{v_{\beta }q}{v_{\alpha }^2+v_{\beta }^2}$

  (41.8)

$i_{\beta p}=\frac{v_{\beta }p}{v_{\alpha }^2+v_{\beta }^2}$

  (41.9)

$i_{\beta q}=\frac{-v_{\alpha }q}{v_{\alpha }^2+v_{\beta }^2}$

  (41.10)

From Eqs. (41.3) and (41.4), the values of p and q can be expressed in terms of the dc components and the ac components, that is,

$p=\overline{p}+\tilde{p}$

  (41.11)

$q=\overline{q}+\tilde{q}$

  (41.12)

where

$\overline{p}$ is the dc component of the instantaneous power p and is related to the conventional fundamental active current,

$\tilde{p}$ is the ac component of the instantaneous power p; it does not have average value and is related to the harmonic currents caused by the ac component of the instantaneous real power.

$\overline{q}$ is the dc component of the imaginary instantaneous power q and is related to the reactive power generated by the fundamental components of voltages and currents.

$\tilde{q}$ is the ac component of the instantaneous imaginary power q and it is related to the harmonic currents caused by the ac component of instantaneous reactive power.

In order to compensate reactive power (displacement power factor) and current harmonics generated by nonlinear loads, the reference signal of the shunt active power filter must include the values of $\tilde{p}$, $\overline{q}$, and $\tilde{q}$. In this case, the reference currents required by the shunt active power filters are calculated with the following expression:

$\left[\begin{array}{c}\hfill i_{c,\alpha }^{⁎}\hfill \\ \hfill i_{c,\beta }^{⁎}\hfill \end{array}\right]=\frac{1}{v_{\alpha }^2+v_{\beta }^2}\cdot \left[\begin{array}{cc}\hfill v_{\alpha }\hfill & \hfill v_{\beta }\hfill \\ \hfill v_{\beta }\hfill & \hfill -v_{\alpha }\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill {\tilde{p}}_L\hfill \\ \hfill {\overline{q}}_L+{\tilde{q}}_L\hfill \end{array}\right]$

  (41.13)

The final compensating currents including the zero-sequence components in a, b, c reference frame are the following:

$\left[\begin{array}{c}\hfill i_{c,a}^{⁎}\hfill \\ \hfill i_{c,b}^{⁎}\hfill \\ \hfill i_{c,c}^{⁎}\hfill \end{array}\right]=\sqrt{\frac{2}{3}}\cdot \left[\begin{array}{ccc}\hfill \frac{1}{\sqrt{2}}\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill \frac{1}{\sqrt{2}}\hfill & \hfill \frac{-1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill \\ \hfill \frac{1}{\sqrt{2}}\hfill & \hfill \frac{-1}{2}\hfill & \hfill \frac{-\sqrt{3}}{2}\hfill \end{array}\right]\cdot \left[\begin{array}{c}\hfill -i_0\hfill \\ \hfill i_{c,\alpha }^{⁎}\hfill \\ \hfill i_{c,\beta }^{⁎}\hfill \end{array}\right]$

  (41.14)

where the zero-sequence current component i₀ is equal to $\frac{1}{\sqrt{3}}\left(i_a+i_b+i_c\right)$. The block diagram of the circuit required to generate the reference currents defined in Eq. (41.14) is shown in Fig. 41.10.

The advantage of instantaneous reactive power theory is that real and reactive power associated with fundamental components are dc quantities. These quantities can be extracted with a low-pass filter. Since the signal to be extracted is dc, filtering of the signal in the $\alpha -\beta$ reference frame is insensitive to any phase-shift errors introduced by the low-pass filter, improving compensation characteristics of the active power filter. The same advantage can be obtained by using the synchronous reference frame method, proposed in [2]. In this case, transformation from a, b, c axes to d-q synchronous reference frame is done.

Effects of the low-pass filter design characteristics in compensation performance

A Butterworth filter is normally used due to the adequate frequency response. A second-order filter offers an appropriate relation between the transient response and the required attenuation characteristic. Higher-order filters achieve better filtering characteristic, but the settling time is increased. Since the low-pass filter cannot eliminate completely the low-frequency harmonic contained in the p and q signals, the shunt active power filter cannot compensate the entire low-frequency harmonic contained in the load current. Normally, the cutoff frequency is equal to 127 Hz with an attenuation factor of 15 dB for the first ac component to be eliminated, which means an 82.2% attenuation of the fifth and seventh harmonic components (Fig. 41.11). The expression that relates the system line current harmonic distortion with the LPF cutoff frequency and load power factor is shown in Eq. (41.15):

$\begin{array}{l}TH{D}_{\mathit{isys}}\\ =\frac{\sqrt{{\displaystyle \sum _{h=6k}^{\infty }}\frac{1}{{\left(f_n/f_c\right)}^4+1}\left[\left(\left(h^2+1\right)/\left(h^2-1\right))-\left(1/\left(h^2-1\right)\right)cos(2\phi \right)\right]}}{cos\left(\phi \right)}\end{array}$

  (41.15)

with k=1, 2, 3,… and f_h is the frequency of the harmonic component of order h and f_c is the LPF cutoff frequency.

Fig. 41.12 shows the total harmonic distortion (THD) in the line currents introduced by the second-order Butterworth filtering characteristics, as a function of the cutoff frequency and considering a 50 Hz ac mains frequency. The harmonic distortion of the compensated line current depends on the load displacement power factor, as shown in Eq. (41.15). If the cutoff frequency of the low-pass filter is changed, the active power filter compensation performance is affected and the transient response of the control scheme.

Effects of the supply voltage distortion in compensation performance

One of the most important characteristics of the instantaneous imaginary power concept is that in order to obtain the current reference signal required to compensate reactive and harmonic current components, the system phase-to-neutral voltages are used. In general, purely sinusoidal voltages are considered in previously reported analysis. In case voltage is purely sinusoidal, the dc components of p and q in the α–β plane are related with the fundamental components in the real a, b, c reference system. This is not the case if the system voltages are distorted or unbalanced, as it is demonstrated below.

It is assumed that the supply voltages have harmonic distortion, and these are represented by

$\begin{array}{c}{v}_a\left(t\right)=V_{1}cos\left(\omega t\right)+V_{h}cos\left[h\left(\omega t-{\delta }_h\right)\right]\\ v_b\left(t\right)=V_{1}cos\left(\omega t-\frac{2\pi }{3}\right)+V_{h}cos\left[h\left(\omega t-{\delta }_h-\frac{2\pi }{3}\right)\right]\\ v_c\left(t\right)=V_{1}cos\left(\omega t+\frac{2\pi }{3}\right)+V_{h}cos\left[h\left(\omega t-{\delta }_h+\frac{2\pi }{3}\right)\right]\end{array}$

  (41.16)

Since the harmonic voltage component introduces a dc component in p and q, compensation performance of the shunt active power filter is reduced, as shown in Fig. 41.13. The larger the harmonic distortion in the system voltage is, the active power filter performance is more affected.

Effects of the supply voltage unbalance in compensation performance

Voltage unbalance also affects the active power filter compensation performance. In this analysis, the phase-to-neutral voltages of the ac supply are equal to

$\begin{array}{c}{v}_a\left(t\right)=V_{1}cos\left(\omega t\right)\\ v_b\left(t\right)=V_1\left(1+m\right)cos\left(\omega t-\frac{2\pi }{3}\right)\\ v_c\left(t\right)=V_1\left(1-m\right)cos\left(\omega t+\frac{2\pi }{3}\right)\end{array}$

  (41.17)

with $0<m<1$.

If the low-pass filter is considered as ideal and the active filter follows exactly the current references, the compensated supply current (phase a) is

$\begin{array}{c}{i}_{Sa}=I_{1}cos\left(\phi \right)cos\left(\omega t\right)+\frac{\sqrt{3}}{3}m\left[I_{1}sin\left(\phi \right)cos\left(3\omega t\right)\right.\\ +\left.I_1,,cos,,\left(3\omega t-\phi \right)\right]\end{array}$

  (41.18)

In this case, the active power filter is not able to fully compensate the system line current, since compensation performance depends on the voltage unbalance magnitude.

The unbalance is defined as a function of the positive and negative-sequence component as

$\begin{array}{c}{V}_{a1}=\frac{1}{3}\left[V_a+a^2{V}_b+a{V}_c\right]\\ V_{a2}=\frac{1}{3}\left[V_a+a{V}_b+a^2{V}_c\right],a=1\angle 120\mathrm{degrees}\\ \mathrm{Unbalance}=\frac{\left|V_{a2}\right|}{\left|V_{a1}\right|}=\frac{\sqrt{3}}{3}m\end{array}$

  (41.19)

The harmonic distortion is equal to

$TH{D}_i=\frac{\sqrt{3}}{3}m$

  (41.20)

The relation between line current harmonic distortion and voltage unbalance is shown in Fig. 41.14.

Effects of the time delay introduced by the dsp in compensation performance

If a DSP is used to derive the reference generation signals, a time delay, T, associated with the processing time is introduced in the calculation of the reference signals. Considering a simultaneous sampling, the ac mains compensated line current with an ideal low-pass filter in the control scheme (phase a) is

$\begin{array}{c}{i}_{Sa}=I_{1}cos\left(\omega t\right)\left[cos\left(\phi \right)+sin\left(\omega T\right)\right]+\mathrm{\Delta }{V}_{DC}cos\left(\omega t\right)\\ +{\displaystyle \sum _{n=2k-1}^{\infty }}{I}_n\left[1-cos\left(n\omega T\right)\right]cos\left[n\left(\omega t-\phi \right)\right]\\ +{\displaystyle \sum _{n=2k-1}^{\infty }}{I}_{n}sin\left(n\omega T\right)sin\left[n\left(\omega t-\phi \right)\right]\end{array}$

  (41.21)

The time delay introduced by the DSP affects the compensation performance, especially when fast changes are present in the load current. Nevertheless, when T is small ($T>50\mathrm{s}$), the effect in compensation performance is negligible.

In four-wire systems, unbalances in the load current also affect the generation of the adequate current reference signal that will assure high-performance active power filter compensation. This assumption can be proved by considering the following load currents:

$\begin{array}{c}{i}_{La}=I_{1}cos\left(\omega t-\phi \right)+{\displaystyle \sum _{n=2k-1}^{\infty }}{I}_{n}cos\left[n\left(\omega t-\phi \right)\right]\\ \begin{array}{c}{i}_{Lb}=I_1\left(1+l\right)cos\left(\omega t-\phi -\frac{2\pi }{3}\right)\\ +{\displaystyle \sum _{n=2k-1}^{\infty }}{I}_n\left(1+l\right)cos\left[n\left(\omega t-\phi -\frac{2\pi }{3}\right)\right]\end{array}\\ \begin{array}{c}{i}_{Lc}=I_1\left(1-l\right)cos\left(\omega t-\phi +\frac{2\pi }{3}\right)\\ +{\displaystyle \sum _{n=2k-1}^{\infty }}{I}_n\left(1-l\right)cos\left[n\left(\omega t-\phi +\frac{2\pi }{3}\right)\right]\end{array}\\ i_{LN}=i_{La}+i_{Lb}+i_{Lc}\end{array}$

  (41.22)

with $0<l<1$, k=1, 2, 3,…

Eq. (41.23) shows the presence of low-frequency even harmonics in the active power expression introduced by the load current unbalanced. These low-frequency current components force to reduce the cutoff frequency of the low-pass filter, close to 100 Hz, which affects the transient response of the active power filter:

$\begin{array}{c}p=\frac{3}{2}{V}_1\left[I_{1}cos\left(\phi \right)+{\displaystyle \sum _{n=2k-1}^{\infty }}{I}_{n}cos\left[\left(n+1\right)\omega t-n\phi \right]\right.\\ +\left.{\displaystyle \sum _{m=2k-1}^{\infty }},I_m,,cos,,\left[\left(m-1\right)\omega t-m\phi \right]\right]\mathrm{with}k=1,2,3,\dots \end{array}$

  (41.23)

Synchronous reference frame algorithm

The block diagram of a current reference generator that uses the synchronous reference frame algorithm is shown in Fig. 41.15.

In this case, the real currents are transformed into a synchronous reference frame [2]. The reference frame is synchronized with the ac mains voltage and is rotating at the same frequency. The transformation is defined by

$\left[\begin{array}{c}\hfill i_d\hfill \\ \hfill i_q\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill cos\left(\omega t\right)\hfill & \hfill sin\left(\omega t\right)\hfill \\ \hfill sin\left(\omega t\right)\hfill & \hfill cos\left(\omega t\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_{\alpha }\hfill \\ \hfill i_{\beta }\hfill \end{array}\right]$

  (41.24)

As for the instantaneous reactive power theory, d and q terms are composed by a dc and multiple ac components, such as $i_d=i_{ddc}+i_{dac}$ and $i_q=i_{qdc}+i_{qac}$. The compensation reference signals are obtained from the following expressions: $i_{\mathit{dref}}=-i_{dac}$ and $i_{\mathit{qref}}=-i_{qdc}-i_{qac}$. The compensated currents generated by the shunt active power filter are obtained from Eq. (41.25):

$\left[\begin{array}{c}\hfill i_{\mathit{aref}}\hfill \\ \hfill i_{\mathit{bref}}\hfill \\ \hfill i_{\mathit{cref}}\hfill \end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\hfill \frac{1}{\sqrt{2}}\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill \frac{1}{\sqrt{2}}\hfill & \hfill -\frac{1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill \\ \hfill \frac{1}{\sqrt{2}}\hfill & \hfill -\frac{1}{2}\hfill & \hfill -\frac{\sqrt{3}}{2}\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill cos\left(\omega t\right)\hfill & \hfill -sin\left(\omega t\right)\hfill \\ \hfill 0\hfill & \hfill sin\left(\omega t\right)\hfill & \hfill cos\left(\omega t\right)\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_0\hfill \\ \hfill i_{\mathit{dref}}\hfill \\ \hfill i_{\mathit{qref}}\hfill \end{array}\right]$

  (41.25)

One of the most important characteristics of this algorithm is that the reference currents are obtained directly from the load currents without considering the source voltages. This is an important advantage since the generation of the reference signals is not affected by voltage unbalance or voltage distortion, therefore increasing the compensation robustness and performance. However, in order to transform from the $\alpha -\beta$ plane to the d-q synchronous reference frame, sine and cosine signals synchronized with the respective phase-to-neutral voltages are required. A phase-locked loop (PLL) per each phase, as the one shown in Fig. 41.16, must be used.

Since the algorithm used to obtain the reference current presents the same mathematical procedure and operations that the ones required in the instantaneous reactive power concept, the effects introduced by the filter and DSP time delay are similar. Unbalanced load currents generate a different harmonic spectrum in the synchronous reference frame, and low-order harmonic components appear in the reference signal. In order to separate these uncharacteristic low-frequency current components, the cutoff frequency of the low-pass filter must be reduced.

Peak detection method

There are other possibilities to generate the current reference signal required to compensate reactive power and current harmonics. Basically, all the different schemes try to obtain the current reference signals that include the reactive components required to compensate the displacement power factor and the current harmonics generated by the nonlinear load. Fig. 41.17 shows another scheme used to generate the current reference signals required by a shunt active power filter. In this case, the ac current generated by the inverter is forced to follow the reference signal obtained from the current reference generator. In this circuit, the distorted load current is filtered, extracting the fundamental component, i_l1. The band-pass filter is tuned at the fundamental frequency (50 or 60 Hz), so that the gain attenuation introduced in the filter output signal is zero and the phase-shift angle is 180 degrees. Thus, the filter output current is exactly equal to the fundamental component of the load current but phase shifted by 180 degrees. If the load current is added to the fundamental current component obtained from the second-order band-pass filter, the reference current waveform required to compensate only harmonic distortion is obtained. In order to provide the reactive power required by the load, the current signal obtained from the second-order band-pass filter I_l1 is synchronized with the respective phase-to-neutral source voltage (see Fig. 41.17) so that the inverter ac output current is forced to lead the respective inverter output voltage, thereby generating the required reactive power and absorbing the real power necessary to supply the switching losses and also to maintain the dc voltage constant.

The real power absorbed by the inverter is controlled by adjusting the amplitude of the fundamental current reference waveform, I_l1, obtained from the reference current generator. The amplitude of this sinusoidal waveform is equal to the amplitude of the fundamental component of the load current plus or minus the error signal obtained from the dc voltage control unit. In this way, the current signal allows the inverter to supply the current harmonic components, the reactive power required by the load, and to absorb the small amount of active power necessary to cover the switching losses and to keep the dc voltage constant (Fig. 41.18).

The main characteristic of this method is the direct derivation of the compensating component from the load current, without the use of any reference frame transformation [1,2]. Nevertheless, this technique presents a low-frequency oscillation problem in the active power filter dc bus voltage. To improve this technique, a modification of the previous scheme (Fig. 41.17) is shown in Fig. 41.19. The scheme is necessary for each phase. The expression for i_Ma is

$\begin{array}{c}{i}_{\mathit{Ma}}=\frac{I_{1}cos\left(\phi \right)}{2}+\frac{I_{1}cos\left(2\omega t-\phi \right)}{2}\\ +{\displaystyle \sum _{n=2k-1}^{\infty }}\frac{I_n}{2}\left\{cos\left[\left(n-1\right)\omega t-n\phi \right]\right.\\ +\left.cos,,\left[\left(n+1\right)\omega t-n\phi \right]\right\}\end{array}$

  (41.26)

with k=1, 2, 3,…

Fig. 41.20 shows the current harmonic distortion introduced by the low-pass filter used in the control scheme and the associated cutoff frequency. The current distortion of the compensated current depends on the phase angle of the fundamental load current component.

The supply voltage has no effect on the reference current generation. Synchronization with the ac mains voltage is the important issue in this scheme and in the synchronous reference frame theory. Unbalanced loads do not affect the reference generation. Nevertheless, the method cannot achieve active power balance in four-wire systems. The control circuit implementation of the peak detection method is simple and does not require complex calculation, so the processing time on a DSP is lower than the required in the two previous implementations, (T<10 s). The use of this method minimizes the distortion introduced on current harmonics.

Technical comparison of the three different techniques

In order to validate the effectiveness of the proposed analysis, a common industrial system is considered. The results are obtained using the previous equations and Matlab simulation results. The DSP delay introduced is using the processing time according to the DSP ADSP2187. The parameters considered are

$\begin{array}{c}TH{D}_{iL}=29\%\\ TH{D}_V=5\%\\ cos\left(\phi \right)=0.87\\ \mathrm{Unbalance}=3\%\end{array}$

Table 41.2 shows the numeric results.

Table 41.2

Results of the considered case

Technique	THDiS(%)	FP	Transient delay td (ms)
PQ	8.2	0.99	18.1
DQ	3.1	0.99	18.1
DPVM	2.3	0.99	56.8

Table 41.3 shows a comparison of the three different techniques analyzed. The effects of the nonideal conditions are described for each technique.

Table 41.3

Comparison of the techniques

Parameter	PQ	SRF	PDM
Load PF required to achieve full compensation	$PF>0.3$	$PF>0.3$	$PF>0.2$
Harmonic distortion voltage effect on the compensated current	$TH{D}_i\approx TH{D}_v$	0	0
Unbalanced voltage effect	$TH{D}_i\approx \mathrm{Unbalance}$	0	0
Dynamic response under load changes (in function of fc, $t_d=5\tau$)	Fast (load balanced)	Fast (load balanced)	Slow
Capability of load balance	Yes (It is necessary to reduce cutoff frequency fc)	Yes (it is necessary to reduce cutoff frequency fc)	No
DSP delay time introduced	Minimum	Minimum	$\approx 0$

In conclusion, it can be mentioned that the compensation performance of the different techniques is similar under ideal conditions, but under the presence of unbalanced and voltage distortion, the synchronous reference frame algorithm presents the best performance, since it is insensitive to voltage perturbations. It is fundamental to consider adequately the cutoff frequency in the filter used to extract the ac component in the different techniques. If the frequency is changed, the compensation performance is affected as well as the transient response of the control scheme. In four-wire systems, unbalance in the load current also affects the generations of the adequate current reference. Unbalanced load currents generate a different harmonic spectrum in the dc reference frame, and low-order harmonic components appear in the reference signal; then, the cutoff frequency of the filter must be reduced.

41.2.3.2 Current Modulator

The effectiveness of an active power filter depends basically on the design characteristics of the current controller, the method implemented to generate the reference template and the modulation technique used. Most of the modulation techniques used in active power filters are based on PWM strategies. In this chapter, four of these methods, whose characteristics are their simplicity and effectiveness, are analyzed: periodical sampling control, hysteresis band control, triangular carrier control, and vector control. The first three methods have been tested with different waveform templates sinusoidal, quasisquare, and rectifier compensation current and were compared in terms of the harmonic content and distortion at the same switching frequency [3]. The analysis shows that for sinusoidal current generation the best method is triangular carrier, followed by hysteresis band and periodical sampling. For other types of references, however, one strategy may be better than the others. Also, it was shown that each control method is affected in a different way by the switching time delays present in the driving circuitry and in the power semiconductors.

Periodical sampling

The periodical sampling method switches the power transistors of the converter during the transitions of a square-wave clock of fixed frequency (the sampling frequency). As shown in Fig. 41.21, this type of control is very simple to implement since it requires a comparator and a D-type flip-flop per phase. The main advantage of this method is that the minimum time between switching transitions is limited to the period of the sampling clock. However, the actual switching frequency is not clearly defined.

Hysteresis band

The hysteresis band method switches the transistors when the current error exceeds a fixed magnitude: the hysteresis band. As can be seen in Fig. 41.22, this type of control needs a single comparator with hysteresis per phase. In this case, the switching frequency is not determined, but it can be estimated.

Triangular carrier

The triangular carrier method, shown in Fig. 41.23, compares the current error with fixed-amplitude and fixed-frequency triangular wave (the triangular carrier). The error is processed through a proportional-integral (PI) gain stage before the comparison with the triangular carrier takes place. As can be seen, this control scheme is more complex than the periodical sampling and hysteresis band. The values for the PI control gain k_p and k_i determine the transient response and steady-state error of the triangular carrier method. It was found empirically that the values for k_p and k_i shown in Eqs. (41.27) and (41.28) give a good dynamic performance under transient- and steady-state operating conditions:

$k_p^{⁎}=\frac{\left(L+L_o\right)\cdot {\omega }_c}{2V_{\mathit{dc}}}$

  (41.27)

$k_i^{⁎}={\omega }_c{k}_p^{⁎}$

  (41.28)

where $L+L_o$ is the total series inductance seen by the converter; ω_c is the triangular carrier frequency, whose amplitude is 1 V peak-peak; and V_dc is the dc supply voltage of the inverter.

Vector control technique

This current control technique proposed in [4] divides the $\alpha -\beta$ reference frame of currents and voltages in six regions, phase shifted by 30 degrees (Fig. 41.24); identifies the region where the current vector error Δi is located; and selects the inverter output voltage vector V_inv that will force Δi to change in the opposite direction, keeping the inverter output current close to the reference signal.

Fig. 41.25 shows the single-phase equivalent circuit of the shunt active power filter connected to a nonlinear load and to the power supply.

The equation that relates the active power filter currents and voltages is obtained by applying Kirchhoff law to the equivalent circuit shown in Fig. 41.25:

$V_{inv}=L\frac{d{i}_{gen}}{dt}+E_0$

  (41.29)

The current error vector Δi is defined by the following expression:

$\mathrm{\Delta }i=i_{ref}-i_{gen}$

  (41.30)

where i_ref represents the inverter reference current vector defined by the instantaneous reactive power concept. By replacing Eq. (41.30) in Eq. (41.29),

$V_{inv}=L\frac{d}{dt}\left(i_{ref}-\mathrm{\Delta }i\right)+E_0\Rightarrow L\frac{d\mathrm{\Delta }i}{dt}=L\frac{d{i}_{ref}}{dt}+E_0-V_{inv}$

  (41.31)

If $E=L\left(d{i}_{ref}/dt\right)+E_0$, then Eq. (41.31) becomes

$L\frac{d\mathrm{\Delta }i}{dt}=E-V_{inv}$

  (41.32)

Eq. (41.32) represents the active power filter state equation and shows that the current error vector variation dΔi/dt is defined by the difference between the fictitious voltage vector E and the inverter output voltage vector V_inv. In order to keep dΔi/dt close to zero, V_inv must be selected near E₀.

The selection of the inverters' gating signals is defined by the region in which Δi is located and by its amplitude. In order to improve the current control accuracy and associate time response, depending on the amplitude of Δi, the following actions are defined:

− If $\mathrm{\Delta }i\le \delta$, the gating signals of the inverter are not changed.

− If $h\le \mathrm{\Delta }i\le \delta$, the inverter gating signals are defined following mode a.

− If $\mathrm{\Delta }i>h$, the inverter gating signals are defined following mode b.

where δ and h are reference values that define the accuracy and the hysteresis window of the current control scheme.

Mode a: Small changes in $\mathrm{\Delta }i\left(h\le \mathrm{\Delta }i\le \mathbf{\delta }\right)$

The selection of the inverter switching mode a can be explained with the following example. Assuming that the voltage vector E is located in region I (Fig. 41.26A) and the current error vector Δi is in region 6 (Fig. 41.26B), the inverter voltage vectors, V_inv, located closest to E are V₁ and V₂. The vectors $E-V_2$ and $E-V_1$ define two vectors LdΔi/dt, located in regions III and V, respectively, as shown in Fig. 41.26A, so in order to reduce the current vector error Δi, LdΔi/dt must be located in region III. Thus, the inverter output voltage has to be equal to V₁. In this way, Δi will be forced to change in the opposite direction reducing its amplitude faster. By doing the same analysis for all the possible combinations, the inverter switching modes for each location of Δi and E can be defined (Table 41.4).

Table 41.4

Inverter switching modes

E region	△i Region
	1	2	3	4	5	6
I	V1	V2	V2	$V_0-V_7$	$V_0-V_7$	$V{}_1$
II	V2	V2	V3	V3	$V_0-V_7$	$V_0-V_7$
III	$V_0-V_7$	V3	V3	V4	V4	$V_0-V_7$
IV	$V_0-V_7$	$V_0-V_7$	V4	V4	V5	V5
V	V6	$V_0-V_7$	$V_0-V_7$	V5	V5	V6
VI	V1	V1	$V_0-V_7$	$V_0-V_7$	V6	V6

V_k represents the inverter switching functions defined in Table 41.5.

Table 41.5

Relationship between switching function and inverter output voltage

k	Switch on phase a	Switch on phase b	Switch on phase c	Inverter output voltage Vk
0	4	6	2	0
1	1	6	2	$\frac{2}{3}{V}_{\mathit{dc}}$
2	1	3	2	$\frac{2}{3}{V}_{\mathit{dc}}{e}^{j\pi /3}$
3	4	3	2	$\frac{2}{3}{V}_{\mathit{dc}}{e}^{j2\pi /3}$
4	4	3	5	$\frac{2}{3}{V}_{\mathit{dc}}{e}^{j\pi }$
5	4	6	5	$\frac{2}{3}{V}_{\mathit{dc}}{e}^{j4\pi /3}$
6	1	6	5	$\frac{2}{3}{V}_{\mathit{dc}}{e}^{j5\pi /3}$
7	1	3	5	0

Mode b: Large changes in $\mathrm{\Delta }i\left(\mathrm{\Delta }i>h\right)$

If Δi becomes larger than h in a transient state, it is necessary to choose the switching mode in which the dΔi/dt has the largest opposite direction to Δi. In this case, the best inverter output voltage V_inv corresponds to the value located in the same region of Δi.

The switching frequency may be fixed by controlling the time between commutations and not applying a new switching pattern if the time between two successive commutations is lower than a selected value ($t=1/2f_c$).

Fig. 41.27 shows the block diagram of the inverter vector current control scheme implemented in a microcontroller. In Fig. 41.27, ⁎E represents the region where the vector E is located, ⁎Δi, the region of Δi, k₁ keeps the same value of k (no commutation in the inverter), k₂ selects the new inverter output voltage from Table 41.4, and k₃ selects V_inv in the same region of Δi.

41.2.3.3 Control Loop Design

Active power filters based on self-controlled dc bus voltage require two control loops, one to control the inverter output current and the other to regulate the inverter dc voltage. Different design criteria have been presented in the technical literature; however, a classic design procedure using a PI controller will be presented in this chapter. In general, the design procedure for the current and voltage loops is based on the respective time-response requirements. Since the transient response of the active power is determined by the current control loop, its time response has to be fast enough to follow the current reference waveform closely. On the other hand, the time response of the dc voltage does not need to be fast and is selected to be at least 10 times slower than the current control loop time response. Thus, these two control systems can be decoupled and designed as two independent systems.

A PI controller is normally used for the current and the voltage control loops since it contributes to zero steady-state error in tracking the reference current and voltage signals, respectively.

Design of the current control loop

The design of the current control loop gains depends on the selected current modulator. In the case of selecting the triangular carrier technique, to generate the gating signals, the error between the generated current and the reference current is processed through a PI controller, and then, the output current error is compared with a fixed-amplitude and fixed-frequency triangular wave (Fig. 41.23). The advantage of this current modulator technique is that the output current of the converter has well-defined spectral line frequencies for the switching frequency components.

Since the active power filter is implemented with a voltage-source inverter, the ac output current is defined by the inverter ac output voltage. The block diagram of the current control loop for each phase is shown in Fig. 41.28 where

E	phase-to-neutral source voltage,
Z(s)	impedance of the link reactor,
Ks	gain of the converter, and
Gc(s)	gain of the controller.

The values of K_s and Gc(s) are given in Eqs. (41.33) and (41.34):

$K_s=\frac{V_{\mathit{dc}}}{2\xi }$

  (41.33)

$G_c\left(s\right)=K_p+\frac{K_i}{s}$

  (41.34)

From Fig. 41.27 and Eq. (41.34), the following expression is obtained:

$I_{gen}=\frac{\left[K_s\left(K_p+\left(K_i/s\right)\right)\right]/\left(R_r+s{L}_r\right)}{1+\left[K_s\left(K_p+\left(K_i/s\right)\right)/\left(R_r+s{L}_r\right)\right]}{I}_{ref}-\frac{1/\left(R_r+s{L}_r\right)}{1+\left(K_s\left(K_p+K_i/s\right)/\left(R_r+s{L}_r\right)\right)}E$

  (41.35)

The characteristic equation of the current loop is given by

$1+\frac{\left(K_{p}s+K_i/s\right)}{s\left(R_r+s{L}_r\right)}$

  (41.36)

The analysis of the characteristic equation proves that the current control loop is stable for all values of K_p and K_i. Also, this analysis shows that K_p determines the speed response and K_i defines the damping factor of the control loop. If K_p is too big, the error signal can exceed the amplitude of the triangular waveform, affecting the inverter switching frequency, and if K_i is too small, the gain of the PI controller decreases, which means that the generated current will not be able to follow the reference current closely. The active filter transient response can be improved by adjusting the gain of the proportional part (K_p) to equal one and the gain of the integrator (K_i) to equal the frequency of the triangular waveform.

41.2.4.1 Design of the Synchronous Link Reactor

The design of the synchronous link reactor depends on the current modulator used. The design criteria presented in this section are based considering that the triangular carrier modulator is used. The design of the synchronous reactor is performed with the constraint that for a given switching frequency the minimum slope of the inductor current is smaller than the slope of the triangular waveform that defines the switching frequency. In this way, the intersection between the current error signal and the triangular waveform will always exist. In the case of using another current modulator, the design criteria must allow an adequate value of L_r in order to ensure that the di/dt generated by the active power filter will be able to follow the inverter current reference closely. In the case of the triangular carrier technique, the slope of the triangular waveform, λ, is defined by

$\lambda =4\xi f_t$

where ξ is the amplitude of the triangular waveform, which has to be equal to the maximum permitted amount of ripple current, and f_t is the frequency of the triangular waveform (i.e., the inverter switching frequency). The maximum slope of the inductor current is equal to

$\frac{d{i}_L}{dt}=\frac{V_{\mathit{an}}+0.5V_{\mathit{dc}}}{L_r}$

Since the slope of the inductor current (di_L/dt) has to be smaller than the slope of the triangular waveform (λ) and the ripple current is defined, from Eqs. (41.37) and (41.38), as

$L_r=\frac{V_{\mathit{an}}+0.5V_{\mathit{dc}}}{4\xi f_t}$

41.2.4.2 Design of the DC Capacitor

Transient changes in the instantaneous power absorbed by the load generate voltage fluctuations across the dc capacitor. The amplitude of these voltage fluctuations can be controlled effectively with an appropriate dc capacitor value. It must be noticed that the dc voltage control loop stabilizes the capacitor voltage after a few cycles, but is not fast enough to limit the first voltage variations. The capacitor value obtained with this criteria is bigger than the value obtained based on the maximum dc voltage ripple constraint. For this reason, the voltage across the dc capacitor presents a smaller harmonic distortion factor. The maximum overvoltage generated across the dc capacitor is given by

$V_{Cmax}=\frac{1}{C}{\displaystyle {\int }_{{\theta }_1/\omega }^{{\theta }_2/\omega }}{i}_C\left(t\right)dt+V_{\mathit{dc}}$

where V_C max is the maximum voltage across the dc capacitor, V_dc is the steady-state dc voltage, and i_c(t) is the instantaneous dc bus current. From Eq. (41.40),

$C=\frac{1}{\mathrm{\Delta }V}{\displaystyle {\int }_{{\theta }_1/\omega }^{{\theta }_2/\omega }}{i}_C\left(t\right)dt$

Eq. (41.41) defines the value of the dc capacitor, C, that will maintain the dc voltage fluctuation below ΔV p.u. The instantaneous value of the dc current is defined by the product of the inverter line currents with the respective switching functions. The mean value of the dc current that generates the maximum overvoltage can be estimated by

${\displaystyle {\int }_{{\theta }_1/\omega }^{{\theta }_2/\omega }}{i}_C\left(t\right)dt=I_{inv}{\displaystyle {\int }_{{\theta }_1/\omega }^{{\theta }_2/\omega }}\left[sin\left(\omega t\right)+sin\left(\omega t+120\mathrm{degrees}\right)\right]dt$

In this expression, the inverter ac current is assumed to be sinusoidal. These operating conditions represent the worst case.

41.3.3.1 PWM Voltage-Source Inverter

Since series active power filter can compensate voltage unbalance and current harmonics simultaneously, the rated power of the PWM voltage-source inverter increases compared with other approaches that compensate only current harmonics, since voltage injection of arbitrary phase with respect to the load current implies active power transfer from the inverter to the system. Also, the transformer leakage inductance entails fundamental voltage drop and apparent power, which has to be supported by the inverter, reducing the series active filter inverter rating available for harmonic and voltage compensation. The rated apparent power required by the inverter can be obtained by calculating the apparent power generated in the primary of the coupling transformers. The voltage reflected across the primary winding of each coupling transformer is defined in Eq. (41.43):

$V_{\mathit{series}}={\left[K_1^2{\left\{{\displaystyle \sum _{k\ne 1}}{I}_{sk}^2\right\}}^{1/2}+K_2^2{\left\{V_2+V_0\right\}}^2\right]}^{1/2}$

where V_series is the root-mean-square voltage across the primary winding of the coupling transformer. Eq. (41.43) shows that the voltage across the primary winding of the transformer is defined by two terms. The first one is inversely proportional to the quality factor of the passive LC filter, while the second one depends on the voltage unbalance that needs to be compensated. K₁ depends on the LC filter values, while K₂ is equal to one. The current flowing through the primary winding of the coupling transformer, due to the harmonic currents (Eq. 41.44), can be obtained from the equivalent circuit shown in Fig. 41.35:

$I_{sk}=\frac{Z_{fk}{I}_{lk}}{Z_{fk}+Z_{sk}+K_1}$

where $V_{\mathit{series}}=-K_1{I}_{sk}$. The fundamental component of the primary current depends on the amplitude of the negative- and zero-sequence component of the source voltage due to the system unbalance.

41.3.3.2 Coupling Transformer

The purpose of the three coupling transformers is not only to isolate the PWM inverters from the source but also to match the voltage and current ratings of the PWM inverters with those of the power distribution system. The total apparent power required by each coupling transformer is one-third the total apparent power of the inverter. The turn ratio of the current transformer is specified according to the inverter dc bus voltage, K₁ and V_ref. The correct value of the turn ratio “a” must be specified according to the overall series active power filter performance. The turn ratio of the coupling transformer must be optimized through the simulation of the overall active power filter, since it depends on the values of different related parameters. In general, the transformer turn ratio must be high in order to reduce the amplitude of the inverter output current and to reduce the voltage induced across the primary winding. Also, the selection of the transformer turn ratio influences the performance of the ripple filter connected at the output of the PWM inverter. Taking into consideration all these factors, in general, the transformer turn ratio is selected equal to 1:20.

41.3.3.3 Secondary Ripple Filter

The design of the ripple filter connected in parallel to the secondary winding of the coupling transformer is performed following the method presented by Akagi in [6]. However, it is important to notice that the design of the secondary ripple filter depends mainly on the coupling transformer turn ratio and the current modulator used to generate the inverter gating signals. If the triangular carrier is used, the frequency of the triangular waveform has to be considered in the design of the ripple filter. The ripple filter connected at the output of the inverter avoids the induction of the high-frequency ripple voltage generated by the PWM inverter switching pattern at the terminals of the primary winding of the coupling transformer. In this way, the voltage applied in series to the power system corresponds to the components required to compensate voltage unbalanced and current harmonics. The single-phase equivalent circuit is shown in Fig. 41.36.

The voltage reflected in the primary winding of the coupling transformer has the same waveform as that of the voltage across the filter capacitor. For low-frequency components, the inverter output voltage must be almost equal to the voltage across C_fr. However, for high-frequency components, most of the inverter output voltage must drop across L_fr, in which case the voltage at the capacitor terminals is almost zero. Moreover, C_fr and L_fr must be selected in order not to exceed the burden of the coupling transformer. The ripple filter must be designed for the carrier frequency of the PWM voltage-source inverter. To calculate C_fr and L_fr, the system equivalent impedance at the carrier frequency, Z_sys, reflected in the secondary must be known. This impedance is equal to

$Z_{sys\left(\mathit{secondary}\right)}=a^2\cdot Z_{sys\left(\mathit{primary}\right)}$

For the carrier frequency, the following design criteria must be satisfied:

(i) $X_{Cfr}\ll X_{Lfr}$	to ensure that at the carrier frequency most of the inverter output voltage will drop across Lfr
(ii) XCfr and $X_{Lfr}\ll Z_{sys}$	to ensure that the voltage divider is between Lfr and Cfr

The small-rated LC passive filter exhibits a high-quality-factor circuit because of the high impedance on the output side. Oscillation between the small-rated inductor and capacitor may occur, causing undesirable high-frequency voltage across the ripple filter capacitor, which is reflected in the primary winding of the coupling transformer generating high-frequency current to flow through the power distribution system. It is important to note that this oscillation is very difficult to eliminate through the design and selection of the L_fr and C_fr values. However, it can be eliminated with the addition of a new control loop. The cause of the output voltage oscillation is explained with the help of Fig. 41.37. The transfer function G_p(s) between the input voltage V_i(s) and the output voltage V_C(s) is given by the following equation:

$G_p\left(s\right)=\frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}$

where ${\omega }_n=\sqrt{1/L_{fr}{C}_{fr}}$ and $\xi =r_L/2\sqrt{C_{fr}/L_{fr}}$.

Normally, the damping factor ξ is smaller than one, causing the voltage oscillation across the capacitor ripple filter, C_fr. Generally, relatively low impedance loads are connected to the output terminals of voltage-source PWM inverters. In these cases, the quality factor of the LC filter can be low, and the oscillation between the inductor L_fr and the capacitor C_fr is avoided [7].

41.3.3.4 Passive Filters

Passive filters connected between the nonlinear load and the series active power filter play an important role in the compensation of the load current harmonics. With the connection of the passive filters, the series active power filter operates as a harmonic isolator. The harmonic isolation feature reduces the need for precise tuning of the passive filters and allows their design to be insensitive to the system impedance and eliminates the possibility of filter overloading due to supply voltage harmonics. The passive filter can be tuned to the dominant load current harmonic and can be designed to correct the load displacement power factor.

However, for industrial loads connected to stiff supply, it is difficult to design passive filters that can absorb a significant part of the load harmonic current, and therefore, its effectiveness deteriorates. Specially, for compensation of diode rectifier type of loads, where a small kilovolt-ampere passive filter is required, it is difficult to achieve the required tuning to absorb significant percentage of the load harmonic currents. For this type of application, the passive filter cannot be tuned exactly to the harmonic frequencies because they can be overloaded due to the system voltage distortion and/or system current harmonics.

The single-phase equivalent circuit of a passive LC filter connected in parallel to a nonlinear current source and to the power distribution system is shown in Fig. 41.38. From this figure, the design procedure of this filter can be derived. The harmonic current component flowing through the passive filter i_fh and the current component flowing through the source i_sh are given by the following expressions:

$i_{fh}=\frac{Z_s}{Z_s+Z_f}{i}_h;i_{sh}=\frac{Z_f}{Z_s+Z_f}{i}_h$

At the resonant frequency, the inductive reactance of the passive filter is equal to the capacitive reactance of the filter, that is,

$2\mathit{\pi frL}=\frac{1}{2\mathit{\pi frC}}$

Therefore, the resonant frequency of the passive filter is equal to

$fr=\frac{1}{2\pi \sqrt{LC}}$

The passive filter bandwidth is defined by the upper and lower cutoff frequency, at which values the filter current gain is −3 dB, as shown in Fig. 41.39.

The magnitude of the passive filter impedance as a function of the frequency is shown in Fig. 41.40.

At the resonant frequency, the passive filter magnitude is equal to the resistance. If the resistance is zero, the filter is in short circuit. The quality factor of the passive filter is defined by the following expression:

$Q=\frac{{\omega }_{n}L}{R}$

It is important to note that the operation of the series active power filter with off-tuned passive filter has an adverse impact on the inverter power rating compared with the normal case. The more off-tuned the passive filter, the more rated apparent power is required by the series active power filter.

41.3.3.5 Protection Requirements

Short circuits in the power distribution system generate large currents that flow through the power lines until the circuit breaker operates clearing the fault. The total clearing time of a short circuit depends on the time delay imposed by the protection system. The clearing time cannot be instantaneous due to the operating time imposed by the overcurrent relay and by the total interruption time of the power circuit breaker. Although power system equipment, such us power transformers and cables, are designed to withstand short-circuit currents during at least 30 cycles, the active power filter may suffer severe damage during this short time. The withstand capability of the series active power filter depends mainly on the inverter power semiconductor characteristics.

Since the most important feature of series active power filters is the small-rated power required to compensate the power system, typically 10%–15% of the load rated apparent power, the inverter semiconductors are rated for low values of blocking voltages and continuous currents. This makes series active power filters more vulnerable to power system faults.

The block diagram of the protection scheme described in this section is shown in Fig. 41.41. It consists of a varistor connected in parallel to the secondary winding of each coupling transformer and a couple of antiparallel thyristors [8]. A special circuit detects the current flowing through the varistors and generates the gating signals of the antiparallel thyristors. The protection circuit of the series active power filter must protect only the PWM voltage-source inverter connected to the secondary of the coupling transformers and must not interfere with the protection scheme of the power distribution system. Since the primary of the active power filter coupling transformers is connected in series to the power distribution system, they operate as current transformers, so that their secondary windings cannot operate in open circuit. For this reason, if a short circuit is detected in the power distribution system, the PWM voltage-source inverter cannot be disconnected from the secondary of the current transformer. Therefore, the protection scheme must be able to limit the amplitude of the currents and voltages generated in the secondary circuits. This task is performed by the varistors and by the magnetic saturation characteristic of the transformers.

The main advantages of the series active power filter protection scheme described in this section are the following:

(i) It is easy to implement and has a reduced cost.

(ii) It offers full protection against power distribution short-circuit currents.

(iii) It does not interfere with the power distribution system.

When short-circuit currents circulate through the power distribution system, the low saturation characteristic of the transformers increases the current ratio error and reduces the amplitude of the secondary currents. The larger secondary voltages induced by the primary short-circuit currents are clamped by the varistors, reducing the amplitude of the PWM voltage-source inverter currents. After a few cycles of duration of the short circuit, the PWM voltage-source inverter is bypassed through a couple of antiparallel thyristors, and at the same time, the gating signals applied to the PWM voltage-source inverter are removed. In this way, the PWM voltage-source inverter can be turned off. The principles of operation and the effectiveness of the protection scheme are shown in Fig. 41.42.

The secondary short-circuit currents will circulate through the antiparallel thyristors and the varistors until the fault is cleared by the protection equipment of the power distribution system.

By using the protection scheme described in this subsection, the voltage and currents reflected in the secondary of the coupling transformers are significantly reduced. When short-circuit currents circulate through the power distribution system, the low saturation characteristic of the coupling transformers increases the current ratio error and reduces the amplitude of the secondary voltages and currents. Moreover, the saturated high secondary voltages induced by the primary short-circuit currents are clamped by the varistors, reducing the amplitude of the PWM voltage-source inverter ac currents. Once the secondary current exceeds a predefined reference value, the PWM voltage-source inverter is bypassed through a couple of antiparallel thyristors, and then, the gating signals applied to the PWM voltage-source inverter are removed. The effectiveness of the protection scheme is shown in Fig. 41.43.

By increasing the current ratio error due to the magnetic saturation, the energy dissipated in the secondary of the coupling transformer is significantly reduced. The total energy dissipated in the varistor for the different protection conditions is shown in Fig. 41.44.

41.3.4.1 Reference Signals Generator

The compensation characteristics of series active power filters are defined mainly by the algorithm used to generate the reference signals required by the control system. These reference signals must allow current and voltage compensation with minimum time delay. Also, it is important that the accuracy of the information contained in the reference signals allows the elimination of the current harmonics and voltage unbalance present in the power system. Since the voltage and current control schemes are independent, the equations used to calculate the voltage reference signals are the following:

$\left[\begin{array}{c}\hfill v_{a0}\hfill \\ \hfill v_{a1}\hfill \\ \hfill v_{a2}\hfill \end{array}\right]=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill a\hfill & \hfill a^2\hfill \\ \hfill 1\hfill & \hfill a^2\hfill & \hfill a\hfill \end{array}\right].\left[\begin{array}{c}\hfill v_a\hfill \\ \hfill v_b\hfill \\ \hfill v_c\hfill \end{array}\right]$

The voltages v_a, v_b, and v_c correspond to the power system phase-to-neutral voltages before the current transformer. The reference voltage signals are obtained by making the positive-sequence component, v_a1, zero and then applying the inverse of the Fortescue transformation. In this way, the series active power filter compensates only voltage unbalance and not voltage regulation. The reference signals for the voltage unbalance control scheme are obtained by applying the following equations:

$\left[\begin{array}{c}\hfill v_{\mathit{refa}}\hfill \\ \hfill v_{\mathit{refb}}\hfill \\ \hfill v_{\mathit{refc}}\hfill \end{array}\right]=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill a^2\hfill & \hfill a\hfill \\ \hfill 1\hfill & \hfill a\hfill & \hfill a^2\hfill \end{array}\right].\left[\begin{array}{c}\hfill -v_{ao}\hfill \\ \hfill 0\hfill \\ \hfill -v_{a2}\hfill \end{array}\right]$

In order to compensate current harmonics generated by the nonlinear loads, the following equations are used:

$\left[\begin{array}{c}\hfill i_{\mathit{aref}}\hfill \\ \hfill i_{\mathit{bref}}\hfill \\ \hfill i_{\mathit{cref}}\hfill \end{array}\right]=\sqrt{\frac{2}{3}\cdot }\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill \frac{-1}{2}\hfill & \hfill \frac{\sqrt{3}}{2}\hfill \\ \hfill \frac{-1}{2}\hfill & \hfill \frac{-\sqrt{3}}{2}\hfill \end{array}\right]\cdot {\left[\begin{array}{ll}{v}_{\alpha }\hfill & v_{\beta }\hfill \\ -v_{\beta }\hfill & v_{\alpha }\hfill \end{array}\right]}^{-1}\left[\begin{array}{c}\hfill p_{ref}\hfill \\ \hfill q_{ref}\hfill \end{array}\right]+\frac{1}{\sqrt{3}}\left[\begin{array}{c}\hfill i_0\hfill \\ \hfill i_0\hfill \\ \hfill i_0\hfill \end{array}\right]$

where i₀ is the fundamental zero-sequence component of the line current and is calculated using the Fortescue transformation Eq. (41.50):

$i_0=\frac{1}{\sqrt{3}}\left(i_a+i_b+i_c\right)$

In Eq. (41.40), p_ref, q_ref, v_α, and v_β are defined according to the instantaneous reactive power theory. In order to avoid the compensation of the zero-sequence fundamental frequency current, the reference signal i₀ must be forced to circulate through a high-pass filter generating a current i₀′ that is used to create the reference signal required to compensate current harmonics. Finally, the general equation that defines the references of the PWM voltage-source inverter required to compensate voltage unbalance and current harmonics is the following:

$\begin{array}{c}\left[\begin{array}{c}{V}_{\mathit{refa}}\\ V_{\mathit{refb}}\\ V_{\mathit{refc}}\end{array}\right]=K_1\left\{\sqrt{\frac{2}{3}}\left[\begin{array}{cc}1& 0\\ \frac{-1}{2}& \frac{\sqrt{3}}{2}\\ \frac{-1}{2}& \frac{-\sqrt{3}}{2}\end{array}\right].{\left[\begin{array}{cc}{v}_{\alpha }& v_{\beta }\\ -v_{\beta }& v_{\alpha }\end{array}\right]}^{-1}\left[\begin{array}{c}{P}_{ref}\\ q_{ref}\end{array}\right]\right.\\ +\left.\frac{1}{\sqrt{3}},\left[\begin{array}{c}{i}_0^{\prime }\\ i_0^{\prime }\\ i_0^{\prime }\end{array}\right]\right\}+K_2\left\{\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& a^2& a\\ 1& a& a^2\end{array}\right].\left[\begin{array}{c}-v_{a0}\\ 0\\ -v_{a2}\end{array}\right]\right\}\end{array}$

where K₁ is the gain of the coupling transformer that defines the magnitude of the impedance for high-frequency current components and K₂ defines the degree of compensation for voltage unbalance, ideally K₂ equals 1.

41.3.4.2 Gating Signals Generator

This circuit provides the gating signals of the three-phase PWM voltage-source inverter required to compensate voltage unbalance and current harmonic components. The current and voltage reference signals are added, and then, the amplitude of the resultant reference waveform is adjusted in order to increase the voltage utilization factor of the PWM inverter for steady-state operating conditions. The gating signals of the inverter are generated by comparing the resultant reference signal with a fixed-frequency triangular waveform. The triangular waveform helps to keep the inverter switching frequency constant (Fig. 41.46).

A higher voltage utilization of the inverter is obtained if the amplitude of the resultant reference signal is adjusted for the steady-state operating condition of the series active power filter. In this case, the reference current and reference voltage waveforms are smaller. If the amplitude is adjusted for transient operating conditions, the required reference signals will have a larger value, creating a higher dc voltage in the inverter and defining a lower voltage utilization factor for steady-state operating conditions. The inverter switching frequency must be higher in order not to interfere with the current harmonics that need to be compensated.

41.4.2.1 Effects of the Power System Equivalent Impedance

The influence of the power system equivalent impedance on the hybrid filter compensation performance is related with its effects on the passive filter, since if the system equivalent impedance is lower than the passive filter equivalent impedance at the resonant frequency, most of the load current harmonics will flow mainly to the power distribution system. In order to compensate this negative effect on the hybrid filter compensation performance, K must be increased, as shown in Eq. (41.57), increasing the active power filter rated power.

Fig. 41.57 shows how the system equivalent impedance affects the relation between the system current THD with the active filter gain, K, in a power distribution system with passive filters tuned at the fifth and seventh harmonics. If Z_s decreases, the current system THD increases, so in order to keep the same compensation performance of the hybrid scheme, the active power filter gain, K, must be increased. On the other hand, if Z_s is high, it is not necessary to increase K in order to ensure a low THD value in the system current.

41.4.2.2 Effects of the Passive Filter Quality Factor

The quality factor, Q, defines the passive filter bandwidth. A passive filter with a high quality factor, Q, presents a larger bandwidth and better compensation characteristics. Eq. (41.67) defines the quality factor value in terms of the passive filter parameters L, R, and C. This equation shows that if R or C decreases, Q increases, and the filter bandwith becomes larger, improving the hybrid scheme compensation characteristics:

$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$

Although by decreasing R or C the passive filter quality factor is improved, each element produces a different effect in the hybrid filtering behavior. For example, by increasing R, the filter equivalent impedance at the resonant frequency becomes bigger, affecting the current harmonic compensation characteristics at this specific frequency.

Fig. 41.58 illustrates how the system current THD changes with different values of C while keeping the filter tuned factor, δ, constant. It is important to note that the larger the value of C, the better is the compensation characteristic of the hybrid scheme. An appropriate value of C must be selected, since C cannot be increased too much. If C is too big, the fundamental component of the filter current will increase, overloading the hybrid filter and generating a large amount of reactive power to the power distribution system. A similar analysis is made for the value of the resistance, showing the results in Fig. 41.59.

41.4.2.3 Effects of the Passive Filter Tuned Factor

The tuned factor, δ, is defined in Eq. (41.63). This factor affects the hybrid scheme performance especially at the passive filter resonant frequency, since δ defines the changes in the system frequency, affecting the value of the passive filter resonant frequency. Fig. 41.60 shows how the system current THD changes with respect to the active power filter gain, K, for different values of passive filter tuned factors. The larger the value of δ, the filtering performance of the hybrid filter becomes worst.

41.5.1.1 Current Reference Generator

This unit is designed to generate the required current reference that is used to compensate the undesirable load current components. In this case, the system voltages, the load currents, and the dc voltage converter are measured, while the neutral output current and neutral load current are generated directly from these signals.

41.5.1.2 Prediction Model

The converter model is used to predict the output converter current. Since the controller operates in discrete time, both the controller and the system model must be represented in a discrete-time domain. The discrete-time model consists of a recursive matrix equation that represents this prediction system. This means that for a given sampling time T_s, knowing the converter switching states and control variables at instant kT_s, it is possible to predict the next states at any instant [k+1]T_s. Due to the first-order nature of the state equations that describe the model in (41.68) and (41.69), a sufficiently accurate first-order approximation of the derivative is considered:

$v_o=v_{xn}-R_{\mathit{eq}}{i}_o-L_{\mathit{eq}}\frac{d{i}_o}{dt}$

$v_{xn}=S_x-S_n{v}_{\mathit{dc}},x=u,v,w,n\text{.}$

Therefore, the 16 possible output current predicted values can be obtained from (41.68) as

$i_o\left[k+1\right]=\frac{T_s}{L_{\mathit{eq}}}\left(v_{xn}\left[k\right]-v_o\left[k\right]\right)+\left(1-\frac{R_{\mathit{eq}}{T}_s}{L_{\mathit{eq}}}\right)i_o\left[k\right]$

41.5.1.3 Cost Function

In order to select the optimal switching state that must be applied to the power converter, the 16 predicted values obtained for i_o[k+1] are compared with the reference using a cost function g, as follows:

$\begin{array}{c}g\left[k+1\right]={\left(i_{ou}^{⁎}\left[k+1\right]-i_{ou}\left[k+1\right]\right)}^2\\ +{\left(i_{ov}^{⁎}\left[k+1\right]-i_{ov}\left[k+1\right]\right)}^2\\ +{\left(i_{ow}^{⁎}\left[k+1\right]-i_{ow}\left[k+1\right]\right)}^2\\ +{\left(i_{\mathit{on}}^{⁎}\left[k+1\right]-i_{\mathit{on}}\left[k+1\right]\right)}^2\end{array}$

The output current (i_o) is equal to the reference (i_o⁎) when g=0. Therefore, the optimization goal of the cost function is to achieve a g-value close to zero. During each sampling state, the voltage vector v_xn that minimizes the cost function is found and then applied to the converter in the next sampling state.