2. Modelling of the voltage source converter

The total DC current Idc is flowing to the DC side of VSC. Vdc is the instantaneous voltage across the capacitor, and id is the instantaneous DC current. The quality factor of DC capacitor is assumed to be high, so the series resistance is neglected. The instantaneous power is supplied by the renewable energy source, i.e. wind turbine. The instantaneous current Is is supplied by the external source. It is equal to zero when VSC operates as a reactive power compensator.

The block diagram of VSC control structure is depicted in Figure 1. The control system consists of (i) voltage control, (ii) a phase lock loop (PLL), (iii) current reference calculation block and (iv) current control. The real power reference is

Figure 1. Control structure of grid-connected wind converters. Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design DOI: http://dx.doi.org/10.5772/intechopen.89167

calculated by the PI controller, which considers the DC voltage and desired active power through the VSC to the grid. The instantaneous reactive power is calculated by the separate loop, which may consider either the desired power factor or the reference voltage. The dq frame is synchronised with the positive sequence fundamental voltage of the grid at PCC with the help of PLL. It converts three phase voltages Va, Vb and Vc into Vd+ and Vq+, which are converted in the alpha-beta reference frame voltages Vα and Vβ. Then current reference is obtained by the α-β to abc transformation. These reference currents are compared with actual current, and modulating signals md and mq are generated. These signals are finally transformed into ma, m<sup>b</sup> and mc to generate pulse width modulated (PWM) signal.

Both switching frequency and grid voltage distortion can cause poor power quality. A filter design is a subject that requires trade-off between filter performance and the control bandwidth. Filters are required to meet power quality standard, avoid parallel resonance and improve power quality.

Inverter for grid interfacing will need to incorporate interface filters to attenuate the injection of current harmonics.

## 3. Impedance-based stability analysis

This method was proposed in [7]. The system impedance is partitioned into source and grid impedance. The source impedance is either represented by Thevenin's equivalent circuit or Norton equivalent circuit (Figures 2 and 3). Thevenin's equivalent circuit consists of ideal voltage source in series with the series impedance (Zs), whereas the load impedance is modelled by series impedance (Zl). Since the converter circuit is non-linear, it is represented by the small signal circuit. This linear representation of circuit is valid only for the small perturbation of signal. With this assumption, the current (I) flowing from source to load is given by

$$I(\mathfrak{s}) = \frac{V\_{\mathfrak{s}}(\mathfrak{s})}{Z\_{l}(\mathfrak{s}) + Z\_{\mathfrak{s}}(\mathfrak{s})} \tag{1}$$

Figure 2. Thevenin's equivalent circuit.

Figure 3. Norton's equivalent circuit.

conditions in practical. In a relatively weak system, VSCs are subject to various power quality disturbances, such as unbalance voltage, voltage swell and swag, notches, etc. [3]. The occurrence of such disturbances causes various problems like ripple in torque of generator and motor, increased losses, abnormal tripping of protective devices, malfunction of sophisticated control system, reduction in the

There are two types of harmonics generated by VSCs. One is characteristics harmonics, which are related with the switching operations of the IGBTs inside the VSC. And second is non-characteristics harmonics. The voltage ripple on DC side of

VSC generates harmonics on its AC side current. According to [5], the noncharacteristics harmonics are generated by the unbalanced voltage in the AC side. However the quantification of magnitude of such harmonics is not simple and cannot be done with deterministic method. The non-characteristics harmonics are considered as the steady-state low-frequency components which would not appear

if the grid voltage is balanced. The unbalance grid voltage has fundamental frequency negative sequence component and third-order positive sequence

Though it is possible to eliminate zero sequence third harmonic component using transformer of proper vector group, the non-characteristics third-order positive sequence harmonics cannot eliminate transformers with delta-connected

In [6], the author has proposed DC voltage control to eliminate DC oscillating voltage when AC side is unbalanced. To achieve this, VSC has to operate with constant AC power control. However, the effect of control on AC current is not discussed. It is important to analyse the distorted and unbalanced AC side current, when such control is implemented. In this case, the currents of AC side of converter contain non-characteristics low-frequency component such as fundamental negative sequence and third harmonic components of positive and negative sequence.

The total DC current Idc is flowing to the DC side of VSC. Vdc is the instantaneous voltage across the capacitor, and id is the instantaneous DC current. The quality factor of DC capacitor is assumed to be high, so the series resistance is neglected. The instantaneous power is supplied by the renewable energy source, i.e. wind turbine. The instantaneous current Is is supplied by the external source. It is

The block diagram of VSC control structure is depicted in Figure 1. The control system consists of (i) voltage control, (ii) a phase lock loop (PLL), (iii) current reference calculation block and (iv) current control. The real power reference is

equal to zero when VSC operates as a reactive power compensator.

expected lifetime of equipment, etc. [4].

Wind Solar Hybrid Renewable Energy System

2. Modelling of the voltage source converter

component.

winding.

Figure 1.

70

Control structure of grid-connected wind converters.

Which is rearranged as,

$$I(\mathfrak{s}) = \frac{V\_s(\mathfrak{s})}{Z\_l(\mathfrak{s})} \frac{\mathbf{1}}{\mathbf{1} + Z\_l(\mathfrak{s}) \not\!\!/ \_{\mathbb{Z}^{(\mathfrak{s})}}} \tag{2}$$

grid [5]. The grid-connected converter used in renewable sources is modelled as a current source in parallel with an impedance, i.e. Norton's equivalent circuit [6]. The stability of grid-connected inverter can be determined by Nyquist criterion [8]. The control structure of most of the VSCs is developed in the rotating dq reference frame [7]. The phase lock loop is used to synchronise converter with grid [9]. The use of complex multiple-loop control structure introduces non-linearities. These are generally overlooked in the simplified low-order modelling [10]. On the flip side, the detailed model introduces complexity and cross-couplings between various terms, which makes the determination of output impedance cumbersome. The trade-off way suggested in some literatures is to linearise the model by small signal

Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design

The impedance of converter-interconnected generator is affected by various factors such as control parameters, PLL, switching delays and converter harmonic filters. The converter is basically controlled by output current signal. In Figure 4, i1 is the converter current and i2 is the grid current. The converter is controlled either by i1 or i2. If the grid current is the control variable, then the current control loop is

proportional-integral-type current controller and GDð Þs is the switching delay. Here, the converter output voltage is considered as pure sinusoidal; if there is a noise in the voltage, then it needs to be considered as a disturbance signal. If, the converter is controlled by taking converter current i1, then Y21 is replaced by output conduc-

GPIðÞ¼ s kp þ

Higher-order controller also can be used for current control, but here simple PI controller is considered for the sake of simplicity. The delay, estimated by Pade's

> 1 1 þ 1:5Tds

Z3<sup>f</sup> Z1fZ<sup>2</sup> þ Z1fZ3<sup>f</sup> þ Z2Z3<sup>f</sup>

> ki s

where Y21 is the forward trans-admittance of the filter, GPIð Þs is the

tance Y11. The forward transconductance Y21 is given by

Y21ðÞ¼ s

approximation technique, is given by

margin (PM).

Figure 4.

73

Converter output circuit (Source: [11]).

The simple transfer function of current controller is given by

GDðÞ¼ s

impedance. The phase at the intersection of two curves gives the phase

The frequency response of output impedance is plotted here with the line

YoðÞ¼ s GPIð Þs GDð Þs Y21ð Þs (7)

(8)

(9)

(10)

analysis technique.

DOI: http://dx.doi.org/10.5772/intechopen.89167

System stability analysis is based on the assumption that the source voltage and load impedance remain stable. So, V(s) and 1/Z(s) are stable. So, the stability depends on the extreme right-hand side of Eq. (2). It is given by

$$H(\mathfrak{s}) = \frac{\mathbf{1}}{\mathbf{1} + \mathbf{2}\_{\mathfrak{r}}(\mathfrak{s})\_{\mathbf{Z}\_{\mathfrak{l}}(\mathfrak{s})}} \tag{3}$$

The close observation of Eq. (3) reveals the important characteristics. It is a transfer function with unity gain and feedback equal to Zs(s)/Zl(s). According to the linear control theory, the H(s) is stable, if and if only, when ration Zs(s)/Zl(s) meets the requirement of Nyquist stability criterion [7].

In the above analysis for stability, it is assumed that the source is ideal voltage source and it remains stable under unloaded condition. However, the gridconnected inverters are usually current controlled. Hence above analysis is not much useful. So, the source should be represented as current source. To arrive at the equivalent current source, the same small signal voltage source is modified. The voltage across load is given by

$$V(\mathfrak{s}) = \frac{I\_{\mathfrak{s}}(\mathfrak{s})}{Y\_{l}(\mathfrak{s}) + Y\_{\mathfrak{S}}(\mathfrak{s})} \tag{4}$$

by rearranging

$$V(s) = \frac{I\_s(s)}{Y\_l(s)} \frac{I\_s(s)}{1 + \frac{Y\_S(s)}{Y\_l(s)}} \tag{5}$$

Similar to the above analysis, the current source is assumed to be stable under unloaded condition. The load is stable when connected to ideal current source. Under this condition, I(s) and 1/Yl(s) are stable. Under this condition, stability of V(s) depends on stability of second term of Eq. (5). This again resembles the closedloop transfer function with negative feedback. The gain is unity and the feedback factor is Ys(s)/Yl(s). Therefore the system is stable, if, along with above conditions, it meets the Nyquist criterion. In Eq. (5) admittances are used instead of impedance, though the analysis still can be carried out in terms of the impedances. In this case, Eq. (5) becomes

$$V(s) = I\_{\iota}(s) \, Z\_{\iota}(s) \frac{1}{1 + \frac{Z\_{\iota}(s)}{Z\_{\iota}(s)}} \tag{6}$$

It is important to note the requirement of stability. In the voltage source model, the output impedance of source should be as low as possible (ideally zero); whereas in the current source model, the output impedance should be as high as possible (ideally infinite).

## 4. Grid-connected inverters

The modelling of impedance of grid-connected VSC has very important use in analysis of stability and resonance phenomena when converter is integrated into the Harmonic Resonance Analysis for Wind Integrated Power System and Optimized Filter Design DOI: http://dx.doi.org/10.5772/intechopen.89167

grid [5]. The grid-connected converter used in renewable sources is modelled as a current source in parallel with an impedance, i.e. Norton's equivalent circuit [6]. The stability of grid-connected inverter can be determined by Nyquist criterion [8]. The control structure of most of the VSCs is developed in the rotating dq reference frame [7]. The phase lock loop is used to synchronise converter with grid [9]. The use of complex multiple-loop control structure introduces non-linearities. These are generally overlooked in the simplified low-order modelling [10]. On the flip side, the detailed model introduces complexity and cross-couplings between various terms, which makes the determination of output impedance cumbersome. The trade-off way suggested in some literatures is to linearise the model by small signal analysis technique.

The impedance of converter-interconnected generator is affected by various factors such as control parameters, PLL, switching delays and converter harmonic filters. The converter is basically controlled by output current signal. In Figure 4, i1 is the converter current and i2 is the grid current. The converter is controlled either by i1 or i2. If the grid current is the control variable, then the current control loop is

$$\mathbf{Y\_{o}(s)} = \mathbf{G\_{PI}(s)} \mathbf{G\_{D}(s)} \mathbf{Y\_{21}(s)}\tag{7}$$

where Y21 is the forward trans-admittance of the filter, GPIð Þs is the proportional-integral-type current controller and GDð Þs is the switching delay. Here, the converter output voltage is considered as pure sinusoidal; if there is a noise in the voltage, then it needs to be considered as a disturbance signal. If, the converter is controlled by taking converter current i1, then Y21 is replaced by output conductance Y11. The forward transconductance Y21 is given by

$$\mathbf{Y\_{21}(s)} = \frac{\mathbf{Z\_{3f}}}{\mathbf{Z\_{4f}}\mathbf{Z\_{2}} + \mathbf{Z\_{4f}}\mathbf{Z\_{3f}} + \mathbf{Z\_{2}}\mathbf{Z\_{3f}}} \tag{8}$$

The simple transfer function of current controller is given by

$$\mathbf{G}\_{PI}(\mathbf{s}) = \mathbf{k}\_p + \frac{\mathbf{k}\_i}{\mathbf{s}} \tag{9}$$

Higher-order controller also can be used for current control, but here simple PI controller is considered for the sake of simplicity. The delay, estimated by Pade's approximation technique, is given by

$$\mathbf{G}\_{\rm D}(s) = \frac{\mathbf{1}}{\mathbf{1} + \mathbf{1}.5T\_d s} \tag{10}$$

The frequency response of output impedance is plotted here with the line impedance. The phase at the intersection of two curves gives the phase margin (PM).

Figure 4. Converter output circuit (Source: [11]).

Which is rearranged as,

Wind Solar Hybrid Renewable Energy System

voltage across load is given by

by rearranging

Eq. (5) becomes

(ideally infinite).

72

4. Grid-connected inverters

I sðÞ¼ Vsð Þ<sup>s</sup>

depends on the extreme right-hand side of Eq. (2). It is given by

the requirement of Nyquist stability criterion [7].

load impedance remain stable. So, V(s) and 1/Z(s) are stable. So, the stability

H sðÞ¼ <sup>1</sup>

The close observation of Eq. (3) reveals the important characteristics. It is a transfer function with unity gain and feedback equal to Zs(s)/Zl(s). According to the linear control theory, the H(s) is stable, if and if only, when ration Zs(s)/Zl(s) meets

In the above analysis for stability, it is assumed that the source is ideal voltage

source and it remains stable under unloaded condition. However, the gridconnected inverters are usually current controlled. Hence above analysis is not much useful. So, the source should be represented as current source. To arrive at the equivalent current source, the same small signal voltage source is modified. The

V sðÞ¼ Isð Þ<sup>s</sup>

V sðÞ¼ Isð Þ<sup>s</sup> Ylð Þs

V sðÞ¼ Isð Þs :Zlð Þs

It is important to note the requirement of stability. In the voltage source model, the output impedance of source should be as low as possible (ideally zero); whereas in the current source model, the output impedance should be as high as possible

The modelling of impedance of grid-connected VSC has very important use in analysis of stability and resonance phenomena when converter is integrated into the

Similar to the above analysis, the current source is assumed to be stable under unloaded condition. The load is stable when connected to ideal current source. Under this condition, I(s) and 1/Yl(s) are stable. Under this condition, stability of V(s) depends on stability of second term of Eq. (5). This again resembles the closedloop transfer function with negative feedback. The gain is unity and the feedback factor is Ys(s)/Yl(s). Therefore the system is stable, if, along with above conditions, it meets the Nyquist criterion. In Eq. (5) admittances are used instead of impedance, though the analysis still can be carried out in terms of the impedances. In this case,

Ylð Þþs YSð Þs

Isð Þs <sup>1</sup> <sup>þ</sup> YSð Þ<sup>s</sup> Ylð Þs

> 1 <sup>1</sup> <sup>þ</sup> Zlð Þ<sup>s</sup> Zsð Þs

Zlð Þ<sup>s</sup> : <sup>1</sup> 1 þ Zsð Þ<sup>s</sup>=Zlð Þs

System stability analysis is based on the assumption that the source voltage and

1 þ Zsð Þ<sup>s</sup>=Zlð Þs (2)

(3)

(4)

(5)

(6)
