Wind Energy System with Matrix Converter

Abstract

Keeping in mind the end goal to address energy issues, considering financial and natural variables, a Wind Energy Conversion System (WECS) is bit-by-bit gaining interest as the suitable source of renewable energy. In this chapter, a direct AC-AC Matrix Converter (MC) is utilized to interface an isolated static load fed from the WECS. Due to the advantage of lower cost, maintenance simplicity and short circuit self-protection, the self-excited induction generator is used to generate electric energy. The MC is used instead of the conventional rectifier-inverter converter because it does not contain the bulky size capacitor, the ability of controlling the input displacement factor and the unity factor can be achieved. Moreover, the MC can be provided in a simple construction, and gives a wide range of output frequency. In comparison with cycloconverter, the MC can give an output frequency that may be equal, less or greater than input frequency. This chapter introduces a modified algorithm for space vector modulation that reduces the total harmonic distortion of the output voltage so the size and cost of the required filter can be reduced. Moreover, a modified open loop control of matrix converter with indirect space vector modulation is introduced to provide constant frequency and output voltage even if the wind speed changed.

5.1 Introduction

Nowadays, there are more and more concerns over fossil fuel exhaustion and the environmental problems caused by fossil fuel based conventional power generation. To solve those problems renewable energy is widely accepted as one of the appropriate alternatives, in particular, the wind and the solar energies. Some load centers like ships, islands and remote villages require their own electric power supply like stand-alone electric generators for the excitation required for their local loads. These requirements encourage researchers to search and develop a method for providing the required excitation by a simple technique for longest time like renewable sources of energy. The significant favorable advantage of utilizing renewable sources is the absence of harmful emissions. The Wind Energy Conversion System (WECS) is bit-by-bit gaining interest as a suitable source of renewable energy. There are two types of Wind Turbine (WT): horizontal axis configuration and vertical axis configuration. A Squirrel Cage Induction Generator (SCIG) is preferable in isolated load application as it is cost-effective. The SCIG also has the ability to self-protection against short circuits and does not require routine maintenance. So, it is used to convert the captured mechanical energy from the WT into electrical energy. Different types of AC-AC power converters will be introduced in this chapter. A Matrix Converter (MC) is used to control the rms value and frequency of the load voltage. An indirect space vector modulation (ISVM) technique will be introduced, in addition to showing how to transform from the indirect matrix converter to direct ones. Moreover, a modified algorithm for space vector modulation (SVM) will be introduced [1]. The advantage of the modified algorithm, as well as the ability of MC to give a wide range of output frequency, controlling the phase angle between the input current and voltage and achieving a unity input displacement factor will be discussed [2, 3]. A modified open loop controller for the SVM of MC provides output voltage and frequency with constant values even if the wind speed changes. Finally, experimental and simulation results are discussed for each part in this chapter (Fig. 5.1).

Fig. 5.1

Block diagram of matrix converter in wind energy system

5.2 Wind Energy Conversion Systems

This part introduces a short description of WT types, WTs and SCIG’s advantages, and disadvantages, characteristics and modeling.

5.2.1 Wind Turbine

The function of the wind turbine is transforming the wind kinetic energy into mechanical energy. A WT can spins about either a vertical or a horizontal axis. Therefore, there are two types of WT, the vertical axis and the horizontal-axis turbines.

5.2.1.1 Horizontal Axis–Wind Turbine

Figure 5.2a shows the construction of the horizontal-axis wind turbines (HAWTs). The electrical generator is at the top of the tower. To obtain a suitable speed to drive the electrical generator a gearbox must be used to turn the blades slower rotation into quick rotation [2]. HAWTs have a tall tower base which enables to collect a large amount of energy compared to the short tower base. Every 10 meters up the ground, the wind speed increases by 20% and the output power increases by 34%. A HAWT has a variable blade pitch, which enable the blades of the turbine to have the optimum angle so that it can collect the maximum amount of wind energy. Moreover, it has a high efficiency, as the moving of the turbine blades is perpendicularly to the wind. The disadvantage of HAWT is the tower requires enormous construction so that it can hold the gearbox, generator and the heavy blades. Moreover, it has a high cost as it requires very tall and expensive cranes to carry the shaft of the turbine and the generator. HAWTs cause disrupting of the environmental landscapes due to their tall height. It affects radar installations and creating signal clutter due to the reflections from the tall tower. An additional control mechanism is required to turn the blades toward the wind [3, 4].

Fig. 5.2

a HAWT, b VAWT [1]

5.2.1.2 Vertical Axis–Wind Turbine

The rotor shaft of vertical-axis wind turbines (VAWTs) is arranged perpendicularly to the ground. The advantage of VAWT is it does not need an additional control mechanism to move the blades to the suitable side for the wind. The generator and gearbox are in the ground as shown in Fig. 5.2b. The advantage of that is making the maintenance process easy. The VAWT can install in different places such as highways and roofs. It has a low speed compared to HAWT and low speed means less noise. It does not kill wildlife and birds as it is more visible and has a low speed. The disadvantages of VAWTs are the lower efficiency compared to HAWTs, lower speed rotation because the turbine blades are near the ground and this does not take the advantage of higher speed as in HAWT, so the output power in the VAWT is small compared to the HAWT [5,6,7,8].

5.2.1.3 Modelling of Wind Turbine

The produced mechanical power generated from the WT is calculated from Eq. (5.1).

$$P = \frac{1}{2}\rho C_{p} A_{r} v_{w}^{3}$$

(5.1)

The power coefficient $C_{p }$ depends on the rotor blade pitch angle $\upbeta$ and the tip speed ratio $\uplambda$ according to Eq. (5.2)

$$C_{p} \left( {\lambda ,\beta } \right) = 0.73\left( {\frac{151}{{\lambda_{i} }} - 0.58\beta - 0.002\beta^{2.14} - 13.2} \right)$$

(5.2)

$$\lambda_{i} = \frac{1}{{\frac{1}{\lambda - 0.02\beta } - \frac{0.003}{{\beta^{3} + 1}}}}$$

(5.3)

$$\lambda = \frac{{\omega_{r } R_{r} }}{{v_{w} }}$$

(5.4)

where, P is the mechanical power, ρ is the density of air in ${\text{g}}/{\text{m}}^{3}$, $A_{r}$ is the rotor area of WT in ${\text{m}}^{2}$, (${\text{A}}_{\text{r}} =\uppi{\text{R}}_{\text{r }}^{2}$, where ${\text{R}}_{\text{r}}$ is the radius of rotor blade) and ${\text{v}}_{\text{w}}$ is the wind speed in m/s. Figure 5.3 illustrates the calculated wind turbine power-speed characteristics at different wind speeds. Figure 5.4 shows the implementation of a wind turbine using MATLAB Simulink [3, 4].

Fig. 5.3

Wind turbine power-speed characteristics

Fig. 5.4

Simulink model of wind turbine torque

5.2.1.4 Self-excited Induction Generator

The induction machine is a popular machine, which may be utilized as a motor or generator. An induction machine will generate electrical power if it is driven at a speed slightly above its synchronous speed while being connected to the supply. This part introduces principles of operation of the induction machine, equivalent circuit, torque-speed characteristics and torque equation. The value of the generated voltage is dependent on the value of the capacitance required to provide reactive power for the excitation [3, 5]. The SCIG is suitable for isolated operation due to its ruggedness, low cost of construction, less maintenance, and it is an inexpensive alternative to synchronous ones. For an isolated generation in remote areas, a variable capacitor is required to build up the voltage in a SCIG [3].

5.2.1.5 Excitation Method

Induction motors and generators require reactive power for their excitation. They can take the required reactive power for excitation from the grid if they are grid connected. An induction generator usually used to supply the remote loads that are not connected to the grid, and the generator can take the required reactive power for excitation from a capacitor bank. Due to the residual magnetism, when the rotor speed increases above synchronous speed, small voltage and current will be produced in the capacitors that are connected to the terminal of the stator. The function of the capacitor bank is to provide the reactive power required for excitation in addition to the reactive power needed for the load. The suitable value for the capacitor bank required for SCIG was calculated as described in [2] (Fig. 5.5).

Fig. 5.5

Torque-speed characteristics for induction machine

5.2.1.6 Modeling of Induction Generator

All electrical machines can be expressed by the same equation sets regardless they operate as motors or generators. These equations can be divided in two groups; torque equations and voltage equations. The following assumption is used to deal with the machine equations: neglecting the saturation of iron [6], the magnetic permeability of iron is assumed to be infinitely compared to the air gap permeability, which means that the magnetic flux is radial to the gap. Neglecting all iron losses, symmetric and balanced three-phase induction machine and constant air gap, stator and rotor windings represent distributed windings that always generate a sinusoidal magnetic field distribution in the gap.

All the explained hypotheses enable to use the following set of equations which describe the dynamic behavior of the induction machine [7, 8].

$$\left\{ {\begin{array}{*{20}c} {V_{s}^{abc} } \\ {V_{r}^{abc} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {r_{s}^{abc} } & 0 \\ 0 & {r_{r}^{abc} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {i_{s}^{abc} } \\ {i_{r}^{abc} } \\ \end{array} } \right\} + \frac{d}{dt}\left\{ {\begin{array}{*{20}c} {\lambda_{s}^{abc} } \\ {\lambda_{r}^{abc} } \\ \end{array} } \right\}$$

(5.5)

where, $V_{s}^{abc}$ is the voltage vector of stator’s winding, $V_{r}^{abc}$ is voltage vector of rotor’s winding, ${\text{i}}_{\text{s}}^{\text{abc}}$ is current vector of stator’s winding, ${\text{i}}_{\text{r}}^{\text{abc}}$ is current vector of rotor’s winding, $\uplambda_{\text{s}}^{\text{abc}}$ is stator’s winding flux linkage vector, $\uplambda_{\text{r}}^{\text{abc}}$ is rotor’s winding flux linkage vector. ${\text{r}}_{\text{s}}^{\text{abc}}$ and ${\text{r}}_{\text{r}}^{\text{abc}}$ are the resistance vectors of stator’s and rotor’s windings. The flux linkage can be described as a function of stator and rotor currents as given in Eq. (5.6) (Fig. 5.6).

Fig. 5.6

One-phase equivalent circuit of induction generator

$$\left\{ {\begin{array}{*{20}c} {\lambda_{s}^{abc} } \\ {\lambda_{r}^{abc} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {L_{ss}^{abc} } & {L_{sr}^{abc} } \\ {L_{rs}^{abc} } & {L_{rr}^{abc} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {i_{s}^{abc} } \\ {i_{r}^{abc} } \\ \end{array} } \right\}$$

(5.6)

where, each term represents a 3-dimensional matrix or a three-dimensional vector. The vector can be described as:

$$\begin{aligned} & V_{s}^{abc} = \left[ {\begin{array}{*{20}c} {v_{sa} } \\ {v_{sb} } \\ {v_{sc} } \\ \end{array} } \right] ,\quad V_{r}^{abc} = \left[ {\begin{array}{*{20}c} {v_{ra} } \\ {v_{rb} } \\ {v_{rc} } \\ \end{array} } \right] ,\quad i_{s}^{abc} = \left[ {\begin{array}{*{20}c} {i_{sa} } \\ {i_{sb} } \\ {i_{sc} } \\ \end{array} } \right],\quad i_{r}^{abc} = \left[ {\begin{array}{*{20}c} {i_{ra} } \\ {i_{rb} } \\ {i_{rc} } \\ \end{array} } \right] \\ & L_{ss}^{abc} = \left[ {\begin{array}{*{20}c} {L_{ss} + L_{ls} } & {L_{sm} } & {L_{sm} } \\ {L_{sm} } & {L_{ss} + L_{ls} } & {L_{sm} } \\ {L_{sm} } & {L_{sm} } & {L_{ss} + L_{ls} } \\ \end{array} } \right],\;\quad r_{s}^{abc} = \left[ {\begin{array}{*{20}c} {r_{s} } & 0 & 0 \\ 0 & {r_{s} } & 0 \\ 0 & 0 & {r_{s} } \\ \end{array} } \right] \\ & L_{rr}^{abc} = \left[ {\begin{array}{*{20}c} {L_{rr} + L_{lr} } & {L_{rm} } & {L_{rm} } \\ {L_{rm} } & {L_{rr} + L_{lr} } & {L_{rm} } \\ {L_{rm} } & {L_{rm} } & {L_{rr} + L_{lr} } \\ \end{array} } \right], \quad r_{r}^{abc} = \left[ {\begin{array}{*{20}c} {r_{r} } & 0 & 0 \\ 0 & {r_{r} } & 0 \\ 0 & 0 & {r_{r} } \\ \end{array} } \right] \\ & L_{sr}^{abc} = \left\{ {L_{rs}^{abc} } \right\}^{t} = L_{sr} \left[ {\begin{array}{*{20}c} {\cos \,\theta_{r} } & {\cos \left( {\theta_{r} + \frac{2\pi }{3}} \right)} & {\cos \left( {\theta_{r} - \frac{2\pi }{3}} \right)} \\ {\cos \left( {\theta_{r} - \frac{2\pi }{3}} \right)} & {\cos \theta_{r} } & {\cos \left( {\theta_{r} + \frac{2\pi }{3}} \right)} \\ {\cos \left( {\theta_{r} + \frac{2\pi }{3}} \right)} & {\cos \left( {\theta_{r} - \frac{2\pi }{3}} \right)} & {\cos \theta_{r} } \\ \end{array} } \right] \\ \end{aligned}$$

(5.7)

where, ${\text{L}}_{\text{ss}}$: the stator winding self-inductance, ${\text{L}}_{\text{sm}}$: stator winding mutual inductance, ${\text{L}}_{\text{rr}}$: the rotor winding self-inductance, ${\text{L}}_{\text{rm}}$: rotor winding mutual inductance, ${\text{L}}_{\text{sr}}$: mutual inductance between stator and rotor winding maximum value, ${\text{L}}_{\text{ls}}$: stator winding leakage inductance, ${\text{L}}_{\text{lr}}$: rotor winding leakage inductance.

5.2.2 AC-AC Converter

An AC-AC converter can be classified into two groups as depicted in Fig. 5.7. The first group is an AC Voltage regulator which can control the rms value of voltage at a constant frequency. The second one is the frequency converter which can control the voltage rms in addition to control the frequency. The frequency converter can be classified in two groups, firstly a cycloconverter which is a converter that has naturally commutation with the ability of bidirectional power flow and it does not have a limitation on its size like in SCR inverter with commutation elements. The cycloconverter main limitations are:

Fig. 5.7

AC-AC Converters

• 1. The range of frequency for sub harmonic-free with efficient operation is limited; and

• 2. Input displacement factor at low output voltages is poor.

The second one is the MC by which the output frequency can be controlled with values that may be equal, greater or less than the input frequency, in addition to, can control the rms value of the load voltage and. the phase angle between the input current and voltage as well as the value of the input displacement factor with a unity value could be achieved [9].

5.3 Matrix Converter

The MC is a set of a nine-bidirectional switches which help to connect the load directly with the three-phase input voltage with no need for any dc link. Therefore, it can be designed in a compact and simple form. The MC provides the advantage of power flow in both direction and the displacement factor at its input with a unity value can be provided. Also, it has minimal energy storage requirement, which permits to dispose of massive and lifetime-constrained capacitor, but the MC does not take its suitable place in the industry due to the disadvantages of the limited input output voltage transfer ratio which is 0.866 [10]. Because of the bi-directional switch, some MC types need more number of switches compared to the conventional rectifier inverter type [11]. Input filters are required to reduce the high frequency harmonics and clamping circuits are needed to protect switches from over voltages due to energy stored in inductive loads. The main structure of MC as shown in Fig. 5.8 consists of:

Fig. 5.8

The main structure of MC

• 1. Matrix switches.

• 2. Input filter.

• 3. Clamping circuit.

5.3.1 Matrix Switches

A bi-directional switch is required for MC. It has the ability to connect the current in both directions, but these switches are not available in our markets so far. Thus, conventional unidirectional can be used to obtain bi-directional switches as presented in Fig. 5.9. The bi-direction switch used in this prototype is shown in Fig. 5.9a [12].

Fig. 5.9

a Bridge of diodes with single IGBT, b Common emitter bi-directional switch, c Common collector bi-directional Switch

5.3.1.1 Diode Bridge with a Single Switch

This switch consists of four ultra-fast diodes with a controllable unidirectional switch as described in Fig. 5.5a. The advantage of this switch is that it has a simple construction and requires only insulated-gate bipolar transistor (IGBT). The disadvantage of this switch is it requires 36 ultra-fast diodes and nine isolated power supply as shown in Table 5.1 [2].

Table 5.1 The difference between bi-directional switches

5.3.1.2 Common Emitter Bi-directional Switch

This switch consists of two ultra-fast diodes with two controllable unidirectional switches as presented in Fig. 5.9b. The advantage of this switch is that it can control the current through switch. The disadvantage of this switch is it requires more gate drives power supplies and more IGBTs as shown in Table 5.1 [3].

5.3.1.3 Common Collector Bi-directional Switch

This switch consists of a two ultra-fast diode with two controllable unidirectional switches (IGBT) as shown in Fig. 5.9c. This switch has the same advantage of the common emitter one, but it requires only six isolated gate power supplies. The disadvantage of this switch, compared to diode bridge with a single switch, is it requires more IGBTs.

5.3.2 Input Filter

As all power electronic-based converters, the MC injects harmonics into the grid, which affects other equipment connected to the same system. The power converter must meet the requirements given in IEEE 519. This standard refers to the allowable injected harmonic contents into the electrical network. Therefore, the MC requires a suitable input filter to reduce these harmonic components. To reduce the current harmonics in the supply, an LC low-pass filter can be included, as shown in Fig. 5.10. The input filter has to meet the following main requirements: high efficiency, small size and low cost and very small voltage drop across the filter inductance [13].

Fig. 5.10

a Input filter, b Clamp circuit

5.3.3 Clamp Circuit

The clamp circuit protects matrix switches from over voltages. Figure 5.10b shows the clamp circuit used to protect matrix switches. When all switches of the MC are turned off, the inductive load has to discharge the energy stored in it without making any dangerous over voltages. Therefore, the energy stored in the inductive loads can be discharged through the clamp circuit [6].

5.3.4 Control of the Matrix Converter

The MC comprises of nine-bi-directional switches which allow connecting all input lines to connect with all output lines. If the switches of MC are arranged as shown on Fig. 5.11a, the MC power at its input must be the same output power as there is no any energy storage element. An ISVM technique is the control method used with the MC [10, 14].

Fig. 5.11

a Direct topology of MC, b Indirect topology of IMC

5.3.4.1 Transformation from Indirect to Direct MC

Figure 5.12b shows that the ISVM technique deals with the MC as a rectifier-inverter converter with a virtual dc link. The indirect MC consists of two stages, the first stage is a current source rectifier based on switches S1–S6, second stage is voltage source inverter, which has a standard three phase voltage source topology based on six switches S7–S12 [15, 16].

Fig. 5.12

Transformation from Indirect MC to Direct MC in phase A

$$V_{DC} = E{*}V_{abc} ,\quad V_{ABC} = N{*}V_{DC} \;V_{ABC} = {\text{N}}*{\text{E}}*V_{abc} ,\quad {\text{K}} = {\text{N}}*{\text{E}}$$

(5.8)

$$E = \left[ {\begin{array}{*{20}c} {S_{1} } & {S_{3} } & {S_{5} } \\ {S_{2} } & {S_{4} } & {S_{6} } \\ \end{array} } \right],\;N = \left[ {\begin{array}{*{20}c} {S_{7} } & {S_{8} } \\ {S_{9} } & {S_{10} } \\ {S_{11} } & {S_{12} } \\ \end{array} } \right],\;K = \left[ {\begin{array}{*{20}c} {S_{aA} } & {S_{bA} } & {S_{cA} } \\ {S_{aB} } & {S_{bB} } & {S_{cB} } \\ {S_{aC} } & {S_{bC} } & {S_{cC} } \\ \end{array} } \right]$$

(5.9)

$$\left[ {\begin{array}{*{20}c} {{\text{S}}_{\text{aA}} } & {{\text{S}}_{\text{bA}} } & {{\text{S}}_{\text{cA}} } \\ {{\text{S}}_{\text{aB}} } & {{\text{S}}_{\text{bB}} } & {{\text{S}}_{\text{cB}} } \\ {{\text{S}}_{\text{aC}} } & {{\text{S}}_{\text{bC}} } & {{\text{S}}_{\text{cC}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\text{S}}_{7} } & {{\text{S}}_{8} } \\ {{\text{S}}_{9} } & {{\text{S}}_{10} } \\ {{\text{S}}_{11} } & {{\text{S}}_{12} } \\ \end{array} } \right]\;\left[ {\begin{array}{*{20}c} {{\text{S}}_{1} } & {{\text{S}}_{3} } & {{\text{S}}_{5} } \\ {{\text{S}}_{2} } & {{\text{S}}_{4} } & {{\text{S}}_{6} } \\ \end{array} } \right]$$

(5.10)

where, the matrix N represent the transfer function of the inverter, matrix E represent the transfer function of the rectifier and K represent transfer function of MC. This method deals with MC as a rectifier-inverter converter. Therefore, the space vector of the inverter output voltage and space vector of the rectifier input current can be decoupled to control the direct MC. As shown in Eq. (5.11), the output phases can be compounded by the product and sum of input phases through rectifier and inverter switches ${\text{S}}_{1} - {\text{S}}_{6}$ and ${\text{S}}_{7} - {\text{S}}_{12}$, respectively. The output phase A can be obtained from the input phases a, b and c for direct MC as shown in the first row of the matrix in (5.12) using the ISVM and this can be illustrated again in graphical viewpoint as shown in Fig. 5.12 [17, 18].

$$\left[ {\begin{array}{*{20}c} {V_{A} } \\ {V_{B} } \\ {V_{C} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{7} } & {S_{8} } \\ {S_{9} } & {S_{10} } \\ {S_{11} } & {S_{12} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {S_{1} } & {S_{3} } & {S_{5} } \\ {S_{2} } & {S_{4} } & {S_{6} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {v_{a} } \\ {v_{b} } \\ {v_{c} } \\ \end{array} } \right]$$

(5.11)

$$\left[ {\begin{array}{*{20}c} {V_{A} } \\ {V_{B} } \\ {V_{C} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{7} S_{1} + S_{8} S_{2} } & {S_{7} S_{3} + S_{8} S_{4} } & {S_{7} S_{5} + S_{8} S_{6} } \\ {S_{9} S_{1} + S_{10} S_{2} } & {S_{9} S_{3} + S_{10} S_{4} } & {S_{9} S_{5} + S_{10} S_{6} } \\ {S_{11} S_{1} + S_{12} S_{2} } & {S_{11} S_{3} + S_{12} S_{4} } & {S_{11} S_{5} + S_{12} S_{6} } \\ \end{array} } \right]*\left[ {\begin{array}{*{20}c} {v_{a} } \\ {v_{b} } \\ {v_{c} } \\ \end{array} } \right]$$

(5.12)

5.3.4.2 Indirect Space Vector Modulation

The main idea of the ISVM is to decouple the controlled output voltage of the inverter and the input current for the source rectifier. In this section, SVMs for current sources rectifier and voltage source inverter are introduced, then the two modulations can be decoupled to control the direct MC [19, 20].

5.3.4.2.1 Space Vector of the Current Source Rectifier

The current source rectifier consists of six switches S1–S6, as shown in Fig. 5.13. The rectifier has to generate constant dc voltage from three phase input voltage. The virtual dc link voltage and input currents can be calculated from the rectifier transfer function as follows [21, 22].

Fig. 5.13

Current source rectifier

$$\left[ {\begin{array}{*{20}c} {V_{DC}^{ + } } \\ {V_{DC}^{ - } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{1} } & {S_{3} } & {S_{5} } \\ {S_{2} } & {S_{4} } & {S_{6} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {V_{a} } \\ {V_{b} } \\ {V_{c} } \\ \end{array} } \right]$$

(5.13)

$$\left[ {\begin{array}{*{20}c} {I_{a} } \\ {I_{b} } \\ {I_{c} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{1} } & {S_{2} } \\ {S_{3} } & {S_{4} } \\ {S_{5} } & {S_{6} } \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} {I_{p} } \\ {I_{n} } \\ \end{array} } \right]$$

(5.14)

The input current space vector ${\text{I}}_{\text{IN}}$ can be expressed as:

$$I_{IN} = \frac{2}{3}\left( {I_{a} + a.I_{b} + a^{2} .I_{c} } \right)$$

(5.15)

There are only allowed nine switching states for the virtual rectifier so that an open circuit can be avoided in rectifier dc link. These nine switching states can be classified into three zero input current vectors ${\text{I}}_{0}$ and six active input current vectors ${\text{I}}_{1} - {\text{I}}_{6}$ as described in Fig. 5.14a. The current space vector state ${\text{I}}_{1}$ (a b) means that input phase a is connected to the positive terminal of the virtual dc link (V_DC+) and input phase b is connected to the negative terminal (V_DC−). Table 5.2 lists the possible switching states and relevant switching vectors. In addition, amplitude and angle of the input current space vector are evaluated for six active vectors and three zero vectors [3, 23]. Figure 5.14a shows the configuration of the discrete seven space vectors of the input current in a hexagon complex plane and the reference input current vector ${\text{I}}_{\text{IN}}^{ *}$ within a sector of the input current hexagon. The ${\text{I}}_{\text{IN}}^{ *}$ can be obtained by impressing the adjacent active vectors ${\text{I}}_{\upgamma}$ and ${\text{I}}_{\updelta}$ with the duty cycles ${\text{d}}_{\upgamma} \;{\text{and}}\;{\text{d}}_{\updelta}$, respectively, as shown in Fig. 5.14b. By using the current–time product sum of the adjacent active vectors, the reference input vector can be expressed as follows [2, 24].

Fig. 5.14

a Space vector of current source rectifier, b Composition of the reference input current

Table 5.2 Switching states and vectors for current source rectifier

$$I_{IN}^{*} = d_{\gamma } I_{\gamma } + d_{\delta } I_{\delta } + d_{oc} I_{0}$$

(5.16)

The duty cycle of the active vectors can be described as in [2]:

$$d_{\gamma } = m_{c} .\sin \left( {\frac{\pi }{3} - \theta_{c} } \right)$$

(5.17)

$$d_{\delta } = m_{c} .\sin \left( {\theta_{c} } \right)$$

(5.18)

$$d_{0c} = 1 - \left( {d_{\gamma } + d_{\delta } } \right)$$

(5.19)

where, the angle of the reference vector for input current can be represented by $\uptheta_{\text{c}}$. The ${\text{m}}_{\text{c}}$ represent the modulation index of the required input current vector and define such as;

$$m_{c} = \frac{{I_{IN}^{*} }}{{I_{DC} }}$$

(5.20)

5.3.4.2.2 Space Vector of the Voltage Source Inverter

The Voltage Source Inverter (VSI) consists of six switches ${\text{S}}_{7} - {\text{S}}_{12}$ as shown in Fig. 5.15. This VSI has to give three-phase output voltages from constant virtual DC input voltage [25, 26]. The output voltage can be represented as a function of DC input voltage and transfer function of the inverter as given in Eq. (5.21). The dc link current can be derived by using the transposed of the transfer function of the inverter as in Eq. (5.22)

Fig. 5.15

Voltage source inverter

$$\left[ {\begin{array}{*{20}c} {V_{A} } \\ {V_{B} } \\ {V_{C} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{7} } & {S_{8} } \\ {S_{9} } & {S_{10} } \\ {S_{11} } & {S_{12} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\frac{{V_{DC} }}{2}} \\ {\frac{{ - V_{DC} }}{2}} \\ \end{array} } \right]$$

(5.21)

$$\left[ {\begin{array}{*{20}c} {I_{p} } \\ {I_{n} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {S_{7} } & {S_{8} } \\ {S_{9} } & {S_{10} } \\ {S_{11} } & {S_{12} } \\ \end{array} } \right]^{T} \left[ {\begin{array}{*{20}c} {I_{A} } \\ {I_{B} } \\ {I_{C} } \\ \end{array} } \right]$$

(5.22)

The output voltage space vector ${\text{V}}_{\text{out}}$ can be expressed as follows:

$$V_{out} = \frac{2}{3}\left( {V_{A} + a.V_{B} + a^{2} .V_{C} } \right),\quad a = 1*e^{j120}$$

(5.23)

There are only eight switching states allowed for the inverter switches, so the output must not be short and the load must not be opened at any instant [27, 28]. These eight permitted combinations can be classified into six active nonzero output voltage vectors ${\text{V}}_{1} - {\text{V}}_{6}$ and two zero output voltage vectors ${\text{V}}_{\text{z}}$. The voltage space vector ${\text{V}}_{1}$ (100) means that the output phase ${\text{V}}_{\text{A}}$ is connected to the positive terminal of the virtual dc link (V_DC+) and the phases ${\text{V}}_{\text{B}} ,{\text{V}}_{\text{C}}$ are connected to the negative terminal (V_DC−). Table 5.3 lists the possible switching states and relevant switching vectors. In addition, the amplitude and angle of the output voltage space vector are evaluated for six active vectors and two zero vectors [24, 29].

Table 5.3 Switching states and vectors for voltage source inverter

Figure 5.16a describes the configuration of the hexagon of the discrete seven space vectors of the inverter in a complex plane. The vector of the reference output voltage ${\text{V}}_{\text{O}}^{ *}$ can be generated from the vector sum out of the seven discrete vectors,${\text{V}}_{1} - {\text{V}}_{6} \;{\text{and}}\;{\text{V}}_{\text{z}}$. This hexagon can be classified into six sectors. The duty cycles ${\text{d}}_{\upalpha}$ and ${\text{d}}_{\upbeta}$ for active vectors ${\text{V}}_{\upalpha}$ and ${\text{V}}_{\upbeta}$, respectively. The reference voltage vector ${\text{V}}_{\text{O}}^{ *}$ within a sector of the voltage hexagon can be derived from Fig. 5.16b.

Fig. 5.16

a Inverter voltage hexagon, b Reference output voltage vector composition

$$V_{o}^{*} = d_{\alpha } V_{\alpha } + d_{\beta } V_{\beta } + d_{z} V_{z}$$

(5.24)

$$d_{\alpha } = \frac{{T_{\alpha } }}{{T_{s} }} = m_{v} \;.\;\sin \;\left( {\frac{\pi }{3} - \theta_{v} } \right)$$

(5.25)

$$d_{\beta } = \frac{{T_{\beta } }}{{T_{s} }} = m_{v} \,.\,\sin \left( {\theta_{v} } \right)$$

(5.26)

$$d_{z} = \frac{{T_{z} }}{{T_{s} }} = 1 - \left( {d_{\alpha } + d_{\beta } } \right)$$

(5.27)

where, ${\text{T}}_{\upalpha} ,{\text{T}}_{\upbeta} \;{\text{and}}\;{\text{T}}_{\text{z}}$ are the total duration times of the vectors ${\text{V}}_{\upalpha} ,{\text{V}}_{\upbeta} \;{\text{and}}\;{\text{V}}_{\text{z }}$, respectively, and $\uptheta_{\text{v}}$ indicates the angle of the reference output voltage vector within the sector of the hexagon. The ${\text{m}}_{\text{v}}$ represents the modulation index of the vector of the output voltage and it is defined such as [30];

$$m_{v} = \frac{{\sqrt 3 v_{o,max} }}{{V_{DC} }}$$

(5.28)

where, ${\text{V}}_{\text{o}}$ is the desired output line voltage, ${\text{T}}_{0} = \frac{{{\text{T}}_{\text{z}} }}{2}$

5.4 Modified Symmetric Sequence Algorithm

This section proposes a modified symmetric sequence algorithm for SVM. The proposed algorithm reduces THD of the output voltage. When the required reference vector for the output voltage of the inverter lies in sector 1 as shown in Fig. 5.16b, the inverter switches ${\text{S}}_{7} - {\text{S}}_{12}$ does not have a state that represents this position, so this position can be represented by adjacent vectors ${\text{V}}_{\upalpha}$, ${\text{V}}_{\upbeta} \;{\text{and}}\;{\text{V}}_{\text{z}}$ with duty cycles ${\text{d}}_{\upalpha} ,\;{\text{d}}_{\upbeta} \;{\text{and}}\;{\text{d}}_{\text{z}}$. The main distinction between PWM algorithms that utilize adjacent vectors is zero vector selection, sequence in which the adjacent vectors are applied and splitting of the duty cycle of each adjacent vector.

5.4.1 Conventional Symmetric Sequence Algorithm

One of SVM algorithms is symmetric sequence algorithm that has a low THD as shown in Fig. 5.17a. The duty cycles of each vector ${\text{V}}_{\upalpha}$, ${\text{V}}_{\upbeta} \;{\text{and}}\;{\text{V}}_{\text{z}}$ (${\text{d}}_{\upalpha} ,{\text{d}}_{\upbeta} \;{\text{and}}\;{\text{d}}_{\text{z}}$) is calculated during each switching time ${\text{T}}_{\text{s}}$. In the conventional symmetric sequence algorithm, the duty cycle of vector ${\text{V}}_{\upalpha}$ (${\text{d}}_{\upalpha}$) is divided to two equal periods, ${\text{d}}_{\upbeta} \;{\text{also}}\;{\text{and}}\;{\text{d}}_{\text{z}}$ is divided to three periods $\frac{{{\text{d}}_{\text{z}} }}{2},\frac{{{\text{d}}_{\text{z}} }}{4}\;{\text{and}}\;\frac{{{\text{d}}_{\text{z}} }}{4}$. The sequence in this method is ${\text{V}}_{\text{z}} - {\text{V}}_{\upalpha} - {\text{V}}_{\upbeta} - {\text{V}}_{\text{z}} - {\text{V}}_{\upbeta} - {\text{V}}_{\upalpha}$ [2].

Fig. 5.17

a Conventional symmetric sequence algorithm, b Modified symmetric sequence algorithm

5.4.2 Modified Symmetric Sequence Algorithm

In this section, a modified symmetric sequence algorithm will be proposed, the sequence in which the vectors are applied. The modification in this algorithm will be the number of divisions of each duty cycle for each vector. In the modified algorithm the duty cycle of ${\text{V}}_{\upalpha}$ (${\text{d}}_{\upalpha}$) is divided to four equal periods, ${\text{d}}_{\upbeta} \;{\text{also}}\;{\text{and}}\;{\text{d}}_{\text{z}}$ is divided to five periods $\frac{{{\text{d}}_{\text{z}} }}{4},\frac{{{\text{d}}_{\text{z}} }}{4},\frac{{{\text{d}}_{\text{z}} }}{4},\frac{{{\text{d}}_{\text{z}} }}{8}\;{\text{and}}\;\frac{{{\text{d}}_{\text{z}} }}{8}$. The sequence in the proposed algorithm is as follow ${\text{V}}_{\text{z}} - {\text{V}}_{\upalpha} - {\text{V}}_{\upbeta} - {\text{V}}_{\text{z}} - {\text{V}}_{\upalpha} - {\text{V}}_{\upbeta} - {\text{V}}_{\text{z}} - {\text{V}}_{\upbeta} - {\text{V}}_{\upalpha} - {\text{V}}_{\text{z}} - {\text{V}}_{\upbeta} - {\text{V}}_{\upalpha} - {\text{V}}_{\text{z}}$ as shown in Fig. 5.17b.

5.4.3 Implementation of Modified Symmetric Sequence Algorithm

This part introduces how to implement the SVM for VSI with modified symmetric sequence algorithm by using MATLAB Simulink. Figure 5.18a shows how to transform the reference output voltage into vector angle and vector amplitude. By using the angle of the reference output voltage, we can know number of sector in which the reference output voltage is located. If the required reference vector for the output voltage of the inverter lies in sector 1 as shown in Fig. 5.18a the inverter switches ${\text{S}}_{7} - {\text{S}}_{12}$ does not have a state that represents this position, so this position can be represented by adjacent vectors ${\text{V}}_{\upalpha}$, ${\text{V}}_{\upbeta} \;{\text{and}}\;{\text{V}}_{\text{z}}$ with duty cycles ${\text{d}}_{\upalpha} ,\;{\text{d}}_{\upbeta} \;{\text{and}}\;{\text{d}}_{\text{z}}$. After calculating the duty cycles of each adjacent vectors, the modified symmetric sequence algorithm will be used. Figure 5.18b shows the trend of implementation modified symmetric sequence algorithm in MATLAB Simulink. The first step after calculating duty cycles for each adjacent vector in the modified symmetric sequence algorithm is to apply the zero vectors for a time equal to one-fourth of its period. This can be achieved by comparing the one-fourth of zero vectors duty cycle with a ramp signal with a switching time equal to 2 µs as shown in Fig. 5.18b. The second step is to apply the first vectors for a time equal to one-fourth of its period. This can be achieved by comparing the ramp signal with two signals as shown in Fig. 5.18b. The third step is to apply the second vectors for a time equal to one-fourth of its period. This can be achieved by comparing the ramp signal with two signals as shown in Fig. 5.18b and so on.

Fig. 5.18

Implementation of the modified symmetric sequence algorithm

Fig. 5.19

THD for 25 Hz output voltage for conventional symmetric sequence algorithm

5.4.4 Simulation Results for Symmetric Sequence Algorithm

Simulations were done using the MATLAB/Simulink software package. The simulation results for a MC interfaced 50 Hz three-phase supply with an isolated R-L load (R = 144 Ω, L = 0.25 H) will be presented. The modified symmetric sequence algorithm is used with indirect space vector to control the MC. The modified algorithm reduces the THD for the output voltage as shown in Fig. 5.20, also THD for output voltage with conventional symmetric sequence is shown in Fig. 5.19. The simulation results showed that the THD of output voltage is reduced with the proposed algorithm. The proposed method decreases the THD of the output voltage, so decreases cost and size of the required filter. The THD of the output voltage is shown in Table 5.4.

Fig. 5.20

THD for 25 Hz output voltage for modified symmetric sequence algorithm

Table 5.4 THD for conventional and modified symmetric sequence algorithm

5.5 Modified Open Loop Control of MC

The MC can control the rms value of output voltage and frequency, but the output voltage of it in case of open loop control is a percent of the input voltage. If q (the ratio between output voltage and input voltage) = 0.4 in the open loop control, input voltage = 100 V, the output voltage will be 40 V but with the required frequency. If the input voltage changes from 100 to 50 V, the output voltage will be 20 V. If we need to obtain constant output voltage, the q ratio must be changed from 0.4 to 0.8, so the q ratio must depend on the input voltage and this can be achieved by modified open loop control. The modified open loop control takes a signal from three-phase input voltage and q ratio can be calculated from Eq. (5.29).

$$q = \frac{{V_{out}^{*} }}{{V_{IN} }}$$

(5.29)

From Table 5.5 and Eq. (5.29), the q ratio in case of modified open loop control depends on the input voltage where if the input voltage decreases, the q ratio increases so that a constant output voltage is obtained. Figure 5.21 shows the proposed modified open loop control of ISVM. Figure 5.21a shows the block diagram of the ISVM with the modification and Fig. 5.21b shows the modification in the MATLAB Simulink model.

Table 5.5 Modified open loop control and open loop control

Fig. 5.21

Modified open loop control of ISVM

5.5.1 Simulation Results for Modified Open Loop Control

Simulations were done using MATLAB/Simulink software package where the MC interfaced wind energy conversion system is used to feed an isolated R-L load (R = 2 Ω, L = 1 mH). The used output LC filter has a value of L = 2.3 mH, C = 100 µf [3]. To obtain the desired output voltage and frequency, the MC is controlled using ISVM with a modified open loop control. Figures 5.22 and 5.23 show the simulation results at different wind speeds with the open loop control. In the modified open loop control, the output voltage and frequency remains constant to 220 V, 50 Hz even if the wind speed changed. In case of the open loop control, the output voltage is a ratio of input voltage and therefore, the output voltage changed with speed as shown Fig. 5.22c. Table 5.6 gives the magnitude of voltages and frequency with the variation of wind speeds. Figures 5.22a and 5.23a show the wind velocity with time, where the speed of wind changed from 7 to 12 m/s at t = 0.3 s, from 12 to 9 m/s at t = 0.6 s, from 9 to 6 m/s at t = 1 s. Figure 5.22b shows the input voltage to MC (generated voltage), Fig. 5.22c shows the simulation results of output voltage of 50 Hz with the open loop control. It is clear from Fig. 5.22c that the magnitude of the output voltage is not constant.

Fig. 5.22

Simulation results for open loop control of a matrix converter

Fig. 5.23

Simulation results for modified open loop control of a matrix converter

Table 5.6 Variation of generated voltage and frequency with wind velocity

Figure 5.22d shows the simulation results of the output current at 50 Hz with the open loop control. Figure 5.23b shows the input voltage to MC (generated voltage), where for t = 0:0.3 s the generated voltage is 320 V, 25 Hz, at t = 0.3:0.6 s the generated voltage is 460 V, 46 Hz, at t = 0.6:1 s the generated voltage is 380 V, 34 Hz, at t = 1:1.5 s the generated voltage is 280 V, 21 Hz. Figure 5.23c shows the simulation results for the desired output voltage of 220 V, 50 Hz with the modified open loop control. It is clear from Fig. 5.23c that the magnitude of the output voltage is constant and equal to 220 V. Figure 5.23d shows the simulation results of output current at 50 Hz with the modified open loop control.

5.5.2 Experimental Results for Modified Open Loop Control

Experimental results were performed using DSP1104, with an isolated, static load of (R = 20 Ω, L = 40 mH). The field and the armature voltage of dc motor is controlled to control its speed to simulate the wind turbine. Figure 5.24 shows the experimental results for change in speed at a time 20 s in case of the open loop control with 50 Hz output frequency, where Fig. 5.24a shows the change in speed where it from 1535 to 1675 rpm by increasing the armature voltage. Figure 5.24b shows the input voltage, where the rms value and the frequency of the input voltage increase with the increase in speed and the input frequency increases from 30 to 35 Hz as shown in the zoomed view of the input voltage in Fig. 5.24c. Figure 5.24d shows the output voltage for the open loop control and Fig. 5.24e shows the zoomed view of output voltage which has 50 Hz frequency. The output frequency does not change even if the input frequency changes. The rms value of the output voltage is a percent of the input voltage, so the rms value of the output voltage changes with a change in input voltage as shown in Fig. 5.24e.

Fig. 5.24

Experimental results with open loop control with output frequency 50 Hz

Figure 5.25 shows the experimental results for a change in speed at t = 18.5 s in case of the modified open loop control with a 50 Hz output frequency, where Fig. 5.25a shows the change in speed where the speed changes from 1528 to 1653 rpm by increasing the armature voltage. Figure 5.25b shows the input voltage where the rms value and frequency of the input voltage increase with the increase in the speed. The input frequency increases from 29.3 to 33.5 Hz approximately as shown in the zoomed view of the input voltage in Fig. 5.25c. Figure 5.25d shows the output voltage in case of the modified open loop control. Figure 5.25e shows the zoomed view of output voltage which has frequency of 50 Hz. The output frequency does not change even if the input frequency changes. The rms value of the output voltage is a percent of the input voltage, so to obtain a constant rms value of the output voltage with the change in input voltage as shown in Fig. 5.25e; the reference output voltage must be inversely changed with change in input voltage.

Fig. 5.25

Experimental results with modified open loop control with output frequency 50 Hz

5.6 Conclusion

This chapter introduces performance improvement of the MC fed from a wind energy system. A theoretical analysis is carried out supported by numerical analysis. The results have been carried out using a designed complete prototype setup. According to the initiating problems and aims mentioned in the introduction, the following are the achieved four scenarios. The first analysis has been done for using the MC to control the voltage and frequency of a static R-L load fed from WECS. The angle between the input voltage and current of MC is controlled and the input displacement factor with a unity value is achieved. The model produces a very good waveform on the output side with a wide range of frequency changes. Secondly, the analysis of transforming from the indirect to direct MC is introduced; in addition to introducing analysis for the ISVM. Thirdly, a modified symmetric sequence algorithm for SVM is proposed. The proposed method is compared with the conventional algorithm and it has a lower output voltage THD. Fourthly, a modified open loop control of the ISVM is proposed which improved the performance of the MC with a variable speed operation of the wind turbine. Finally, all measured results showed good correspondence with those obtained by simulation.