Renewable Energies
GEL632
Parts II

Tilda AKIKI

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Part I - Introduction to Renewable Energy Resources Hydrogen Solar and Photovoltaic Geothermal Biomass Ethanol Wind Water / Tides Part II - Modeling converters in microgrid power systems DC/AC inverters DC/DC converters AC/DC rectifiers Part III - Microgrid solar energy systems Photovoltaic power conversion Modeling of a Photovoltaic Module Part IV - Microgrid wind energy systems Wind power - Modeling of induction machines Load Flow Analysis of power grids and microgrids Part V – Solar water heating systems Part VI – Fuel Cells: Components and operation Part VII - Renewable Energy Impacts Methodology of Energy Planning / Life-cycle Analysis / Greenhouse Warming / Ecological Sustainability 2

Part II – Modeling converters in microgrid power systems Single-phase DC/AC inverter with 2 switches (single-phase half-bridge inverter) P: 94
Objective: Convert power from DC to AC at the system frequency of operation. • The PWM (Pulse Width Modulation) technique is used to achieve an AC voltage with a fundamental frequency of 60 or 50Hz. • The two power switches are sequentially turned on an off. • If SW1+ is on, potential in a is the same as potential of the positive DC bus, that is: Van = Vidc. • If SW1- is on, potential in a is the same as that of the negative DC bus, that is: Van = 0. • To generate a time-varying voltage at point a, a sine wave with the desired frequency is compared with a triagular wave to determine the switching policy. • The turn on and off sequence of the 2 switches is decided by the relative value of the sine wave with respect to the triangular wave. 3 (Fig. 3.4, P: 95, Ali Keyhani)

Part II – Modeling converters in microgrid power systems Single-phase DC/AC inverter with 2 switches (single-phase half-bridge inverter)
• Two waves are being compared: A sine wave Vc designated as controlled voltage. A triangular wave VT with amplitude and freq higher than Vc. • If Vc > VT, SW1+ is on, SW1- is off, and Van = Vidc • If Vc < VT ,SW1- is on, SW1+ is off, and Van = 0. • Van is not a pure sinusoid, but it has a DC component, a fundamental AC voltage at the same freq as the control reference voltage and the harmonics voltages. • The DC component of Van is Vidc/2. • The magnitude of the fundamental of AC output voltage is directly proportional to the ratio of the peaks of Vc and VT • This ratio is defined as the amplitude modulation index Ma and varies between 0 and 1: Ma = VC(max) / VT(max) • The peak of the fundamental component of the output voltage is Van = Ma x Vidc / 2 • The instantaneous value of the output voltage will be: 4 Van = Vidc/2 + (Vidc / 2) x Ma x sin wet + harmonics

Part II – Modeling converters in microgrid power systems Single-phase DC/AC inverter with 2 switches and load connected to the Center Tap position
• The inverter has 2 capacitors of equal capacitance with available center tap. (Fig. 3.6, P: 97, Ali Keyhani) • When SW1+ is on, SW1- is off, Van = Vidc and Vao = +Vidc/2. • When SW1- is on, SW1+ is off, Van = 0 and Vao = -Vidc/2. • The load voltage is between + and – values of Vidc/2. • The DC component is 0 and Vao is alternating. (Fig. 3.7, P: 98, Ali Keyhani) • If Vc > VT, SW1+ is on, SW1- is off, and Vao = Vidc/2 • If Vc < VT ,SW1- is on, SW1+ is off, and Vao = -Vidc/2. • The instantaneous value of the output voltage will be: Vao = (Vidc / 2) x Ma x sin wet + harmonics • The fundamental freq. of the output voltage is the same as the frequency of the sine voltage Vc. • Thus, by adjusting the peak of the sine wave, the amplitude of the output voltage can be varied. • Similarly, by changing the frequency of the sine wave Vc, the output 5 frequency is varied.

Part II – Modeling converters in microgrid power systems
Single-phase DC/AC inverter with 2 switches and load connected to the Center Tap position

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The output of the inverter should be as close to the sine wave as possible. The harmonic contents in the voltage should be minimized. To achieve low harmonic distorsion, freq of triangular wave is increased relatively to that of Vc. The freq modulation index is increased. Mf = fs / fe [fs = freq of VT, fe = freq of Vc] The harmonic content of the output voltage for a different Mf with Ma = 0.6 [% of amplitude]
Order of harmonic 1 3 5 7 Mf = 3 (%) 100 163 61 73 Mf = 5 (%) 100 22 168 25 Mf = 7 (%) 100 0.42 22 168 Mf = 9 (%) 100 0.05 0.38 22

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37

62

22

168

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If Mf is high, the low-order harmonic content of the voltage will be very small and the high-order harmonics will be filtered out giving near sinusoidal currents. [the load on an inverter is usually inductive, which acts like a low-pass filter, therefore, high-order harmonics are already easily filtered out]. 6 [When the load is inductive, the current lags behind the voltage]. Ex. P:102, Ali Keyhani (papers 2.2).

Part II – Modeling converters in microgrid power systems Single-phase DC/AC inverter with a four-switch bipolar switching method (Fig. 3.12, P: 106, Ali Keyhani)
• SW1+ and SW2- and SW1- and SW2+ are switched in pairs. • If Vc > VT, SW1+ and SW2- are on, other swiches are off, Van = Vidc and Vbn = 0, and Vab = Van – Vbn = Vidc. • If Vc < VT, SW1- and SW2+ are on, other swiches are off, Van = 0 and Vbn = Vidc, and Vab = Van – Vbn = -Vidc. (Fig. 3.12, P: 107, Ali Keyhani) • This modulation technique is known as bipolar sine PWM as the output voltage jumps between + and – values. • The output voltage is: Vab = Vidc x Ma x sinwet + harmonics • The DC offset voltage is 0. • The amplitude of the fundamental component is decided by the sine wave by varying the amplitude modulation index: Ma = Vc(max)/VT(max) • The harmonic content can be reduced by increasing the modulation index: Mf = fs/ fe 7
• Exercises p: 108, Ali Keyhani, (papers 2.3)

Part II – Modeling converters in microgrid power systems
Pulse Width Modulation with Unipolar Voltage Switching for a single-phase fullbridge (Fig. 3.12, P: 106, Ali Keyhani) • In the bipolar PWM scheme, the output PWM voltage jumps between +Vidc and –Vidc. The load is subjected to high-voltage fluctuations. • The unipolar PWM method allows the output voltage to jump between +Vidc and 0 or –Vidc and 0. • Two sine waves , 180 apart from each other, are compared with the triangular wave. (Fig. 3.15, P: 109, Ali Keyhani) • The unipolar switching policy is: • If Vc > VT and –Vc < VT : SW1+ and SW2- are on, other switches are off and Van = Vidc and Vbn = 0 so Vab = Vidc. • If Vc < VT and –Vc < VT : SW1- and SW2- are on, other switches are off and Van = 0 and Vbn = 0 so Vab = 0. • If Vc < VT and –Vc > VT : SW1- and SW2+ are on, other switches are off and Van = 0 and Vbn = Vidc so Vab = -Vidc. • If Vc > VT and –Vc > VT : SW1+ and SW2+ are on, other switches are off and Van = Vidc and Vbn = Vidc so Vab = 0. • The output voltage is given by: Vab = Vidc x Ma x sinwet + harmonics • Similar to the other PWM schemes, the freq and the magnitude of the output voltage are controlled by the reference sine wave and the harmonic content 8 of the output voltage is determined by the frequency modulation index.

Part II – Modeling converters in microgrid power systems Three-phase DC/AC inverters (Fig. 3.20 P: 119, Ali Keyhani) • A three-phase inverter has 3 legs, one for each phase. • The sine wave signal is supplied to a Digital Signal Processor (DSP) controller to control the output AC voltage, power, and frequency. • The DSP controller sends a sequence of switching signals to control the six power switches to produce the desired output AC power.

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Part II – Modeling converters in microgrid power systems Three-phase DC/AC inverters
• By turning on the upper switch , the output node (a, b, or c) acquires a voltage of the upper DC line. • When the lower switch of a limb is on, the output node of that limb attains a voltage of the lower DC line. The node voltage oscillates between the upper and the lower DC line voltages. • The RMS value of the fundamental of output line voltage is: VL, rms = Ma . √(3/2) . Vidc/2 • The modulated output voltage of each leg has the same waveform of a single-phase converter.

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Part II – Modeling converters in microgrid power systems Three-phase DC/AC inverters
• Frequency of triangular wave desirable to be very high : cause this will result in low harmonics in output voltage and currents.

BUT • High freq of triangular wave increases switching freq which will cause high switching losses. • With a high switching freq, switches may fail to turn on and off properly. • In residential and commercial systems, switching freq has to be out of the audible freq range (20 to 20,000 Hz) to reduce high-pitch audible noise. • Switching freq is selected to be below 6 kHz or above 20 kHz. • Residential: below 6 kHz.
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Part II – Modeling converters in microgrid power systems Three-phase DC/AC inverters
• Vc(for phase a) = Vc(a) = Ma . sin (wet) Vc(b) = Ma . sin(wet - 2π/3) Vc(c) = Ma . sin(wet - 4π/3) • The triangular wave of VT(t) has 2 lines: or x1(t) and x2(t) • The algorithm PWM voltage is: (Fig. 3.21, P: 122, Ali Keyhani) if Vc(a) >= x1(t) or x2(t), then Van = Vidc if Vc(b) > = x1(t) or x2(t), then Vbn = Vidc If Vc(c) > = x1(t) or x2(t), then Vcn = Vidc Otherwise, Van = 0, Vbn = 0, and Vcn = 0.
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Part II – Modeling converters in microgrid power systems
• Example 3.5, p: 121, Ali Keyhani (paper 2.4) Compute the minimum DC input voltage if the switching frequency is set at 5kHz. Assume the three phase inverter is rated 207.6V AC, 60Hz, 100kVA. • Exercise 3.6, P: 125, Ali Keyhani (paper 2.5) Consider the system that I will give you and assume Load1 is a three-phase load of 5kW at a power factor of 0.85 leading, and Load 2 is a three-phase load of 10kW at a power factor of 0.9 lagging at rated voltage of 110V with a 10% voltage variation. Assume a PV source voltage is rated 120V DC. Compute the transformer ratings for an amplitude modulation index Ma = 0.9 (consider an ideal transformer).

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Part II – Modeling converters in microgrid power systems
Pulse Width Modulation Methods: I – The triangular method (Fig. 3.18, P: 113, Ali Keyhani) This method can be modeled by equations of lines Y = mx + b t1 = 0:Step size:Ts/4 t2 = Ts/4 + Step size : Step size : Ts/2 t3 = Ts/2 + Step size : Step size : 3Ts/4 t4 = 3Ts/4 + Step size : Step size : Ts • The Step size is user defined and should be reasonably small. • One period of the triangular wave can be broken into 4 regions as follows: Line1 : VTmax . t1 / (Ts/4) Line2 : … Line3 : … Line4 : … • In a MATLAB simulation, Time = [t1 t2 t3 t4] FullLine = [Line1 Line2 Line3 Line4] Plot(Time, FullLine) • The above codes can only create one period but this can be remedied…

Part II – Modeling converters in microgrid power systems Pulse Width Modulation Methods: II – The identity method • This method uses identity mapping by assigning a number x to the same number x. x = 1 then y = 1, x = 2 then y = 2, etc … • The function that accomplishes this for a triangular wave can be expressed as F(x) = sin-1(sin x) To be checked

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Part II – Modeling converters in microgrid power systems • • • • • • • • The DC/DC converters in green energy systems The DC/DC converter is a three-terminal device(paper 2.6) The input voltage is converted to a higher or lower output voltage as the switching frequency is controlled. Depending on whether the output voltage is lower or higher than the input value, the converter is called a buck or boost or buck-boost converter. The components of DC/DC converters are an inductor, a capacitor, a controllable switch, a diode, and a load resistance. Energy stored in an inductor = ½ L iL2 Energy stored in a capacitor = ½ C vC2 vL = L diL/dt iC = C dvC/dt

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Part II – Modeling converters in microgrid power systems • • • • • • • • • The Step-up converter (boost converter) A PV module is a variable DC power source. Its output depends on the sun. The same is true for a variable-speed wind energy source. A DC/DC boost converter allows capturing a wider range of DC power by boosting the DC voltage. A step-up converter consists of an inductor L, a capacitor C, a controllable switch S, a diode D, and a load resistance (battery). (paper 2.6) The inductor draws energy from the source and stores it as a magnetic field when switch S is on, and VL = Vin When switch S is off, this energy and additional energy from input is transferred to the capacitor and VL = Vin - Vo [note that L acts as a generator now]. L is used as a temporary storage element. The duty ratio D is defined as: D = Ton / TS = Ton / Ton + Toff Also: Toff = TS – Ton = (1- D) TS

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Part II – Modeling converters in microgrid power systems The Step-up converter (boost converter) • When steady-state is reached, the output voltage will be higher than the input voltage and the magnitude depends on the duty ratio of the switch. Vo = Vin / (1- D) and Iin = Io / (1-D) D is less than 1 • The ripple in the inductor current Δ IL= (Vin / L).DTS • The max and min values of the inductor current are: ILmax= IL+ Δ IL/2 and ILmin= IL- Δ IL/2

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Part II – Modeling converters in microgrid power systems

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The Step-down converter (Buck converter) The buck converter converts the input DC voltage level to another and lower voltage level. The main components are a semiconductor switch S, a diode D, an inductor filter L, and a capacitor filter C. (paper 2.8) Switch S closed: D is reverse biased, inductor L is being charged from the source and VL = Vin - Vo Switch S opened: the inductor L forces the diode D into conduction, inductor L delivers its energy to the capacitor C and the load R and VL = - Vo
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Part II – Modeling converters in microgrid power systems
The Step-down converter (Buck converter) • When steady-state is reached, we have: Vo = D.Vin and Iin = D.Io • The ripple in the inductor current Δ IL= (Vo / L).(1 – D)TS • The max and min values of the inductor current are: ILmax= IL+ Δ IL/2 and ILmin= IL- Δ IL/2 • At steady-state, the average capacitor current is zero and I L = Io .

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Part II – Modeling converters in microgrid power systems Exercice Assume a PV system with DC bus voltage of 120Vdc. A DC/DC buck converter, which has a duty ratio of the switch of 0.75 with a switching frequency of 5 kHz, is supplied by the PV system DC bus. If L = 1mH, C = 100 μF and the battery system is assumed to act as a load resistance of 4 Ω. Compute: The output voltage and the output current The average inductor current The rating of the switch and the diode (paper 2.9)

1. 2. 3.

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Part II – Modeling converters in microgrid power systems

The Buck-Boost converter
• It consists of Vin, L, C, D, and S. (paper 2.10) • S closed: D is reverse biased, L starts to increase its stored energy, VL = Vin • S opened: D is forced to conduct the inductor current, VL = -Vo. • It is found that: Vo = D. Vin / (1-D) and Iin = D.Io / (1-D) • The average inductor current is: Iin = D.IL

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Part II – Modeling converters in microgrid power systems

The Buck-Boost converter
• The ripple in the inductor current Δ IL= (Vin / L).DTS • The max and min values of the inductor current are: ILmax= IL+ Δ IL/2 and ILmin= IL- Δ IL/2 • If D < ½, Step-down converter • If D > ½, Step-up converter

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Part II – Modeling converters in microgrid power systems • The quality of the output voltage is determined by the amount of ripple in it. • The smaller the ripple, the better the quality of the output voltage is. • The ripples in the output voltage can be reduced by increasing the capacitance C or by increasing the switching frequency fs. • Increasing the capacitance increases the size of the capacitor, which increases the size and cost of the converter. • Increasing fs would increase the switching losses and decreases the efficiency of the converter. • A tradeoff must be reached between the quality of the output voltage and the size of capacitor and switching frequency. Tilda Akiki - Renewable Energies 24

Part II – Modeling converters in microgrid power systems • If the ripple in inductor current is high, the max current in the switches will increase, increasing their rating and cost. • To limit the ripple in inductor current, inductance should be as high as possible BUT • The size and cost of converter increases with higher size of inductor. • A tradeoff must be reached between the size of inductor, capacitor, switching frequency, and the cost and size of the converter and its efficiency.

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Part II – Modeling converters in microgrid power systems Exercice Design a buck-boost converter that is supplied from a PV source whose voltage varies from 80 to 140 V [depending on the available sun irradiant energy] and supplies a load of 10 kW at fixed 120 VDC. Give the range of duty ratio needed to supply the load at the rated voltage. (paper 2.11)

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Part II – Modeling converters in microgrid power systems
• • • • Rectifiers A rectifier is a power converters that converts AC voltage into DC. A three-phase diode bridge rectifier consists of six diodes arranged in three branches. See figure (paper 2.12) The input voltage to the rectifier is three-phase AC voltage and the output is DC voltage. The following voltages are supplied to the three-phase rectifer Va = √(2).V. sin (wt) Vb = √(2).V. sin (wt - 2π/3) Vc = √(2).V. sin (wt - 4π/3) At any given moment, two diodes will be conducting: one from a branch at the upper side and the other from another branch at the lower side. Which of the two diodes conducts depends on which of the available line voltages has the highest positive value at that moment. (page 156, figure 3.55, Ali Keyhani)
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Part II – Modeling converters in microgrid power systems Rectifiers • The average value of the output DC voltage is a DC wave with ripples. • The output voltage that appears across the output terminals of the rectifier is: Vidc = 3√(2) /π. Vab,rms • The capacitor C is added to filter out the ripples and make the voltage uniform.

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Part II – Modeling converters in microgrid power systems The sizing of an inverter for microgrid operation • The switches of the inverter should be rated at the voltage of the DC line and should be able to carry current for the rated condition. • The frequency modulation index should be chosen from the commutation characteristics of the switches used. • For a three-phase DC/AC inverter, the RMS value of the fundamental of output line voltage is: Vab, rms = Ma . √(3/2) . Vidc/2
Vac = 120V VDC (V) 200 Ma 0.98

250 300

0.78 0.65
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Part II – Modeling converters in microgrid power systems

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The sizing of a rectifier for microgrid operation The switches of the rectifier should be rated at the voltage of the DC line and the frequency modulation index is determined from the commutation time of the switches used. Similar to the inverter, the harmonic content of the current is less if the sampling freq is high. However, a higher sampling freq results in higher switching losses. A tradeoff is reached between the switching loss and the harmonic content to decide upon the sampling frequency.
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Part II – Modeling converters in microgrid power systems

The sizing of DC/DC converters for microgrid operation • The output voltage depends on the duty ratio of the switch.
Vo = Vin / (1- D) : Boost converter Vo = D.Vin : Buck converter Vo = D. Vin / (1- D) : Buck-Boost converter

• For the buck converter and for the boost converter: the switch should have a rating equal to that of the high voltage side. • For the buck-boost converter: the switch should be rated to a voltage level higher than the sum of the input and output voltages.
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Part II – Modeling converters in microgrid power systems

2 exercices Papers 2.14 and 2.15

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