An-Najah National UniversityFaculty of Engineering

Electrical Engineering Department

Power factor correction by Static Variable Compensator

Prepared by:Mohammed Abdallateef Abdaljawwad 11005554Amjad Mohammad Adarba 11001713

Eyad Riad Amer 11001996

Supervised by:

Dr. Kamil Salih

Acknowledgement

we take this opportunity to express our profound gratitude and deep regards to our guide Dr.Kamil Salih for his exemplary guidance, monitoring, continued support and constant encouragement throughout the course of this project.

Table of Contents

Chapter 1...................................................................................................1

Section 1.1 :Introduction..........................................................................1

Section 1. 2 . Objectives............................................................................2

Chapter 2...................................................................................................4

Section 2.1 : Literature review.................................................................4

Section 2.2 :Theoretical background..................................................... 5

Chapter 3: components of the project ……............................................8

Section 3.1 : zero crossing detector ………….…………………………8

Section 3.2 Triac ………….………………………………….…………9

Section 3.3 : Arduino ……………...…………………..…….…………10

Chapter 4 :Methodology…………………………………….……….….11

Section 4.1: Static Variable Compensation in closed loop control..….11

Section 4.2: : Static Variable Compensation in open loop control …..14

Section 4.2.1: PM meter ……………………………………………….15

Section 4.2.2: determine the firing angle …………………………….18

Section 4.2.3: Generating the firing angle………………….................24

Chapter 5: results and conclusions…………………………………...26

References ………………………………………………………..33

Appendix A……………………………………………………….34

Appendix B ……………………………………………………….37

List of Figures

Figure.1: Load connected to the Generator through a Transmission line..............................................................................................................5

Figure.2: Lag between voltage and current..........................................6

Figure.3: Compensated load on The Transmission line.......................6

Figure.4: Schematic of the relation between PF with the Apparent power, Power angle and Reactive power................................................6

Figure 5: The current and voltage citation after correction................7

Figure 6: circuit of zero crossing detection……………………………….8

Figure 7: the Triac symbol and a simplified cross section of the device………………………………………………………………………………………9

Figure 8: simulation of closed loop system ………………………….12

Figure 9 : power factor behavior at closed loop system ……………13

Figure 10 : zero cross detector output ……………………………….15

Figure 11 : comparing between the voltage signal and ZCD ……….16

Figure 12 : triac operation ……………………………………………19

Figure 13: the shape of firing equation ……………………………..20

Figure 14 : a graphical depiction of the bisection method…………21

Figure 15: the code of bisection method to determine firing angle..22

Figure 16: changing alfa with changing load condition……………24

Figure 17: arduino results 1 ……………………………………….26

Figure 18: arduino results 2 ……………………………………….27

Figure 19 :arduino results 3 ………………………………….…….28

Figure 20 :arduino results 4 ………………………………….…….29

Figure 21: arduino results 5 ………………………………….…….30

Figure 22: comparing between alfa 1 and current ZCD …………...31

Figure 23: comparing between alfa 2 and current ZCD …………...31

Figure 24: comparing between alfa 3 and current ZCD …………...32

Figure 25: comparing between alfa 4 and current ZCD …………...33

List of Tables

Table 1 : the relation between number of iteration and the max error

knowing that the value of alfa included in the interval of [0 , pi]…………23

Nomenclature or list of symbols:

Z = Circuit impedance, Ω.R = load Circuit resistance, Ω.XL = load inductive reactance, Ω..Xc= capacitor reactance L= inductance value (Henry)C= capacitive value (Farad) I = Load current, A.PF= power factor Vrms= rms value of voltage at the shunt compensatorVm= peak value of the voltage supplied by the source P= real power consumed by the resistive part of the loadQc= reactive power generated by the capacitor branchQl= reactive power consumed by the inductor (load branch)S= apparent power transmitted from the source to the load α= firing angle of the Triac= phase shift between voltage and current at the Ɵend of the transmission lineW= radial frequency of the source f= frequency of the source in HzTon= the time of the periodic signal that gives ONE value

ZCD= zero cross detector

ABSTRACT

The objective of this project is to improve power factor of transmission

lines using SVC (Static Variable Compensator). Static VAR Compensation under

FACTS uses TSC (ThyristorSwitched Capacitors) based on shunt compensation duly

controlled from a programmed microcontroller.

Prior to the implementation of SVC,power factor compensation was done by

large rotating machines such as synchronous condenser or switched capacitor banks.

These were inefficient and because of large rotating parts they got damaged quickly.

This proposed system demonstrates power factor compensation using thyristor

switched capacitors.

Shunt capacitive compensation – This method is used to improve the

powerfactor. Whenever an inductive load is connected to the transmission line, power

factor lags because of lagging load current. To compensate for this, a shunt capacitor

is connected which draws current leading the source voltage. The net result is

improvement in power factor. The time lag between the zero voltage pulse and zero

current pulse duly generated by suitable operational amplifier circuits in comparator

mode are fed to two interrupt pins of the 8 bit microcontroller of Arduino family.

Thereafter program takes over to actuate appropriate number of opto-isolators duly

interfaced to back to back SCRs. This results in bringing shunt capacitors into the

load circuit to get the power factor till it reaches unity .

Further the project can be enhanced to thyristor controlled triggering for

precise PF correction instead of thyristor switching in steps.

Chapter 1

Section 1.1 : Introduction

Modern civilization depends mostly on electrical energy for agricultural, commercial,domestic, industrial and social purposes [1]. The electrical energy isexclusively generated, transmitted and distributed in the form of alternating current(a.c.). Any load can be presented basically in three elements which are resistor , inductor and capacitor. The resistor consumes active energy in which the electrical energy takes a new form of energy (eg. Heat , mechanical, illumination …etc) while the inductor and capacitor store the electrical energy in magnetic field and electric field respectively which means the electrical energy still in its original form.

The actual amount of power being used, or dissipated, in a circuit is called real power. Reactive loads, inductors and capacitors dissipate zero power, yet they drop voltageand draw current giving the deceptive impression that they actually do dissipate power. This “phantom power” is called reactive power[2]. More precisely power dissipated by a load is referred to as real power where as power merely absorbed and returned in load due to its reactive properties is referred to as reactive power. However in nature, most of the loads are inductive loads consuming reactive power and resulting in low lagging power factor, on the other hand capacitive load (capacitor banks) generating reactive power and resulting in leading power factor. So the capacitors and inductors loads have a opposite effect on power factor . Effects of reactive power flow in line network1- Poor transmission efficiencyLosses in all power system elements from the power station generator to the utilization devices increase due to reactive power drawn by the loads, thereby reducing transmission efficiency.2- Poor voltage regulationDue to the reactive power flow in the lines(higher current), the voltage drop in the lines increases due to which low voltage exists at the bus near the load and makes voltage regulation poor.3- Low power factorThe operating power factor reduces due to reactive power flow in transmission lines.4- Need of large sized conductorPower factor correction allows to obtain advantages also for cable sizing. In fact, as previously said, at the same output power, by increasing the power factor the current diminishes. This reduction in current can be such as to allow the choice of conductors with lower cross sectional area. 5- Increase in KVA rating of the system equipment Generators and transformers are sized according to the apparent power S. At the same active power P, the smaller the reactive power Q to be delivered, the smaller the apparent power. Thus, by improving the power factor of the installation, these machines can be sized for a lower apparent power, but still deliver the same active power.

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6- Reduction in the handling capacity of all system elements Reactive component of the current prevents the full utilization of the installed capacity of all system elements and hence reduces their power transfer capability. A power system is expected to operate under both normal and abnormal conditions and under these conditions it is desired that the voltage must be controlled for system reliability, the transmission loss should be reduced and power factor should be improved (Rajesh Rajaramanet.al., 1998). In this paper the effect of line reactive power flow on transmission efficiency, voltage regulation and power factor with and without VAR compensation techniques are presented. 7- Penalties on consumers. Consumers pay extra fees on bills for poor factor, e.g. in Palestine there is a penalty for PF < 0.92

Section 1.2 : Objectives

The objective of the project is to minimize the effects of reactive power flow in the line network, where the project will contributes the generator in supplying the load by reactive power, so the project will reduce the power flow from generators to load, therefore reducing the current, which in turn reduce the effects of reactive power flow in the line network.

Now starting with the common methods which are used for solving poor PF of loads problem, which are Capacitor Banks and Synchronous Condenser.[3]

Capacitor Banks is a method of adding capacitors in parallel at the load to generate a part of the needed reactive power rather than the reactive power totally generated by the power supply, they are a group of capacitors that are connected together depending on how poor the PF load is, which reflects that the load requires more reactive power. The main disadvantage of this method is that it requires high load to sense the change on the power factor, therefore the generated reactive power by the capacitor banks changes in steps and not in a smooth way, so we can't achieve a specific PF at some loads.

Synchronous Condenser is a Synchronous motor which is over excited and most of the time it's at no load or low load, itgenerates reactive power controlled accurately by the field current, and the PF can be raised smoothly to achieve the required value. Synchronous Condenser usually used for heavy load due its high cost. [3]

Taking into consideration the previous methods, the project tried to take the advantages of both methods and avoid the disadvantages as much as possible. The method of this project uses one large capacitor bank in series with a Triac. The firing angle of the Triac

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controls the flow of reactive power from the capacitor. The controller specifies the firing angle of the Triac to supply a part of reactive power of the load depending on the reference PF. The main advantages of this method are low cost with respect to the mentioned methods, this method is sensitive to any changes at the load even for small changes and by this method a specific PF can be achieved.

components blocks of the project

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Chapter 2

Literature review

Section 2.1: Citation relevant work and results

There are too many research on the ordinary methods of PF correction. Arteche is a huge company in Spain which is a major manufacturer of reactive compensation and harmonic mitigation products, this company published a research about many types of capacitor compensation, their research shows different types of compensation to different types of load, and also shows the importance of reactive power compensation.

A group of companies called ABB is a leader in power and automation technologies that enable utility and industry customers to improve performance while lowering environmental impact. ABB inserted the TSC (thyristor switched capacitors ) as one of the mean technique in PF correction methods and mentioned that this method could replace synchronous condenser, where this method can correct the PF smoothly. This method suffer from harmonics and ABB gave visualization for solving this problem in their published paper (Technical Application Papers No.8 Power factor correction and harmonic filtering in electrical plants).

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Section 2.2

Theoretical background

From the expertise that have been gained through past courses it was easier to deal with the project. The PF correction has been mentioned in different courses such as Electrical Circuits, Fundamental Machines and Power Systems. Many other courses helped us in performing this project such as Power Electronic, 'Signals and Systems', Drive of Electrical Machines and modeling using Matlab.

The following figuer.1 shows a load connected to a generator through a transmission line, where Vs is the terminal voltage of the generator.

Figure.1

S L=P+ jQl

The power factor present the ratio between the real power consumed at the load to the apparent power delivered to the load:

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PF=PS= P

√P2+Ql2

Also the PF gives an indicator of the phase shift between current wave and voltage wave, where PF is the cosine of the angle between current and voltage, a lagging PF means that the current lags the voltage by an angle ( cos−1PF) and a leading PF means that the current leads the voltage by an angle ( cos−1PF). As much as the angle become bigger the power factor becomes smaller and vice versa.In real life the loads always have lagging power factor due to inductive loads Figure.2 shows a schematic for a current lags voltage with relatively low PF:

Figure.2

For adding a shunt compensation (shunt capacitor branch ) to the load it will contribute the generator in generating reactive power to the load .Figure.3 shows the compensated load:

Figure.3

*note that the load will consume the same amount of real and reactive powerS L=P+ jQl

Shunt Compensation will generate reactive power to the load at a value of Qc that will correct the PF by reducing the amount of reactive power delivered by the generator which in turn reduce the apparent power :

PF=PS= P

√P2+ (Ql−Qc )2

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Figure.4: schematic the relation between PF with apparent power , power angle and reactive power Where

Figure.4

Figure.5 shows the current and voltage citation after correction:

Figure.5

We notice that the enhanced PF the smaller phase shift be.

Now the importance of PF correction is that the reactive power transmitted will be smaller so the apparent power will be reduced

S .old=√P2+Ql2

S .new=√P2+(Ql−Qc )2

Which in turn reduce the current at the transmission line

I . old=S .oldV

=√P2+Ql2

V

I .new=S .newV

=√P2−(Ql−Qc )2

V

Then the voltage drop will decrease , the losses at the transmission line will reduce , which enhance the efficiency of the transmission

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line ,avoid extra fees on bills and the rated power of the system equipment will be reduced ,that’s means we need smaller sized and cheaper equipment and transmission lines.

Chapter 3: main components of the project

Section 3.1: zero crossing detector

The zero cross detection circuit is the most critical part for designing a PF meter in our project. This circuit will watch the input voltage and current waveforms and detect when this waveforms cross the zero axis .

Zero cross detection circuits are mainly used in cases when the PF needs to be measured by micro controller. In that case, the micro-controller needs to know the zero cross detection point of the voltage waveform, so that it can calculate the angle offset to send the trigger pulse to the gate of the triac.

Here is an example calculation. Suppose that the AC power oscillates in a 50Hz cycle. This means that each cycle will take 1/50Hz = 20 mSec to be completed. During those 20mSec, the waveform will cross the zero point two times, one at the beginning and one in the middle of the cycle, that will be after 20/2 = 10mSec.

If we want the capacitor to inject reactive power from applying a half waveform of the voltage , then the microcontroller needs to send a pulse in the middle of each semi-cycle. Thus, a pulse must be sent after 5mSec after each time the waveform passes the zero point. For this to be done, the microcontroller will watch the zero cross detection circuit (ZCD) for a pulse. When the ZCD send this pulse, the micro controller will count 5 mSec and then will trigger the gate of the triac.

The following circuit will perform a Zero Cross Detection circuit. This circuit is very stable and accurate, and has a controllable pulse width. Another great advantage is that because of the transformer, this circuit has a complete galvanic isolation with the mains supply so that it makes it completely safe and risk free of destroying the microcontroller due to power peaks.[4]

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Figure 6

Section 3.2 : Triac

The Triac or bi-directional Thyristor, is a device that can be used to pass or block current in either direction. It is therefore classed as an AC power control device. It is equivalent to two Thyristor in anti-parallel with a common gate electrode. As only one device is required there are cost and space savings. 

 

Figure 7.

The Triac has two main terminals. TE1/ TE2 (power in and load out) and a single gate connection. The main terminals are connected to both p and n regions since the current can be conducted in either direction. The gate is similarly connected, since a Triac can be triggered by both negative and positive pulses.

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The ON state voltage or current characteristics resembles a Thyristor. The Triac static characteristics show that the device acts as a bi-directional switch. The condition where terminal TE2 is positive with respect to terminal 1 is denoted by the term TE2+. If the Triac is not triggered the low level of leakage current increases as the voltage increases until the break over voltage V is reached and then the Triac turns ON. The Triac can be triggered below V by a pulse to the gate, provided that the current through the device exceeds the latching current I before the trigger pulse is removed. The Triac has a holding current value below which conductance cannot be maintained.

If terminal 2 is negative with respect to terminal TE2 the blocking and conducting conditions are similar to the TE2+ condition, but the polarity is reversed. The Triac can be triggered in either direction by both negative or positive pulses on the gate. The actual values of gate trigger current and holding current as well as latching current can be slightly different in the different operating quadrants of the Triac due to the internal structure of the device. [5]

Section 3.3: Arduino

The Arduino Uno is a microcontroller board based on the Atmega328 which can be programmed with the Arduino software. It has 14 digital input/output pins (of which 6 can be used as PWM outputs), 6 analog inputs, a 16 MHz ceramic resonator, the Arduino Uno can be powered via the USB connection or with an external power supply. The power source is selected automatically.

The Atmega328 has 32 KB memory (with 0.5 KB used for the boot loader). It also has 2 KB of SRAM and 1 KB of EEPROM (which can be read and written with the EEPROM library).

"Uno" means one in Italian and is named to mark the upcoming release of Arduino 1.0. The Uno and version 1.0 will be the reference versions of Arduino, moving forward. The Uno is the latest in a series of USB Arduino boards, and the reference model for the Arduino platform; for a comparison with previous versions. [6]

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Chapter 4: Methodology

This chapter will explain the used method for calculating the power factor, firing angle for the triac and time of phase shift also dealing with zero cross detector and Arduino programming.

In the last semester it was discussed that the Static Variable Compensation mainly works on measuring the PF of the load then the controller will keep adjusting the amount of reactive power that should be injected to the load by controlling the firing angle of the traic until the new PF will be equal to reference (needed) PF which means that the methodology used a closed loop system.

In this semester, due to the high difficulty finding some of important tools to achieve the project in mentioned methodology (closed loop method), it could be found some of these tools but with a high cost. So we concentrated on the main units of the project which are PF meter and control unit which responsible to generate the firing angle to the triac, so the applied voltage to the capacitor bank is controlled then the amount of reactive power generated from the capacitor and injected to the load also is controlled.

This chapter will include two sections.

At first section we will discuss the closed loop methodology in correcting the power factor which is very efficient and practical method .

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The second section discuss use open loop method which means there is no feedback comes from the load to the controller, because actually to deal with a real load we need instruments tools such as current and potential transformers to convert the real voltages and currents to measurable values suites the control unit, but unfortunately these instruments transformers are difficult to find at the market

Section 4.1: Static Variable Compensation in closed loop control

In the last semester we built simulation on MATlap for static variable compensation in closed loop control manner, see figure

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Figure 8

As seen from the figure the module consist of load, measuring instrument for the load's current and voltage, PF meter, a capacitor bank connected in series with a triac, and most critical part which is the control unit.The control unit will keep adjusting the value of firing angle until that the the measured PF on the load exactly equal the reference PF, where as seen the control unit take the measured PF as feedback. The main element in the control unit is the integration which actually the controller, this controller obviously works in integral manner, where the output of this element is additives depending on the inputs(reference PF and the feedback measured PF), it can be noted that when the reference PF equals the measured PF then the difference between them is zero, therefore the output of the integration unit is constant or DC value, this DC value is translated to the value of firing angle (alfa) for the triac by using a saw tooth oscillator and comparators. The module is able to correct the PF under changing load and achieve the reference PF, the following figure shows how was the module able to correct the PF under changing load and achieving the reference PF(0.9 in our case).

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Figure 9

The main advantages of this method are

The simplicity of its control specially using Arduino In this method there is need only for a PF meter at the load There is no need for KW meter at the load and there is no

need to measure the voltage and the current at the load because the controller specify the value of the firing angle depending on the value of the PF on the load and the reference PF

Ability to supply exact amounts of reactive power to achieve specific reference PF regardless for light or heavy load, where traditional ways to correct the PF face problems to deal with different load sizes

The module is able to deal with the changing the voltage at the bus of the load

Section 4.2: Static Variable Compensation in open loop control

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In this semester we obliged to work in open loop Static Variable Compensation control system because to deal with a real load we need instruments tools such as current and potential transformers to convert the real voltages and currents to measurable values suites the control unit (control’s voltage-level is low), but unfortunately these instruments transformers are difficult to find at the market, and because we couldn’t bring instruments transformers we didn’t try to test our project on a real load bring such as an induction motor.

Open loop method which means there is no feedback comes from the load to the controller, so the accuracy to achieve the reference PF will be less than it for closed loop

In our methodology of open loop Static Variable Compensation we need a PF meter and KW meter, we were able to build a PF meter as will be discussed in this section but we assumed that we have constant consumption of real power instead of KW meter and also a constant voltage at the load, this section will show why these assumptions made. So briefly in our project we will show how the controller changes the value of the firing angle depending on the measured PF.

The used methodology consists of mainly two parts. First part is building a PF meter. Secondly, determining the suitable value of the firing angle, which is supposed to control suitable voltage to control the amount of reactive power the must be generated to the load achieve the reference PF.

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Section 4.2.1 : PF meter

The first part of calculating the PF is to find the phase shift between the voltage and current waveforms, that means we have to find the time difference between when the voltage cross the zero axis and when the current wave cross the zero axis, in order to apply the previous, a Zero Cross Detector Circuit (ZCD) is used.

As mentioned before ZCD circuit gives pulse phase modulation PPM to give an indication when the signal (sinusoidal in the project) start to rise or fall from the zero axis, which give a duplicate the frequency at the output , whenever that occurs the generated pulse gives an

indication that there is a zero cross in this place.

The following figure shows the output signal from ZCD circuit

Figure 10

However, if both voltage and current applied to ZCD circuits, we can find the phase shift between ZCD of voltage and ZCD of current and that means we find the phase shift between the voltage and the current waveforms themselves.

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Note : the current should pass through a resistance in order to convert it to a voltage waveform because the ZCD circuits deals with voltage input , and the resistance makes no phase shift for the current.

The following figure shows the relation between voltage or current waveform with its ZCD

Figure 11

The second part is to use the PPM from the two ZCD circuits to determine the power factor, the ZCD for the voltage and the current will be connected to arduino input, pin for each (D2 for voltage and D3 for current), we wrote a code to find out the time between the ZCD pulses of the voltage and the ZCD pulses of the current then calculate the phase shift between these two inputs, which means that we found the phase shift between the original

waveforms of voltage and current.

There is a function at the arduino called “micros()”, which is a built in timer start to count in micro second since the processer starts to work, we use this function to capture the time stored in the micros function when the voltage crosses the zero, after that we capture the time stored in the micros function when the current crosses the zero, so simply the time of phase shift will be the difference between the captured time of current ZCD and the

captured time of voltage .

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Here is an example of simple micros function code to determine the known time, where in this code we capture the time of the micros and simply apply a delay by 1000 mille second and again captured the stored time at micros then we find the difference between the two captures, the difference will me in micros and to find the result in mille second we divide the result by 1000, finally we showed the result at the monitor to compare between the code result and the applied time delay:

The following shows the monitor output results which shows that the difference in reading the micros function match the supposed delay (1000 Milles).There is a small error in each cycle of the program equal to 20 micro second at most which is high precision.

After calculating the time between these two inputs, the phase shift can be calculated in linear manner where full period occurs after 20 mille seconds (1/freq=1/50 second) correspond 360 degree, so the relation between phase shift and time of phase shift is:

phase shift=time shift∗36020

Then the PF will be equal to cosin the phase shift in radial as in the equations:

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Ɵ=pashe shift∗π360

PF=cosƟ

Section 4.2.2: determine the firing angle

In order to control the generated reactive power from the capacitor bank we have to control the value of either the capacitance or the applied voltage to the capacitor bank according to the following equation:

Qc=Vrms2∗W∗C

Traditional method control the capacitance to control the reactive power generated, in our project we control the applied voltage to the capacitor bank.The main relation is between the firing angle (α) of the triac and the applied voltage according to the following equations:

Vrms=√ 1π∫απ

(VmsinWt )2dwt

¿√Vm2

π ∫α

π

(sinWt )2dwt

¿√Vm2

π ∫α

π12(1−cos2wt )dwt

¿√Vm2

2π[wt− sin 2wt

2];wt=∝¿π

¿√Vm2

2π[ π− sin 2π

2−α+ sin 2α

2]

¿√Vm2

2π[ π−α+ sin 2α

2]

¿√Vm2

2π[ π−α+ sin 2α

2]

Then Vrms=Vm√ 12π

[π−α+ sin 2α2

]

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The following figure shows the relation between the firing angle of the triac and the applied voltage to the capacitor

Figure 12

As mentioned before we assumed that the consumption of real power of the load is constant because it is hard to build KW meter also we assumed that the voltage at the load is constant and equal to the nominal voltage, we also assumed that the worst PF at the load is 0.4.

Now after finding the PF we found the required amount of generated reactive power according to the following equation

Qc=P ¿

Now to generate required reactive power to achieve reference PF the voltage should be set according to the following

Qc=Vrms2∗W∗CThen

Vrms=√ QcW∗C

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To control the applied voltage, the firing angle (α) should be set according to following equations

Vrms=Vm√ 12π

[π−α+ sin 2α2

]

With some manipulations with this equation we reached the following equation

sin 2α−2α=co

Where co=Vrms2

Vm2 ∗4 π−2 π

To solve this equation we use numerical method called Bisection Method The shape of sin 2α−2α will be as followed

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Figure 13

When alfa included inside the interval of [0,π], it can be concluded that the value of “co” has a range between 0 and -6.3

-Bisection methodThe bisection Method, which is alternatively called binary chopping, interval halving, or Bolazan’s method, is one type of incremental search numerical method in which the interval is always divided in half. If a function changes sign over an interval, the function value at the midpoint is evaluated. The location of the roots is then determined as lying at the midpoint if the subinterval within which the sign change occurs. The process is repeated to obtain refined estimates. A simple algorithm for the bisection calculated in the next steps and a

graphical depiction of the method is provided in the following figure [7] .

Figure 14

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The main methodology of programming the bisection method (numerical)

Step 1-Choose lower xl and upper xu guesses for the root such that the function changes sign over the interval. This can be checked by ensuring that f(xl)*f(xu)<0

Step 2-An estimate of the root xr is determined byxr=(xl+xu)/2

Step 3-Make the following evaluations to determine in which subinterval. Therefore set xu=xr and

a) if f(xl)*f(xr)<0, the root lies in the lower subinterval. Therefore, set xu=xr and return to step 2b) if f(xl)*f(xr)>0, the root lies in the upper subinterval. Therefore, set xl=xr and return to step 2

c) if f(xl)*f(xr)=0 , the root equal x; terminate the computation .

Depends on the previous, the following figure shows the bisection used code to estimate the value of alfa in order to control the rms voltage applied to the capacitor to control the injected reactive power to the load to correct the power factor of the load.

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Figure 15

Note: as the number of iteration increase the error will rapidly decrease

following table shows the relation between number of iteration and the max error knowing that the value of alfa included in the interval of [0 , pi].

Number of iterations Max Error)%( Max Error (degree)

1 5090

2 2545

3 12.522.5

4 6.2511.25

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5 3.1255.625

6 1.562.808

7 0.781.404

8 0.390.702

9 0.190.342

10 0.0970.1746

Table 1

Section 4.2.3: Generating the firing angle

This section interested at the used methodologies to generate the firing angle to the triac in order to control the applied voltage at the capacitor bank.

Simply the project use two different methodologies to generate alfa after finding it in numerical method as mentioned in the previous section of this chapter.

-first methodology-:

The first methodology makes a test at every possible” time point” of the period to find out if this is the correct time to generate alfa or not

The test mainly compares between the founded alfa in time domain with the difference between the present time and the time where the ZCD of the voltage start, if they are equal then a high voltage will applied to output pin for a small period of time (0.5 in our project),

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then the program start from the beginning to repeat this generation at every cycle with difference firing angle depends on the new condition of the load.

So if there is a new load power factor, the controller will make a very fast new firing angle to control the amount of injected reactive power.

Figure 16

-second methodology-:

The second methodology has the same test which compare between the founded alfa in time domain with the difference between the present time and the time where the ZCD of the voltage start, also if they are equal then a high voltage will applied to output pin for a small period of time (0.5 in our project),the main difference is that the program generates the same alfa for 50 times (one second) by calculating it once in the second , then the program start from the beginning to repeat this generation at every second with difference firing angle depends on the new condition of the load.

This methodology also has a fast and practical response to the changing in the load conditions,

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Although the second methodology has slower response, it has good accuracy and ever more practical effectiveness.

Chapter 4: results and

conclusions

This chapter shows the ability of the project to correct the PF of the load by generation the firing angle to control the applied voltage of the capacitor bank, the results shows both of the positive and negative parts of the sinusoidal signal have a generated alfa related to the measured PF of the load, As mentioned before. the PF calculated related to the time of phase shift between the voltage and current ZCD which frequency

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duplicated from the sinusoidal signal, this because of the ZCD should applied to both positive and negative part of the sin wave,

Using arduino monitor, we get the result of calculating the time of phase shift, PF and firing angle.The fpllowing figures shows that results for different values :-

Figure 17

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Figure 18

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Figure 19

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Figure 20

31

Figure 21

Also we get results from the oscilloscope to get the results comparing different values of the generated firing angle with respect ZCD of the voltage as shown in the following figures :-

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Figure 22

Figure 23

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Figure 24

Figure 25

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References:

[1 ]B.R.Gupta, )1998(, Power System Analysis And Design, Third Edition , S.Chandand Company Ltd

[2 ]Van Cutsem T., 1991, “A method to compute reactive power margins with respect to voltage collapse”, IEEE Transactions on Power Systems, pp 145

[3 ]Stephen J. Chapman,2004, Electric Machinery Fundamentals, p)363,364(.

[4 ]http://pcbheaven.com/wikipages/Dimmer_Theory

[5]http://www.sprags.com/summary.html

[6] http://arduino.cc/en/Main/arduinoBoardUno

]7[ book-numerical method for engineering, 6th Edition 2009 Chapra Canal, page 124

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Appendix A

int VZD = 2; // voltage zero cross at pin 2 int IZD = 3; // current zero cross at pin 3 int fire =8;// output pin for fireing angle pin 8 float tv; // initial time of shifting calculatio from voltage crossing float ti; // fianl time of shifting calculatio from current crossing float Ton; double theta; const float pi = 3.14; double PF; float alfa; float alfa1; float TF=4000; int flag1=0; int flag2=0; int flag3=0; int flag4=0; double co; double Vrms; int Vm=326; int P = 5000; int C=0.0014; double PFn=0.95; double xl; double xu; double xr; double xrold; int iter; double fxl; double fxr; double test; double ea; int es=2; double Qc; double ea1;

void setup() { Serial.begin(9600); pinMode(VZD,INPUT);pinMode(IZD,INPUT);pinMode (fire,OUTPUT);}

void loop() { // finding the time of voltage ZCDwhile (digitalRead(VZD) != 0 ){ tv = micros() ; flag3=1; // allow to genrate alfa flag4=1; // allow the current code

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}

if ((micros()-tv)>=TF && flag3==1){ // generat alfa if it's in the regon between ZCD of voltage and current digitalWrite (fire , HIGH ); delay (0.5); digitalWrite (fire , LOW ); flag3=0; } //finding the time of current ZCDif (flag4==1){ while (digitalRead(IZD) != 0 ){ ti = micros() ; flag1=1;

}}

if (flag1==1){ // there is a measured time shift between the two inputs flag1=0; Ton = ti-tv; // find the time between tv and ti Ton=Ton/1000; // transfer the time phase shift to milli second if (Ton >0.1 && Ton <4.359){ // the accepted values of Ton related to the accepted values of PF // calculating PF theta = Ton * pi/10 ; // finding the phase shift angle in rad PF=cos(theta ); // calculating the power factor // finding the firing angle using bisectional method xr=0; xl=0; xu=pi; Qc=P*(0.328- tan(theta)); Qc=abs (Qc); Vrms = sqrt(Qc/C*100*pi); co=((Vrms*Vrms)/8419)-(6.28); // co=((Vrms*Vrms)*4*pi/(Vm*Vm))-(2*pi) iter = 0; while (iter <=10 && ea<es ){ xrold=xr; xr=(xl+xu)/2; iter++; if (xr != 0){ ea1=(xr-xrold)/xr; ea =(abs(ea1))*100; } fxl=sin (2*xl)-(2*xl)-co; fxr=sin (2*xr)-(2*xr)-co; test = fxl*fxr;

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if (test<0){ xu=xr; } else if (test >0){ xl=xr; } else { ea=0; } } alfa = xr;// firing angle in rad TF=alfa*10000/pi;// finding the fiering angle as time in micro second flag2=1; // there is allowed values to print } else if (Ton <=0.1 && Ton >=0){ theta = Ton * pi/10 ; PF=cos(theta ); alfa=pi; flag2=1;//there is allowed values to print } else if (Ton <=5 && Ton >=4.359){ theta = Ton * pi/10 ; PF=cos(theta ); alfa=0.01; flag2=1;//there is allowed values to print } alfa1= alfa*180/pi; // firing angle in degree...only for monitoring TF=alfa*10000/pi;// finding the fiering angle as time in micro second

if (flag2==1){ // variables to be prented flag2=0; Serial.println("time of phase shift : "); Serial.println(Ton); Serial.println(""); Serial.println("value of power factor"); Serial.println(PF); Serial.println(""); Serial.println("value of fiering angle"); Serial.println(alfa1); Serial.println(""); }

} if ((micros()-tv)>=TF && flag3==1){ // generat alfa if it's not in the regon between ZCD of voltage and current digitalWrite (fire , HIGH ); delay (0.5); digitalWrite (fire , LOW ); flag3=0; }

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}Appendix B

int VZD = 2; // voltage zero cross at pin 2int IZD = 3; // current zero cross at pin 3int fire =8;// output pin for fireing angle pin 8float tv; // initial time of shifting calculatio from voltage crossingfloat ti; // fianl time of shifting calculatio from current crossingint flag1=0;float Ton;double theta;const float pi = 3.14;int flag2=0;double PF;double alfa;float TF=4;int flag3=0;int counter;double co;double Vrms;int Vm=326;int P = 5000;int C=0.0015;double PFn=0.92;double xl;double xu;double xr;double xrold;int iter;double fxl;double fxr;double test;double ea;int es=2;double Qc;double ea1;int counter ;

void setup() { Serial.begin(9600); pinMode(VZD,INPUT); pinMode(IZD,INPUT);pinMode (fire,OUTPUT);}

void loop() {

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while (digitalRead(VZD) != 0 ){ tv = micros() ; flag3=1; // can search for TF } } while (digitalRead(IZD) != 0 && flag3==1 ){ ti = micros() ;flag1 =1;flag3=0;}

if (flag1==1){ //PF meter flag1 =0; Ton = ti-tv; // find the time between tv and ti Ton=Ton/1000; // transfer the time phase shift to milli second flag2=1; if (Ton > 5){ flag2=0; } if(PF>0.95|| PF<0.2){ flag1=0; } }if (flag1 ==1){//dteremining the value of alfa (firing angle) using neumerical Bisectional metheodtheta = Ton * pi/10 ; // finding the phase shift angle in radPF=cos(theta ); // calculating the power factorPF=abs (PF);

Serial.println("time of phase shift : ");Serial.println(Ton);Serial.println("");Serial.println("value of power factor");Serial.println(PF);Serial.println("");

flag2=0;xr=0;xl=0;xu=pi;Qc=P*(0.328- tan(theta));Vrms = sqrt(Qc/C*100*pi);

co=((Vrms*Vrms)*12.56/(106276))-(6.28); // co=((Vrms*Vrms)*4*pi/(Vm*Vm))-(2*pi)iter = 0;while (iter <=10 || ea<es ){xrold=xr;xr=(xl+xu)/2;iter++;if (xr != 0){ ea1=(xr-xrold)/xr;ea =(abs(ea1))*100;}

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fxl=sin (2*xl)-(2*xl)-co; fxr=sin (2*xr)-(2*xr)-co; test = fxl*fxr; if (test<0){ xu=xr; } else if (test >0){ xl=xr; } else { ea=0; }}alfa = xr;

if(PF>0.92){alfa=pi;}

alfa=alfa*180/pi;

TF=alfa*20/360;// finding the fiering angle as time in mille second

Serial.println("value of fiering angle");Serial.println(alfa);Serial.println("");

}

counter = 0;while (counter<100){ while(digitalRead(VZD) != 0){TF = TF+1;delay (TF);digitalWrite (fire, HIGH);delay (1.1);digitalWrite (fire, LOW);counter ++;

}}}

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