MINI PROJECT IN TET4190 POWER ELECTRONICS FOR

RENEWABLE ENERGY

Redundance in cascaded multilevel H-bridge inverter

By:

Andreas Harstad Hallan,

Krister Haugen,

Lars Martin Hytten og

Siri Minothi

1 Summary

In this project it has been studied possibilities for including redundance in a modular power supply system

for subsea applications. The circuit studied in the project is a three-phase power supply made up by

independent inverter cells connected in series to achieve the three phase voltages. The main scope of this

project was to study how the line to line voltages can be maintained with a loss of phase voltage magnitude

in one or more phases caused by a fault in one or more cells.

A mathematical description, a Matlab® based solution tool and a PSCAD® model has been made for this

report. The MATLAB script was used to calculate the required phase voltages to restore symmetrical line

voltages after inverter cell breakdowns. The different fault situations were simulated in the PSCAD model.

The model testing shows that compensation by manipulation of the control signal gives the same line to

line voltages as in a symmetrical system. Two examples have been studied in detail along with proposed

solution for a range of other fault cases.

The lesser utilization of inverter capacity and higher cost of production are not considered in detail, but

should be a topic in further studies.

If a deeper study of the inverter is to be carried out the following subjects should also be taken into

account:

Utilization of inverters with over modulation

Optimization of phase voltages to prevent over load of one cell

Innhold 1 Summary ............................................................................................................................................. 2

2 Introduction ......................................................................................................................................... 4

3 Theory ................................................................................................................................................. 5

3.1 Cell components ........................................................................................................................... 5

3.1.1 Cell structure ........................................................................................................................ 5

3.1.2 Cell principle ......................................................................................................................... 5

3.1.3 Full Bridge DC-AC Inverter ..................................................................................................... 6

3.2 Maximizing the output ................................................................................................................. 6

3.2.1 Pulse Width Modulation (PWM) with Bipolar Voltage Switching ........................................... 6

3.2.2 Use of third harmonics .......................................................................................................... 7

3.2.3 Cell bypass ............................................................................................................................ 7

3.2.4 Bypass equal number of cells ................................................................................................ 8

3.2.5 Neutral shift .......................................................................................................................... 8

3.3 Fault effect on output voltage ...................................................................................................... 8

3.4 Method of calculation ................................................................................................................ 10

3.4.1 Method devised by Robicon Corporation, US Patent (5986909) .......................................... 10

3.4.2 Method using Law of cosines .............................................................................................. 10

3.5 Description of the Matlab script ................................................................................................. 11

4 The case ............................................................................................................................................. 11

4.1 Pre fault case .............................................................................................................................. 11

4.2 With fault ................................................................................................................................... 12

4.3 Simple case ................................................................................................................................ 13

4.4 Complex case ............................................................................................................................. 14

4.5 Border-line case ......................................................................................................................... 15

5 Introduction to model test ................................................................................................................. 16

5.1 The PSCAD model ....................................................................................................................... 16

5.2 Test procedure ........................................................................................................................... 18

6 Normal operation............................................................................................................................... 19

7 Fault situation 1 ................................................................................................................................. 20

7.1 Fault simulation .......................................................................................................................... 20

7.2 Correcting for unsymmetrical voltages ....................................................................................... 22

8 Fault situation 2 ................................................................................................................................. 23

8.1 Fault simulation .......................................................................................................................... 23

8.2 Correcting for unsymmetrical voltages ....................................................................................... 23

9 Discussion and conclusion .................................................................................................................. 25

10 References ..................................................................................................................................... 28

2 Introduction

For subsea applications of power electronics there are different challenges compared to normal

surroundings. High pressure, large distance from land and the difficulty to repair any faults that may occur

causes big challenges. To maintain or replace broken components, the power supply unit must be brought

to the surface. This requires specialized vessels and also requires the production to stop. This is very

expensive, so the maintenance of the equipment must be minimized. What is demanded is a reliable power

supply that can run for 5‐7 years without interruptions due to maintenance.

Siemens’ solution to this problem is to create a modular power supply system where each of the three

phases of the power input consists of several independent cells. If one of the cells break down, it can be

bypassed automatically while the power supply keep delivering balanced three‐phase power at an

acceptable voltage level. By including redundancy in the system, it can still work properly even if some

components fail.

In the project it will be studied how the breakdown of cells affects the line voltages of the power supply in a

PSCAD model of the circuit. It will also be studied how to calculate adjustments required for restoring the

system to normal operation from the various fault situations. These solutions will be tested through

simulations in the PSCAD model.

3 Theory

3.1 Cell components

3.1.1 Cell structure

Figure 1: Simplified diagram of a 15 cell converter

The main motivation for this circuit design is to achive redundance in a three-phase power supply for a

motor used subsea. The subsea application requires a power supply that can tolerate faults in some of the

components without a drop in the output voltage. The circuit is made up of independent cells working as

individually controllable AC sources connected in series. If a fault occurs in one of the cells it may be

bypassed within a half second. The other cells may then be adjusted to restore the original line voltages.

The number of cells in each phase will depend on the desired output voltage and the need for redundance.

The phase voltages become the vector sum of the cell voltages of each phase. Figure 1 from [2] shows a

modular three-phase power supply and five cells in each branch.

3.1.2 Cell principle

Figure 2 shows an inverter of the type used in the power supply. In this application a rectified voltage from

a three-phase transformer is used as the DC source, creating a galvanic separated cell. Because each cell is

only exposed for the voltage level given from the DC source, low voltage components may be used to

achieve the high output voltage required for the motor. Low voltage component is an advantage

considering price and number of components on the market.

Figure 2: Typical cell with bypass contactor

3.1.3 Full Bridge DC-AC Inverter

Figure 3: Single-phase full-bridge inverter

Figure 3 shows a basic full bridge inverter, as described in [1].

The full bridge inverter is used to ensure that enough power can be delivered.

Because of the voltage limits of electrolyte capacitors and especially since this inverter will be used for high

voltage applications, it is necessary to use at least two capacitors in series as the capacitor bank.

3.2 Maximizing the output

3.2.1 Pulse Width Modulation (PWM) with Bipolar Voltage Switching

To achieve a sinusoidal output voltage waveform with the desired frequency, a triangular waveform is

compared with a sinusoidal control signal. The comparison of the two signals sets the pulse width that

controls the swiching intervals of the inverter.

Figure 4: PWM with bipolar voltage switching

In Figure 4 from [1] it is shown how PWM can be used to create a sine wave by changing the pulse width,

the pulse shown in figure 8-12 (b) is the output voltage.

3.2.2 Use of third harmonics

By the use of 3rd harmonic modulation it's possible to widen the sinusoidal output voltage, and the

therefore getting more voltage time from the inverter by increasing the RMS value of the voltage. This

gives more voltage with the same current, giving more power delivered. The 3rd harmonic will be cancelled

in the line voltage, which is transferred to the motor, thus it doesn't create any problem for the load. It still

sees a perfect sinusoidal wave. Using this technique is an object for further studies of the cell converter.

3.2.3 Cell bypass

An important part of achieving a high reliability is by the use of cell bypass. Because of the galvanic

separation between the supply voltage and the cell it's possible to disconnect any cell in less than a

second. In figure 4-1 the connection of the bypass contactor is shown. In case of a fault in a cell the bypass

contactor will short the output terminals of the cell, regardless of which component that fails, allowing the

whole system to continue operating at a lower voltage for that string of cells.

3.2.4 Bypass equal number of cells

If one or more cells break down the line voltage will not be the same for all three lines. One way to solve

this problem is to take away more cells so that we end up with the same number of cells in all of the

branches. The shape of the triangle in figure 4-2 will then be restored.

3.2.5 Neutral shift

Another and better way is to exploit the fact that we can move the neutral point. This will have an affect on

the phase voltages and the phase angels, but that is not a problem since the motor only cares about the

line voltages.

3.3 Fault effect on output voltage

In the ideal case we have three phases with the same amplitude shifted 120 degrees as Figure 5

illustrates.

Figure 5 - Phase and line voltages, represented as phasors, in optimal case

The phase voltages are:

With:

Hence the line to line voltages are:

We see that the amplitudes are the same for all line to line voltages with a 120 degree shift. These

vectors rotate and keep the same amplitude and shift to produce a sinusoidal shaped voltage on

the output of the inverter.

In a case where the voltage on one of the phases drop because of a fault in one of the inverter

cells the picture will change drastically. As an example the voltage on phase R is reduced by 40 %

(for example if 2 of 5 inverters are short circuited). The phase shift is the same, but with

amplitudes:

This is illustrated in Figure 6 where the original line to line voltages are dotted lines.

Figure 6 - Phase and line voltages, represented as phasors, after fault

The shows a change in both amplitude and phase angle in the line voltages connected to phase R.

The same calculations as for Figure 5 give:

As the voltage UST is still unchanged this gives a dramatic asymmetry in the line voltages which will

affect the unit connected to the bus, probably in an unfavorable manner.

3.4 Method of calculation

There are many ways of looking at this problem. In this report two different approaches are studied and a

detailed description is given to one of them.

3.4.1 Method devised by Robicon Corporation, US Patent (5986909)

This method is from the patent of Peter W. Hammond and Marc F. Aiello at Robicon Corporation, patented

in 1999, the company was bought by Siemens in 2005. The patent covers the unit described in this report,

and also provides a selection of the table from the report, showing how much output voltage is lost given

the loss of a specific number off cells. The complete table is shown in Appendix I.

3.4.2 Method using Law of cosines

Since there already was a complete solution for comparing the output voltage to the maximum voltage as

you loose cells in each phase. We decided to base our calculations on the principle that the line to line

voltage had to be kept constant, and then try to calculate how the other phases would have to compensate

and phase angels adjusted to keep the required line voltage.

Using the Law of cosines we can easily calculate a missing angle or length of a line in a triangle. To move from one triangle to the next an extra variable β was used to describe the angle as shown in Figure 7. For this method to work we assume that two of the phase voltages can be set by us, and that the line voltages are constant. We build this method around Vc usually being the weakest, Va adjustable but mainly the same as before fault, and then Vb and the phase angels are calculated to complete the triangle. By manually increasing Va it's possible to reduce the pressure on Vb and keep the phase angels closer to

normal. Or the ability to adjust Va could be used to simulate a fault in that phase as well. In this case the

script calculates what Vb must be to keep the line to line.

Using this method it is possible to create some worst case scenarios that the inverters must be able to

handle, and based on that calculate how much voltage you would need to begin with. This provides a more

direct approach to the redundancy problem, and how much of it we need, that the method in section 1.1.

Figure 7: The three-phase triangle with angles

3.5 Description of the Matlab script

The script requires Va, Vc and a line voltage ex. Vab as input. Based on this it returns the phase angels and magnitudes of each phase. It uses the Law of cosine to calculate alpha on line 15, based on the assumption that Va and Vc is known. If

one or both of the line voltages is large enough the angle returned to alpha(3) will be a complex number;

an unphysical solution. The voltage on phase A is then increased in small steps until the two vectors are

long enough to support Vac. Then, using the line voltage Vac, Vc and Va the angle beta(1) is calculated, and

given that the line voltages are balanced we get beta(1)+beta(2)=60°. Using beta(2), Vbc and Vc we can

then calculate Vb, subsequently we have all three phases and line voltages. The script finishes by

calculating the phase angels alpha(2) and placing alpha(1) as the reference.

At the final lines, from 36 and down, some output is generated, mostly to instantly show the result, and

that the phase angels add up to 360°.

4 The case

This section provides a theoretical description of the three phase star coupled system used to create the

line voltages, and gives examples of how the disconnection of the neutral point from the rest of the system

can be used to compensate for the loss of voltage from one or more phases without reducing line to line

voltage or alter the phase angels on the line voltage.

4.1 Pre fault case

In a balanced three phase system the phase voltages and line voltages can be organised as shown in Figure

8.

Figure 8: Three-phase triangle in the pre-fault state

In this case the phase voltages A, B and C are all equal, same for α1, α2 and α3 all equal to 120°. This is the

normal case for a 3 phase system.

4.2 With fault

Figure 9 shows a situation where phase C is reduced to only 80% voltage compered to the normal case in

section 2.1. This reduces the line voltage on Vac and Vbc, it also interferes with the all the phase angels,

leaving the line voltages in a poor state.

Figure 9: Three-phase triangle after fault

To compensate for this it is possible to move the neutral point N, this case is shown in section 1.3.

4.3 Simple case

In this simple case only one of the phase voltages are limited. Vc can only reach 80% capasity, but by

increasing Vb by 26% and keeping Va the same, we change the phase angels to α1=99.36° α2=112.43° and

α3=148.21°, the resulting line voltage is then kept as before with an angle of 120° between the phases. This

case is shown in Figure 10.

Figure 10: Three-phase triangle with phase voltage Vc limited to 80%

4.4 Complex case

In a more complex example we have a case where the voltage on phase C is limited to only 20%. Again it is

possible to compensate for this by increasing the other phase voltages and shifting the neutral point. The

result is shown in Figure 11.

Figure 11: Three-phase triangle with phase voltage Vc limited to 20%

As seen in Figure 11 Va is 53% larger, Vb is 62% larger and Vc at only 20%. To balance this system it's

necessary to change the phase angels to α1=66.60° α2=121.86° and α3=171.54°.

4.5 Border-line case

As an extreme case the lowest sum of voltages from Va and Vb that can still maintain the required output

voltage is when the sum of to phases only are enough to support one of the line voltages. This case is

shown in figure 5. It this case both Va and Vc is reduced by 13.4%.1 To keep the line voltages Vb have to be

increased by 62%, and the phase angles shifted to α1=90° α2=90° and α3=360°.

Altough this is may not be a very realistic case, requirering the phases to opperate at only 62% during

normal opperation, it still shows it possible to handle a 13.4% loss in two phases without loosing line

voltage. As 13.4% is related to normal state, and given an equal redundancy in all three pahses of 62%, the

total voltage reduction is actually 46,5%, almost half the potential voltage. An illustration of the border

case is given in Figure 12.

Figure 12: Three-phase triangle with phase voltage sum of Vc and Va at minimum

In the above figure it can be seen how vulnerable the line voltages become in such an extreme case.

1 13.4% is derived from , Where is the ratio between line and phase voltages, divided between the two phases.

13

2

100%

3

5 Introduction to model test

The mathematical description developed earlier suggests phase voltages to maintain line to line voltages in

case of a fault. This description will now be tested in a model created with the simulation tool PSCAD®. The

scope of the tests is to show that a multi cell inverter system improves reliability in control applications.

There will be presented five cases. The first is the no-fault case where the conditions are as in normal

operation. The next two are different fault conditions with solutions as the two last. The solutions will be

obtained with the Matlab script.

5.1 The PSCAD model

The model consists of two parts. The first is a full-bridge inverter with bipolar switching, and the second is

the generation and comparing of the control signal. In this case the control signal is sinusoidal since we

would like to illustrate the conditions which the controller usually would operate in example connected in

front of an AC motor. The input is a DC-source to idealize a rectified grid voltage. The voltage of the DC

source was set to 12.7 kV to give a line to line voltage of 6.6 kV, as desired in the case studied.

Three inverters are Y-connected with the possibility of changing the DC-input to simulate loss of cells in

Siemens’ multi cell system. Then the control signal can be regulated to maintain the angle and magnitude

of the line to line voltages. Figure 13 shows the layout of the inverters used for simulations as it is drawn in

PSCAD®.

Figure 13: Y-connected switchmode inverters

To control the transistors a circuit is constructed with a one generator giving a sinusoidal wave at the

output, and one with a triangle shaped signal. These signals are compared in a comparator which returns

the value 1 if A>B, and 0 if A<B in Figur 14. To ensure bipolar switching a logic inverter is placed between

signal T1 and T2 in Figur 14. This gives opposite signal on the two pairs of each cell in Figure 13. The

triangular wave signal frequency is what gives the switching frequency of the transistors, whilst the phase

angle, magnitude and frequency of the sine wave give the output voltage characteristics. The control circuit

as it is drawn in PSCAD® is presented in Figure 13.

Figur 14: Control circuit for one cell in the switch mode inverter

We can use this layout and still assume that this is a multi cell construction since a series connection of cells

with a control signal having the same phase angle will effectively result in just adding of voltage

magnitudes. The amplitude modulation, ma, is set to a low value to give possibility of regulation in the

linear range. This will lead to low amplitude of the first harmonic and therefore inefficient use of the

inverter. Control signal with ma > 1 can be used to give a higher voltage-time area, and improved efficiency.

To illustrate the improved redundancy of the multi cell system over modulation is not necessary, but should

be considered in detail if a practical inverter are to be built. Note that the switching frequency is set to be

very low in to improve readability of plots. The frequency modulation, mf, is the same for all experiments.

5.2 Test procedure

The same procedure was followed for all cases, and consists of the following:

1. Lower DC input

To simulate that one or more of the cells in a phase is short circuited.

2. Calculate phase voltages with Matlab script

The desired line voltages are known as the pre fault voltages. The new conditions are inserted into the

model, and a solution is obtained.

3. Insert theoretical values

The control signal phase and amplitude modulation is changed according to the theoretical values. New

phase voltage magnitudes are obtained with the following formula in the linear range:

/

*required

new old

w fault

phase

a a

phase

Vm m

V

4. Register new phase voltages and the resulting line to line voltage.

6 Normal operation

The aim for the first experiment was to study the normal steady state operation of the circuit. The three

DC-voltage sources were set to the initial voltage 12.70 kV, representing the branches with all six cells

intact. The line voltages from the simulations and their fundamental components are plotted in the figures

below.

Figure 15: Eab with fundamental components, normal operation

Figure 16: Ebc with fundamental components, normal operation

Figure 17: Eca with fundamental components, normal operation

Figur 18: Fundamental components of Eab, Ebc and Eca, normal operation

As seen from figure 18 the fundamental components of the three voltages are symmetrical with a phase

shift of 120°. The fundamental components of the three phase voltages have a RMS value of 6.60 kV in

normal operation.

7 Fault situation 1

7.1 Fault simulation

This experiment simulates the loss of one of the six cells in one of the phases. To illustrate this effect, the

DC-voltage of the source in phase C was set to 10.583 kV, which corresponds to a loss of one sixth of the

initial voltage. Phase A and phase B remained at the initial value.

Figure 19: Eab with fundamental component, fault situation 1

Figure 20: Ebc with fundamental component, fault situation 1

Figure 21: Eca with fundamental component, fault situation 1

Figure 22: Fundamental components of Eab, Ebc and Eca with cursors, fault situation 1

As seen from figure 19, the line voltage Eab remains at the pre-fault value after the loss of one cell in phase

C. Ebc and Eca both get the same lower value as showed in figure 20 and figure 21. The horizontal cursors

in figure 22 are included to illustrate the lower amplitude of Ebc and Eca. The numerical RMS values of the

fundamental components are included in Table 1.

7.2 Correcting for unsymmetrical voltages

A MATLAB script (appendix 2) was written to obtain the new phase voltages required for restoring

symmetrical line voltages. The new ma values for the control circuit were calculated based on these

voltages. A new simulation was conducted with the new ma values. The resulting line voltages equal the

initial line voltages with only small deviations due to round-off errors. The calculated phase voltages and

the resulting values for ma are included in Table 1.

8 Fault situation 2

8.1 Fault simulation

This experiment simulates the loss of two of the six cells in phase C and one of six cells in phase A. To

illustrate this effect, the DC-voltage of the source in phase C was set to 8.467 kV, which corresponds to a

loss of one third of the initial voltage. The DC-voltage of the source in phase A was set to 10.583 kV, which

corresponds to a loss of one sixth of the initial voltage.

Figure 23: Fundamental components of Eab, Ebc and Eca with cursors, fault situation 2

As seen from figure 23 all three of the line to line voltages get a lower value after the fault. The horizontal

cursors are included to illustrate the different amplitudes.

The numerical RMS values of the fundamental components are included in Table 1.

8.2 Correcting for unsymmetrical voltages

Again the MATLAB script was used to obtain the new phase voltages required for restoring symmetrical line

voltages. New ma values for the control circuit were calculated based on these voltages, and a new

simulation was conducted with the new ma values. The resulting line voltages were equal the initial line

voltages except from some small round-off errors. The calculated phase voltages and the resulting values

for ma are included in Table 2.

To show that the system can maintain symmetrical line voltages for a broad number of scenarios, two

additional fault situations were simulated. The results are included in Table 2 without further discussion.

Table 1 Post faul line voltages before compensation

Case Post fault line voltages before compensation

Eab Ebc Eca

0 Normal operation 6.594 kV 6.594 kV 6.594 kV

1

Loss of 1 of 6 cells in

phase C 6.594 kV 6.052 kV 6.053 kV

2

Loss of two of six cells

in phase C and one of

six in phase A

6.053 kV 5.531 kV 4.956 kV

Table 2 Phase voltages to restore symmetrical line to line voltages

Case

Post fault condition after compensation

Phase A Phase B Phase C

Va ma Vb ma Vc ma

1

Loss of 1 of 6 cells in phase

C 3.81 / 0° kV 0,600 4.59 / 256.8° kV 0,722 3.18 / 141.6° kV 0,600

2

Loss of two of six cells in

phase C and one of six in

phase A

4.07 / 0° kV 0,769 5.57 / 275.1° kV 0,877 2.54 / 172.6° kV 0,600

Above: Discussed in detail. Below: Additional examples

3

Loss of one cell in phases A

and C 3.44 / 0° kV - 5.50 / 267.8° kV - 3.18 / 172.5° kV -

4

Loss of two cells in phase A

and C and loss of one cell in

B phase

3.32 / 0° kV - 5.32 / 263.1° kV - 3.32 / 166.2° kV -

9 Discussion and conclusion

The tables on the previous page show that loosing cells in one or more of the phases leads to a different

line voltages. In most cases the line voltages will be unsymmetrical.

Table 1 shows the resulting post fault line voltages for fault situations 1 and 2 if no compensation is

applied. Loosing a cell in phase C makes both line voltages Ebc and Eca decrease by 8 % while Eab however

remains at the initial voltage. Loosing two of six cells in phase C and one of six in phase A makes Eab

decrease by 8 %, Ebc by 16% and Eca by 25%.

Table 2 shows how the line voltages may be restored to the initial operation mode from fault situation 1

and 2 in addition to two other fault scenarios. The compensation is done by adjusting the value of ma and

the phase angle between the voltages.

The adjustments may be accomplished in many different ways. There are many combinations of ma values

and phase angles that yields the symmetrical initial voltages from a given fault situation. The combinations

in the table are chosen somewhat arbitrarily. The choice of values may be optimized to some criterion, by

for instance minimizing the highest voltage delivered from a single cell. This is beyond the scope of this

project.

The degree of redundancy of the system depends on the initial value of ma. By setting ma to a low initial

value, the voltage of each cell may be increased much while still being in the linear region of the control

circuit. A low initial value of ma will however lead to low utilization of the equipment. The trade-off

between redundancy and utilization must be balanced for the particular case.

The results from the simulations and the theoretical values calculated with the MATLAB script are

practically equal. The component parameter values used in the PSCAD model were set to the default values

which are almost ideal. The parameter values of the components used in the Siemens circuit were not

known when this report was written. However, electrical components for this kind of applications usually

have an efficiency factor close to unity. It can be assumed that the simulation results will also be applicable

to the actual circuit.

In more detailed studies of this subject there are a lot of other considerations that has to be taken into

account. The line length and on shore grid connection also have great impact on system redundancy. Better

utilization with over modulation an optimization of phase voltages to ensure the lowest possible voltage

over each cell is also subjects that would be interesting in further developments.

Appendix 1

A selection of line output voltage for a 6 cell 3-phase H-bridge inverter from [3]

Number of working cells in pahse Phase angels in degrees

A B C % of Vmax α1 α2 α2

6 6 6 100 0.0 120.0 240.0

6 6 5 94.1 5.4 114.6 240.0

6 6 4 87.8 10.5 109.5 240.0

6 6 3 80-9 15.5 104.5 240.0

6 6 2 73.6 20.4 996 240.0

6 6 1 65.9 25.2 94.8 240.0

6 6 0 57.7 30.0 90.0 272.3

6 5 5 88.5 0.0 113.1 246.9

6 5 4 82.1 5.0 106.5 249.3

6 5 3 75.1 9.8 99.8 253.6

6 5 2 67.2 14.5 92.7 263.2

6 5 1 55.1 20.5 82.4 311.3

6 4 4 75.5 0.0 101.4 258.6

6 4 3 67.3 4.8 91.2 268.4

6 3 3 50.0 0.0 60 300.0

5 5 5 83.3 0.0 120.0 240.0

5 5 4 77.4 6.4 113.6 240.0

5 5 3 70.9 12.5 107.5 240.0

5 5 2 63.8 18.5 101.5 240.0

5 5 1 56.2 24.3 95.7 240.0

Appendix 2

%[V,alpha]=vbEstimate(Va,Vc,Vab); %Calculates Vb in an unbalanced 3 phase system, with the LL voltages

%constant and symetrical with Va beeing as small as possible

% V(1)=Va

% V(2)=Vb

% V(3)=Vc

%

%Eks.:

%[V,alpha]=vbEstimate(1,0.8,sqrt(3));

function [V,alpha]=vbEstimate(Va,Vc,Vab)

Vbc=Vab;

Vac=Vab;

alpha(3)=acosd((Va^2+Vc^2-Vac^2)/(2*Va*Vc));

%Increase phase voltages in phase a and c if they are to low

while(~isreal(alpha(3)))

Va=Va+Va*0.01;

%Vc=Vc+Vc*0.01;

alpha(3)=acosd((Va^2+Vc^2-Vac^2)/(2*Va*Vc));

end

beta(1)=acosd((Vac^2+Vc^2-Va^2)/(2*Vac*Vc));

beta(2)=60-beta(1);

Vb=sqrt(Vc^2+Vbc^2-2*Vc*Vbc*cosd(beta(2)));

alpha(1) = 0; % Set reference voltage angle

alpha(2)=alpha(3) + acosd((Vb^2+Vc^2-Vbc^2)/(2*Vb*Vc));

V(1)=Va;

V(2)=Vb;

V(3)=Vc;

disp(' ');

disp(['Sum of alpha angels = ' , num2str((acosd((Va^2+Vb^2-Vab^2)/(2*Va*Vb))+acosd((Vb^2+Vc^2-

Vbc^2)/(2*Vb*Vc))+acosd((Va^2+Vc^2- … … Vac^2)/(2*Va*Vc)))) , ' degrees']);

if(round(acosd((Va^2+Vb^2-Vab^2)/(2*Va*Vb))+acosd((Vb^2+Vc^2 …

… Vbc^2)/(2*Vb*Vc))+acosd((Va^2+Vc^2-Vac^2)/(2*Va*Vc)))==360) disp('Hurray!');

end

disp(' ');

disp(['Va= ', num2str(V(1)) , ' Vb= ', num2str(V(2)) , ' Vc= ', …

… num2str(V(3))]); disp(['alpha1= ', num2str(alpha(1)) , ' alpha2= ', num2str(alpha(2)) , '… … alpha3 ', num2str(alpha(3))]);

disp(' ');

end

10 References

[1] Mohan, Undeland, Robbins. Power Electronics, Wiley & Sons

[2] ROBICON Perfect Harmony Engineering Manual, Siemens AG 2009

[3] US Patent (5986909), Peter W. Hammond, Marc F. Aiello, Robicon Corporation, 1999.

[4] - http://www.motorlab.com/Motor%20Lab%20Web%20Site_files/05%20-%20PSCAD.htm – Example

used as basis for PSCAD Script