Eee577

EEE577: Term Project

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11/10/2011

EEE577: Term Project STUDY OF TRANSMISSION ADEQUACY BY ARIZONA

TRANSMISSION EXPANSION CO. FOR ADDING PV RESOURCES

A PROJECT REPORT

Submitted by Supriya Chathadi

In partial fulfillment for completion of the course EEE577: Power Engineering Operations/Planning

at

Arizona State University, Tempe


Arizona State University 2

Table of Contents

1. Introduction ......................................................................................................................................... 3

2. The Per Unit System ........................................................................................................................... 3

2.1. Advantages ................................................................................................................................... 3

3. DC Power Flow................................................................................................................................... 4

3.1. Assumptions ................................................................................................................................. 4

3.2. Formulation .................................................................................................................................. 4

4. Actual Power Flows in the System ..................................................................................................... 5

4.1. Bus Terminology ......................................................................................................................... 5

4.2. Initial Calculations ....................................................................................................................... 5

4.3. Power Flow Calculations ............................................................................................................. 5

5. PV Injection ........................................................................................................................................ 6

5.1. Maximization using linprog ......................................................................................................... 6

6. Line Outage Contingencies ................................................................................................................. 6

7. Results ................................................................................................................................................. 7

8. Additional Concerns ........................................................................................................................... 8

9. Conclusion .......................................................................................................................................... 9

10. References ......................................................................................................................................... 9

Appendix 1: System Representation ........................................................................................................ i

Appendix 2: MATLAB Codes ................................................................................................................ ii

A2.1. Base Case: All Lines under Operation ...................................................................................... ii

A2.2. Line Outage Contingency between Flagstaff & Four Corners .................................................. ii

A2.3. Line Outage Contingency between Flagstaff & Palo Verde .................................................... iii

A2.4. Line Outage Contingency between Flagstaff & Blythe ........................................................... iii

A2.5. Line Outage Contingency between Blythe & Imperial Valley ................................................ iv

A2.6. Line Outage Contingency at One of the Lines between Imperial Valley and Palo Verde ........ v

A2.7. Line Outage Contingency between Palo Verde & Bicknell ...................................................... v



1. Introduction

A study is performed by Arizona Transmission Expansion Co. (AZTEx) to obtain the transmission

adequacies at certain parts of Arizona on adding high level solar PV resources at suggested sites.

This report is based on the study where PV resources are added at Bicknell and Blythe.

The excess generation due to PV injection is dissipated at Imperial Valley and Blythe in

proportion to their present load levels.

An approximate representation of the system base loads and generation, suggested PV injections and

the compensating loads are shown in Appendix 1.

2. The Per Unit System It is a common practice in power system studies to represent all system parameters as a ratio of a

defined base quantity in the same units, called the per unit representation.

Per Unit Quantity = units) (ActualQuantity Base

units) (ActualQuantity

Generally, the bases for any two of the system parameters is defined (SB and VB, SB and IB, etc.) are

defined from which all other bases are calculated. For example, when SB and VB are given, the

following expressions can give other base quantities.

IB = 3V

3S

B

B =

3V

S

B

B

ZB = 3S

)3/(V

B

2B

= B

2B

S

V

YB = BZ

1

2.1. Advantages

Per unit quantities for similar components (generators, transformers, lines) will lie within a

narrow range, regardless of their ratings and sizes.

They are the same on either side of a transformer, independent of voltage level.

The relative values of circuit quantities are clearly identified as unnecessary scaling factors

are eliminated.

Use of the constant is reduced in three-phase calculations.

Much simpler for calculations and ideal for computer simulations as well.

It rules out the problem of using different units in different parts of the world by introducing a

universal standard.



3. DC Power Flow There are a number of ways to perform power flow studies in a system. The simplest formulation is

called „DC Power Flow‟. The name is illusive because there is nothing to do with DC in it.

Some important observations are made from actual load flow studies and few assumptions are made

in order to simplify the calculations and still not vary the results in a large scale.

3.1. Assumptions

The transmission lines are assumed to be purely inductive as the resistances are much smaller

than the reactances.

As all machines operate in synchronism, the differences in angles between their voltage

phasors are typically very small (less than 30◦). Thus it is assumed that sin δ ≈ δ.

It is seen that the per unit voltages operate close enough to 1p.u. This comes up with an

assumption that the voltage magnitudes can be set to 1p.u. at all buses.

As reactive power flow between two buses is dependent on the difference in their voltage

magnitudes, an assumption can be made that there is no reactive power flow in the system.

The real power supplied by a bus is much greater than the reactive power supplied and hence

reactive power injection is neglected.

3.2. Formulation

The actual real power flow is given by the equation:

ij

ij

x

sinVVP

ji

ij

Based on the assumptions, the power flow equation can be written as:

ij

ji

ij

ij

xx

ijP

The power flow matrix for a system can be written from the above expression as follows:

busbusbus BP δ

where,

Bbus is called the susceptance matrix, given by the imaginary part of the admittance matrix (Ybus)

nn2n1n

n22221

n11211

busbus

Y...YY

............

Y...YY

Y...YY

imag)imag(YB

Yii = yi0+yi1+yi2+…+yin

Yij = Yji = -yij



where, yij is the primitive admittance between buses i and j, given by ijx

1

4. Actual Power Flows in the System

4.1. Bus Terminology

For ease of calculation, the buses are numbered from 0 to 5, where 0 indicates the reference bus. The

following conventions are used in the following sections:

Palo Verde (P) – 0 (ref)

Four Corners (FC) – 1

Flagstaff (F) – 2

Blythe (BL) – 3

Imperial Valley (I) – 4

Bicknell (BI) – 5

4.2. Initial Calculations

The reactance of each line can be calculated as the product of reactance per km of the conductor used

and the length of the line. The double line is considered as reactances in parallel.

The base and per unit values for all the line are also calculated and tabulated as follows:

Table 1: Impedance Calculations

SB = 100 MVA = 100MW (no reactive power flows)

Reactance (ohms) Base Reactance (ohms) Per Unit Reactance (p.u)

x12 204 x 0.2908j = 59.2008j 3452/100 = 1190.25 59.2008j/1190.25

x20 80 x 0.0496j = 3.968j 3452/100 = 1190.25 3.968j/1190.25

x23 98 x 0.5464j = 53.5472j 3452/100 = 1190.25 53.5472j /1190.25

x34 110 x 0.5464j = 60.104j 3452/100 = 1190.25 60.104j /1190.25

x40 (357 x 0.056j)/2 = 9.996j 5002/100 = 2500 9.996j /2500

x50 41 x 0.2903j = 11.8982j 3452/100 = 1190.25 11.8982j /1190.25

4.3. Power Flow Calculations

The values of δ for all buses are found from the expression, bus1

busbus PB

δ where, Pbus is a

matrix showing power injection at various buses. They are given with reference to Palo Verde as

shown below:

δP 0.0000

δFC -0.3382

δF 0.0100

δBL 0.1000

δI 0.1000

δBI -0.0800

Power flows in the lines are found out by increasing the dimension of δbus by adding a zero for the

reference bus, and using the same formula. But, this time the rows of Bbus have entries only at two

sites between which the power flow is calculated.



Consolidated results are shown in section 7.

5. PV Injection

It is desired to inject maximum PV resources at Blythe and Bicknell without overloading any of the

lines (considering the long term rating). The power injection is compensated at Imperial Valley and

Blythe in proportion to their present loads. Thus, their participation factors are calculated as:

PFI = 926.0225

25

PFBL = 074.0225

2

If the power injected at Blythe P1 and that at Bicknell is P2, they are compensated as 0.926(P1+P2) at

Imperial Valley and 0.074(P1+P2) at Blythe.

Same calculations as done in section 4 are done to obtain power flows in terms of P1 and P2.

Optimal values for P1 and P2 are found using linear programming (linprog) in MATLAB. The codes

are shown in Appendix 2.

5.1. Maximization using linprog

Linear programming is defined as:

Min f(x), subject to

Ax ≤ b; Aeqx = beq

lb ≤ x ≤ ub

where, f is called the objective function, x is a vector consisting of decision variables, Ax ≤ b are the

inequality constraints, Aeqx = beq are the equality constraints, lb and ub are the limits for the decision

variables.

The objective in this case is to maximize P1 and P2, with the line flows not exceeding their limits.

The problem can be formulated as given below:

Max (P1+P2) = -Min (-P1-P2)

line flows ≤ line ratings

0 ≤ P1, P2 ≤ ∞

In the constraints, the line flows can be in either direction, thus they might change signs in some

cases. As the direction of line flows cannot be predicted sometimes, it might be necessary to manually

change the signs for the line flows in order to maximize the PV injection. The resulting values of P1

and P2 are the maximum amounts of power which can be injected in terms of PV resources at Blythe

and Bicknell. Results are tabulated in section 7.

6. Line Outage Contingencies

This section deals with the study of (N-1) line outage contingencies. The procedure is basically the

same as illustrated with all lines under operation, except that the results are going to slightly different

because of the difference in Ybus.



On removing the lines one after another, it is seen that the power generations may not suffice the

loads in some cases such as a line outage between Four Corners and Flagstaff or between Palo Verde

and Bicknell, as the generation in Four Corners or Bicknell would be lost. In such cases, it is assumed

that Palo Verde supplies the power loss due to line outage, for the power flows to converge. This is

based on the given suggestion that equivalent generation could be added to another existing

generation site.

The power flow and maximum PV injection results are tabulated in Section 7. It is seen that a

contingency in one of the lines or both between Palo Verde and Imperial Valley is not allowed as the

lines overload even for the base case generation and loads. Thus the study of PV injection cannot be

done for this case.

7. Results

Table 2 gives the line flows in all the lines for including and not including PV injection. For the case

on injecting PV resources, the highlighted values indicate the line which might start overloading if

little more PV resources are added to the system. It is seen that for a line outage between Palo Verde

and Imperial Valley, line overloading occurs even without injecting PV resources.

Table 2: Per Unit Power Flows in All Lines for with/without PV Injection

Per Unit Power

Flows

(SB = 100MVA)

PFC-F PP-F PF-BL PI-BL PP-I PBI-P

Condition No

PV

With

PV

No PV With

PV

No PV With

PV

No PV With

PV

No

PV

With PV No

PV

With

PV

Line ratings

7.09

7.09

14.8

14.8

4.99

4.99

4.99

4.99

15+15

=30

30

10

10

With all lines

under operation

7

7

3

1.8010

2

2.8010

0.0002

4.9897

25

29.9998

8

8.1988

With line outage

between Flagstaff

& Four Corners

-

-

9.773

5.1902

1.773

2.8098

0.2271

4.5358

25.23

30

8

8.1872

With line outage

between Flagstaff

& Palo Verde

7

7

-

-

1

1

3

1.0100

28

30

8

10

With line outage

between Flagstaff

& Blythe

7

7

1

1

-

-

2

0.9900

27

29

8

10

With line outage

between Blythe &

Imperial Valley

7

7

3

1.3551

2

2.3551

-

-

25

30

8

8.6449

With line outage at

one of the lines

between Imperial

Valley and Palo

Verde

7

-

3.9357

-

2.9357

-

0.9357

-

24.064

(rating

in this

case is 15)

-

8

-

With line outage

between Palo

Verde & Bicknell

7

7

3

1.8162

2

2.8162

0.0002

4.9897

25

29.8162

-

-

Table 3 summarizes the maximum amount of PV resources which can be injected at Bicknell and

Blythe for different cases. The study of injecting PV into the system is not possible when a line outage

occurs between Imperial Valley and Palo Verde because of overloading, as shown in Table 2.



Maximum PV can be injected in the system when all lines are under operation, which is about

1078.79 MW.

Table 3: Consolidated Results for Maximum PV Injection at Blythe and Bicknell

Maximum PV

Injection at Blythe

(MW)

Maximum PV

Injection at Bicknell

(MW)

Total PV Injection

(MW)

With all lines under operation 1058.91 19.88 1078.79

With line outage between

Flagstaff & Four Corners

1010.75 18.72 1029.47


Flagstaff & Palo Verde

230.89 200.00 430.89


Flagstaff & Blythe

338.88 200.00 538.88


Blythe & Imperial Valley

475.46 64.49 539.96

With line outage at one of the

lines between Imperial Valley

and Palo Verde

-

-

-


Palo Verde & Bicknell

1058.96 - 1058.96

8. Additional Concerns

One of the assumptions of DC load flow studies is to consider that the voltage magnitudes at

all buses are 1 p.u. This comes up with a conclusion that there will be no reactive power flows

between any two buses, as difference in voltage magnitudes between them is zero. Thus

inclusion of reactive power flows, voltage magnitudes, their limits and constraints are all

inter-related. Including reactive power flows or voltage magnitudes automatically gives rise

to the other. The study will become more complex and detailed on doing it. The power flow

equations in this case would be the following:

ij

ij

x

ji

ij

VVP

ijx

)VV(VQ

jii

ij

The above equations still assume the following:

The transmission lines are purely inductive.

δ is very small and hence, sin δ ≈ δ.

There is no reactive power injection at any bus.

The active power flow „west of river‟ for the base case is almost negligible. It is 0.0001717

p.u., from Imperial Valley to Blythe. But with PV injection, it is nearly the rating of the line

(4.9897 p.u.). The „worst‟ single line outage is the outage of a line between Imperial Valley

and Palo Verde as one of the lines start overloading for the base case itself. The maximum

active power flow in the line is 0.9357 p.u.



If it is possible to add another transmission line, it is logical to add a line between Flagstaff

and Imperial Valley. A large amount of power produced by Palo Verde and Bicknell could

not be transmitted efficiently because of the relatively lower ratings on the lines between

Flagstaff and Blythe, Blythe and Imperial Valley. A major share of the system load occurs at

Imperial Valley, thus the transmission lines connecting to it must have compatible ratings. If a

transmission line made of type (3) conductor is added between Flagstaff and Imperial Valley,

a transmission path of Palo Verde to Flagstaff to Imperial Valley would be chosen for

effective transmission of the generated power. Yet another option is to add one more line

between Imperial Valley and Palo Verde, if it is not required to supply more loads

anywhere else.

9. Conclusion

It is seen that the maximum PV injection is possible when all lines are in service, and it can

be further improved by adding more transmission lines.

It is seen that the system is not properly designed for an N-1 line outage contingency, and

thus has to be improved

10. References

[1] John J. Grainger, William D. Stevenson, “Power System Analysis”, Tata McGraw-Hill

Publications, January 1994.

[2] Anomynous, “The Power Flow Equations”, available at:

class.ece.iastate.edu/ee458/PowerFlowEquations.doc

[3] Wikipedia, notes appearing on the subject of per-unit systems, available at:

en.wikipedia.org/wiki/Per-unit_system


Arizona State University i

Appendix 1: System Representation

Flagstaff (1)

Four Corners (2)

Blythe (3)

Imperial Valley (4) Palo Verde (0)

Bicknell (5)

7 p.u.

8 p.u.

P1 p.u.

2 p.u.

0.074(P1+P2) p.u.

0.926(P1+P2) p.u.

25 p.u.

20 p.u.

8 p.u.

P2 p.u.

Legend

Generation/Load Site

Transmission Line

Base Case Generation/Load

PV Injection/Load Compensation


Arizona State University ii

Appendix 2: MATLAB Codes

A2.1. Base Case: All Lines under Operation P=[7;-8;-2;-25;8]; Y=[1190.25/59.2008i -1190.25/59.2008i 0 0 0; -1190.25/59.2008i (1190.25/3.968i)+(1190.25/59.2008i)+(1190.25/53.5472i) -1190.25/53.5472i 0 0; 0 -1190.25/53.5472i (1190.25/53.5472i)+(1190.25/60.104i) -1190.25/60.104i 0; 0 0 -1190.25/60.104i (1190.25/60.104i)+(1/(3.9984i*10^-3)) 0; 0 0 0 0 1190.25/11.8982i]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[-20.1053 20.1053 0 0 0 0; 0 299.9622 0 0 0 -299.9622; 0 -22.2281 22.2281 0 0 0; 0 0 -19.8032 19.8032 0 0; 0 0 0 250.10004 0 -250.10004; 0 0 0 0 -100.0361 100.0361]; PP=BB*del1; syms P1; syms P2; Ppv=[7;-8;P1-2-0.074*(P1+P2);-25-0.926*(P1+P2);8+P2]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv; a=vpa(PPpv,4); disp(a);

A=[-3.532*10^(-18) 8.272*10^(-19); -0.4548 0.07523; -0.4548 0.07523; 0.4712 0.001232; 0.4548 0.9248; 0 1]; b=[7.09-7.0;14.8-2.9998;4.99-1.9998;4.99-0.0001717;30-25.0002;10-8]; f=[-1;-1]; lb=[0;0]; [x,fval,exitflag]=linprog(f,A,b,[],[],lb); if exitflag==1 disp(x); end

A2.2. Line Outage Contingency between Flagstaff & Four Corners

P=[-8;-2;-25;8]; Y=[(1190.25/3.968i)+(1190.25/53.5472i) -1190.25/53.5472i 0 0; -1190.25/53.5472i (1190.25/53.5472i)+(1190.25/60.104i) -1190.25/60.104i 0; 0 -1190.25/60.104i (1190.25/60.104i)+(1/(3.9984i*10^-3)) 0; 0 0 0 1190.25/11.8982i]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[299.9622 0 0 0 -299.9622; -22.2281 22.2281 0 0 0; 0 -19.8032 19.8032 0 0; 0 0 250.10004 0 -250.10004; 0 0 0 -100.0361 100.0361];


Arizona State University iii

PP=BB*del1; syms P1; syms P2; Ppv=[-8;P1-2-0.074*(P1+P2);-25-0.926*(P1+P2);8+P2]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv; a=vpa(PPpv,4); disp(a); A=[-0.4548 0.07523;-0.4548 0.07523;0.4712 0.001232;0.4548 0.9248;0 1]; b=[14.8-9.773;4.99-1.773;4.99-0.2271;30-25.23;10-8]; f=[-1;-1]; lb=[0;0]; [x,fval,exitflag]=linprog(f,A,b,[],[],lb); if exitflag==1 disp(x); end

A2.3. Line Outage Contingency between Flagstaff & Palo Verde P=[7;-8;-2;-25;8]; Y=[1190.25/59.2008i -1190.25/59.2008i 0 0 0; -1190.25/59.2008i (1190.25/59.2008i)+(1190.25/53.5472i) -1190.25/53.5472i 0 0; 0 -1190.25/53.5472i (1190.25/53.5472i)+(1190.25/60.104i) -1190.25/60.104i 0; 0 0 -1190.25/60.104i (1190.25/60.104i)+(1/(3.9984i*10^-3)) 0; 0 0 0 0 1190.25/11.8982i]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[-20.1053 20.1053 0 0 0 0; 0 -22.2281 22.2281 0 0 0; 0 0 -19.8032 19.8032 0 0; 0 0 0 250.10004 0 -250.10004; 0 0 0 0 -100.0361 100.0361]; PP=BB*del1; syms P1; syms P2; Ppv=[7;-8;P1-2-0.074*(P1+P2);-25-0.926*(P1+P2);8+P2]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv; a=vpa(PPpv,4); disp(a); A=[-1.13*10^(-16) 2.647*10^(-17);-1.25*10^(-16) 2.927*10^(-17);0.926 -0.074;0 1;0 1]; b=[7.09-7.0;4.99-1;4.99-3;30-28;10.0-8.0]; f=[-1;-1]; lb=[0;0]; [x,fval,exitflag]=linprog(f,A,b,[],[],lb); if exitflag==1 disp(x); end

A2.4. Line Outage Contingency between Flagstaff & Blythe

P=[7;-8;-2;-25;8]; Y=[1190.25/59.2008i -1190.25/59.2008i 0 0 0; -1190.25/59.2008i (1190.25/3.968i)+(1190.25/59.2008i) 0 0 0;


Arizona State University iv

0 0 (1190.25/60.104i) -1190.25/60.104i 0; 0 0 -1190.25/60.104i (1190.25/60.104i)+(1/(3.9984i*10^-3)) 0; 0 0 0 0 1190.25/11.8982i]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[-20.1053 20.1053 0 0 0 0; 0 299.9622 0 0 0 -299.9622; 0 0 -19.8032 19.8032 0 0; 0 0 0 250.10004 0 -250.10004; 0 0 0 0 -100.0361 100.0361]; PP=BB*del1; syms P1; syms P2; Ppv=[7;-8;P1-2-0.074*(P1+P2);-25-0.926*(P1+P2);8+P2]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv; a=vpa(PPpv,4); disp(a); A=[0.926 -0.074;0 1]; b=[2.99;2]; f=[-1;-1]; lb=[0;0]; [x,fval,exitflag]=linprog(f,A,b,[],[],lb); if exitflag==1 disp(x); end

A2.5. Line Outage Contingency between Blythe & Imperial Valley

P=[7;-8;-2;-25;8]; Y=[1190.25/59.2008i -1190.25/59.2008i 0 0 0; -1190.25/59.2008i (1190.25/3.968i)+(1190.25/59.2008i)+(1190.25/53.5472i) -1190.25/53.5472i 0 0; 0 -1190.25/53.5472i (1190.25/53.5472i) 0 0; 0 0 0 (1/(3.9984i*10^-3)) 0; 0 0 0 0 1190.25/11.8982i]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[-20.1053 20.1053 0 0 0 0; 0 299.9622 0 0 0 -299.9622; 0 -22.2281 22.2281 0 0 0; 0 0 0 250.10004 0 -250.10004; 0 0 0 0 -100.0361 100.0361]; PP=BB*del1; syms P1; syms P2; Ppv=[7;-8;P1-2-0.074*(P1+P2);-25-0.926*(P1+P2);8+P2]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv; a=vpa(PPpv,4); disp(a); A=[ -0.926 0.074; -0.926 0.074; 0.926 0.926;


Arizona State University v

0 1]; b=[14.8-3;4.99-2;30-25;10.0-8.0]; f=[-1;-1]; lb=[0;0]; [x,fval,exitflag]=linprog(f,A,b,[],[],lb); if exitflag==1 disp(x); end

A2.6. Line Outage Contingency at One of the Lines between Imperial Valley and

Palo Verde

P=[7;-8;-2;-25;8]; Y=[1190.25/59.2008i -1190.25/59.2008i 0 0 0; -1190.25/59.2008i (1190.25/3.968i)+(1190.25/59.2008i)+(1190.25/53.5472i) -1190.25/53.5472i 0 0; 0 -1190.25/53.5472i (1190.25/53.5472i)+(1190.25/60.104i) -1190.25/60.104i 0; 0 0 -1190.25/60.104i (1190.25/60.104i)+(2500/19.992i) 0; 0 0 0 0 1190.25/11.8982i]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[-20.1053 20.1053 0 0 0 0; 0 299.9622 0 0 0 -299.9622; 0 -22.2281 22.2281 0 0 0; 0 0 -19.8032 19.8032 0 0; 0 0 0 125.05002 0 -125.05002; 0 0 0 0 -100.0361 100.0361]; PP=BB*del1; syms P1; syms P2; Ppv=[7;-8;P1-2-0.074*(P1+P2);-25-0.926*(P1+P2);8+P2]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv; a=vpa(PPpv,4); disp(a);

A2.7. Line Outage Contingency between Palo Verde & Bicknell

P=[7;-8;-2;-25]; Y=[1190.25/59.2008i -1190.25/59.2008i 0 0; -1190.25/59.2008i (1190.25/3.968i)+(1190.25/59.2008i)+(1190.25/53.5472i) -1190.25/53.5472i 0; 0 -1190.25/53.5472i (1190.25/53.5472i)+(1190.25/60.104i) -1190.25/60.104i; 0 0 -1190.25/60.104i (1190.25/60.104i)+(1/(3.9984i*10^-3))]; B=imag(Y); del=inv(B)*P; del1=[del;0]; BB=[-20.1053 20.1053 0 0 0; 0 299.9622 0 0 -299.9622; 0 -22.2281 22.2281 0 0; 0 0 -19.8032 19.8032 0; 0 0 0 250.10004 -250.10004]; PP=BB*del1; syms P1; syms P2; Ppv=[7;-8;P1-2-0.074*P1;-25-0.926*P1]; delpv=inv(B)*Ppv; del1pv=[delpv;0]; PPpv=BB*del1pv;


Arizona State University vi

a=vpa(PPpv,4); disp(a); A=[-3.532*10^(-18); -0.4548; -0.4548; 0.4712; 0.4548]; b=[7.09-7.0;14.8-3;4.99-2;4.99-0.0001717;30.0-25]; f=[-1]; lb=[0]; [x,fval,exitflag]=linprog(f,A,b,[],[],lb); if exitflag==1 disp(x); end

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