optimization principles - electric power

appendix/App-acflow.xlsSensitivity Report 15

Microsoft Excel 8.0e Sensitivity Report

Worksheet: [OPT.POWERFLOW.xls]Sheet1

Report Created: 11/6/01 10:50:30 AM

Adjustable Cells

FinalReduced

CellNameValueGradient

$M$3d1-0.05247509260

$M$4d2-0.04585822410

$M$5d3-0.00771356680

$M$6d4-0.02646935870

$M$7d5-0.02372038580

$M$8d6-0.02553846110

$M$9d7-0.03862768390

$M$10v11.04687954810

$M$11v21.04796046160

$M$12v31.05-0.5440162485

$M$13v41.04779425430

$M$14v51.050.1326051417

$M$15v61.04848931960

$M$16v71.04843382020

$M$17v81.05-1.2411695373

Constraints

FinalLagrange

CellNameValueMultiplier

$B$18P1-0.00000000070

$C$18P2-0.0000000017-40.1769669584

$D$18P33.4999999995-17.8721411214

$E$18P40.0000000003-40.0227334053

$F$18P51.31962653760

$G$18P62.0000000004-5.0146300611

$H$18P7-0.0000000016-40.1185284999

$I$18P83.0000000002-7.8107906733

$H$59Q40.10.0067066801

$B$18P1-0.00000000079.7693372395

$C$18P2-0.00000000170

$D$18P33.49999999950

$E$18P40.00000000030

$F$18P51.31962653760

$G$18P62.00000000040

$H$18P7-0.00000000160

$I$18P83.00000000020

$H$68Q7-0.22-0.0089169981

$H$52Q2-0.1000000001-0.004926778

Sensitivity Report 16



Report Created: 11/27/01 3:38:53 PM

Adjustable Cells

FinalReduced


$M$2d1-0.05243035240

$M$3d2-0.04580346910

$M$4d3-0.00768157520

$M$5d4-0.02641100630

$M$6d5-0.02352887260

$M$7d6-0.02551374470

$M$8d7-0.0386090430

$M$9v11.04589653980

$M$10v21.04700026790

$M$11v31.04974817890

$M$12v41.04715518540

$M$13v51.04818669870

$M$14v61.04815367980

$M$15v71.04806411210

$M$16v81.05-1.7993306946

Constraints

FinalLagrange


$B$18P10.00000000010

$C$18P20.0000000002-40.1360955313

$D$18P33.5-17.8332990657

$E$18P4-0.0000000001-39.9707607913

$F$18P51.31963167670

$G$18P62-4.9159831043

$H$18P70.0000000002-40.0144177852

$I$18P83-7.653398799

$H$59Q40.10.006678363

$B$18P10.00000000019.8111940189

$C$18P20.00000000020

$D$18P33.50

$E$18P4-0.00000000010

$F$18P51.31963167670

$G$18P620

$H$18P70.00000000020

$I$18P830

$H$68Q7-0.22-0.0079352607

$H$52Q2-0.1-0.0049654374

Sensitivity Report 17




Adjustable Cells

FinalReduced


$M$2d1-0.05443003380

$M$3d2-0.04753618420

$M$4d3-0.00766673690

$M$5d4-0.02730391010

$M$6d5-0.02435793640

$M$7d6-0.02643038830

$M$8d7-0.04009066040

$M$9v11.02421180840

$M$10v21.02538426280

$M$11v31.02700718920

$M$12v41.02535560330

$M$13v51.02710089190

$M$14v61.02695158490

$M$15v71.02716899380

$M$16v81.03-1.8213431994

Constraints

FinalLagrange


$B$18P10.00000015090

$C$18P20.000000926-40.1857372891

$D$18P33.5000002218-17.9582180106

$E$18P40.0000003143-40.0319632902

$F$18P51.32046187810

$G$18P62.0000001834-4.9513804097

$H$18P70.0000002451-40.0259239584

$I$18P83.0000005611-7.6081920352

$H$59Q40.10000001010.0065428514

$B$18P10.00000015099.7670016308

$C$18P20.0000009260

$D$18P33.50000022180

$E$18P40.00000031430

$F$18P51.32046187810

$G$18P62.00000018340

$H$18P70.00000024510

$I$18P83.00000056110

$H$68Q7-0.2199999961-0.0074097001

$H$52Q2-0.0999999484-0.0053358782

Sheet1

AC Power Flow Solution

nodes12345678Note

GeneratorsChoice Variables

P103.502200.5Required generationV21.0097495996

loadsV40.9878077897

P11.51.70.5032.10Specified LoadV71.0098342875

Q-0.20.10.25-0.100.10.220Specified loadd2-0.0150913137

d30.0009022743

Voltage11.00974959961.030.987807789711.021.00983428751.03d4-0.0084010584

angle0-0.01509131370.0009022743-0.00840105840.0322043589-0.0285394939-0.0395945965-0.0267943705d50.0322043589

Var injection into network0.000d6-0.0285394939

0.33267201820.0500141398-2.26748775062.5607213505-0.00010421540.3830026490.1100538608-1.2951807725Perf. Indexd7-0.0395945965

Net injection to Networkd8-0.0267943705

P0-1.51.8-0.52-1-2.10.5

Q-0.53267201820.04998586022.5174877506-2.66072135050.0001042154-0.2830026490.10994613921.2951807725Voltage tolerance

1.030.97

B (Imgy.)COMPONENT OF Y

12345678Magnitude of Y

154-2000-200-14012345678

2-2078-150-25-1800154.269328354120.09975124220020.0997512422014.06982586960

30-1586-400-210-10220.099751242278.389029844715.0748134317025.124689052818.08977611800

400-4090-200-3003015.074813431786.428930341640.1995024845021.1047388044010.0498756211

5-20-250-206500040040.199502484590.448880590120.0997512422030.14962686340

60-18-2100740-35520.099751242225.1246890528020.099751242265.3241915373000

7-1400-300089-456018.08977611821.10473880440074.3690795963035.1745646739

800-1000-35-4590714.06982586960030.14962686340089.44389302845.224440295

80010.04987562110035.174564673945.22444029590.4488805901

G (Real) COMPONENT OF Y

ANGLE OF Y

5.4-200-20-1.401-1.4711276725-1.471127679300-1.47112767930-1.47112768140

-27.8-1.50-2.5-1.8002-1.4711276793-1.471127673-1.47112768090-1.4711276783-1.471127679800

0-1.58.6-40-2.10-130-1.4711276809-1.4711276732-1.47112767680-1.4711276790-1.4711276842

00-49-20-30400-1.4711276768-1.4711276732-1.47112767930-1.47112767760

-2-2.50-26.50005-1.4711276793-1.47112767830-1.4711276793-1.4711276728000

0-1.8-2.1007.40-3.560-1.4711276798-1.47112767900-1.4711276730-1.4711276771

-1.400-3008.9-4.57-1.471127681400-1.471127677600-1.4711276732-1.4711276765

00-100-3.5-4.59800-1.471127684200-1.4711276771-1.4711276765-1.4711276732

12345678

FLOW EQUATIONS

Branch PBranch QComponents of

P15-0.6429388196Q150Performance index.

P120.2854881577Q120.1949919924PQ

P170.5469695437P10.1895188818Q170.1376800258Q10.3326720182

P25-1.1660401328Q25-0.2461163578

P21-0.2848381207Q21-0.1968930863

P23-0.2799716586Q230.3067175054

P260.2308453657P2-1.5000045464Q260.1863060784Q20.050014139800.0000000008

P320.2809858243Q32-0.3128686861

P340.5526266294Q34-1.7383190643

P360.6720542606Q36-0.2163000001

P380.2942030241P31.7998697384Q380Q3-2.26748775060.000000017

P470.8695179865Q470.6527383845

P43-0.5451536551Q431.66711176

P45-0.8244451675P4-0.500080836Q450.2408712061Q42.56072135050.0000000065

P510Q510

P521.1719234401Q520.2437399905

P540.8279994145P51.9999228547Q54-0.2438442059Q5-0.00010421540.0000000060.0000000434

P62-0.230320959Q62-0.1881973512

P63-0.6699319736Q630.2142000001

P68-0.0998642999P6-1.0001172325Q680.3570000001Q60.3830026490.0000000137

P71-0.5446180255Q71-0.1390340107

P74-0.8651508456Q74-0.6672933827

P78-0.6903633876P7-2.1001322586Q780.9163812542Q70.11005386080.00000001750.0000000116

P83-0.2933892555Q830

P860.1002254984Q86-0.3605000001

P870.6929602225P80.4997964654Q87-0.9346807724Q8-1.29518077250.0000000414

Sum0.00000010210.0000000558

Sheet2

Sheet3

appendix/app-dcflow.xlsSheet1

DC Load flow formulation

nodes12345678

P203.502.8203Specified Generation

L12.51.73.502.52.10Specified Demand

1-2.51.8-3.52.8-0.5-2.13Net injection

Y Matrix - B (Imgy.)COMPONENT OF Y

12345678

154-2000-200-140Branch 1-2 Flow

2-2078-150-25-18000.6944806636

30-1586-400-210-10

400-4090-200-300

5-20-250-2065000

60-18-2100740-35

7-1400-300089-45

800-1000-35-4590

Y Inverse =Z'PinjdBranch flow

0.03997542470.02355063270.0138433070.01586167770.02623858250.00965703830.01163490201-0.00529285851-20.6944806636

0.02355063270.03270740730.01599369270.01556149320.02461427230.01249460650.0089500410-2.5-0.04001689171-5-0.2508372703

0.0138433070.01599369270.02519674020.01737894510.01575826710.0110407840.008035670201.8-0.01988525181-70.5563566067

0.01586167770.01556149320.01737894510.02695269920.01915884410.00871709090.01158027490-3.5-0.06112676062-3-0.301974599

0.02623858250.02461427230.01575826710.01915884410.03882008220.01045919610.010585454802.80.0072490052-5-1.1816474175

0.00965703830.01249460650.0110407840.00871709090.01045919610.01968593760.00445742990-0.5-0.02213370732-6-0.32189732

0.0116349020.0089500410.00803567020.01158027490.01058545480.00445742990.01696962780-2.1-0.04503261623-41.6496603537

00000000303-60.0472175652

3-8-0.1988525179

Ybr Branch admittance martix4-5-1.3675153122

1-2200000000000004-7-0.4828243341

1-5020000000000006-8-0.7746797547

1-7001400000000007-8-2.0264677274

2-300015000000000

2-500002500000000

2-600000180000000

3-400000040000000

3-600000002100000

3-800000000100000

4-500000000020000

4-700000000003000

6-800000000000350

7-800000000000045

A Node Branch Incidence MatrixY br * A

12345678

1-21-10000001-220-20000000

1-51000-10001-520000-20000

1-7100000-101-71400000-140

2-301-1000002-3015-1500000

2-50100-10002-502500-25000

2-601000-1002-6018000-1800

3-4001-100003-40040-400000

3-600100-1003-6002100-2100

3-80010000-13-800100000-10

4-50001-10004-500020-20000

4-7000100-104-70003000-300

6-80000010-16-800000350-35

7-80000001-17-800000045-45

SF SHIFT FACTOR MATRIX -- = Ybr * A * Z

1-20.3284958393-0.1831354915-0.04300771390.00600368990.0324862045-0.05675136270.05369721850

1-50.2747368429-0.0212727918-0.0382992011-0.0659433289-0.2516299926-0.01604315510.02098894310

1-70.39676731790.20440828340.08130691490.0599396390.2191437880.0727945178-0.07468616160

2-30.14560988570.2507057189-0.1380457123-0.02726177830.13284007880.02180733760.01371556240

2-5-0.06719874550.20232837460.005885641-0.0899337735-0.35514524640.0508852595-0.04088534430

2-60.2500846990.3638304150.08915235740.12319924170.2547913721-0.12944395980.08086700040

3-4-0.08073482660.0172879810.312711805-0.382950167-0.13602308230.0929477238-0.14178418630

3-60.08791164220.0734808110.29727508080.18189893820.1112804905-0.18154822570.07514304650

3-80.138433070.15993692690.25196740180.17378945050.15758267050.11040783950.08035670220

4-5-0.2075380974-0.18105558270.032413560.1558771024-0.393224761-0.03484210440.01989640120

4-70.12680327080.19834356370.2802982450.46117273060.25720167880.1277898283-0.16168058760

6-80.33799634130.4373112260.38642743830.30509817990.36607186270.68900781450.15601004690

7-80.52357058870.40275184710.36160515990.52111236960.47634546680.2005843460.76363325090

7

appendix/Polynomial.fit.xlsSheet1

Polynomial Fit to Data

X0.0001.00010.00015.00020.00025.00030.000

Y0.00018.03724.29310.9314.1851.5530.572

LSQ Approach

a0a1a2a36.28310.99522.92214.8404.393-1.9892.122Computed

6.2835.137-0.4330.0090.00018.03724.29310.9314.1851.5530.572Actual

Choice Variables39.47349.5891.87715.2790.04312.5462.400DIF^2

Objective121.209Sum

Sheet2

Sheet3

appendix/Pseudo.inv.eg.xlsSheet1

Pseudo Inverse - OverSpecified Case.

Least Square Method of Solution

Given

AX - 3 vector=b

123-0.43.0085742

of Rank 34-12-1.42857=4.0057128

-4232.08857145.0085742

5674.0485798

34-1-9.0028514

Compute Pseudo Inverse

Let A+ be choice variable

A+xq = A+ * b

-0.0081294759-0.0108402272-0.0135517749-0.01097192860.0244170678Then-0.4

-0.029059683-0.0387441971-0.0484269975-0.03921463370.0871534853-1.4285700002

0.04246962770.05662539480.07078030810.0573130623-0.1274516622.0885714

Error Analysis

A * xqError

3.00857419970A+ * A

4.005712800200.0211082756-0.0006855747-0.1879447747

5.008574199700.0750588057-0.0029030248-0.6716043569

4.04857979910-0.10993969980.00394620270.9816436953

-9.00285140070

MinimizeSum

index0

Sheet2

Sheet3

Ch2. Matrix Algebra/complex.xlsSheet1

Complex Matrix

20.60.40.5

0.630.150.2

XRealImgyY

X INVY INVYINV * XX INV * Y

0.5319148936-0.106382978740-10020-2760.19680851060.2446808511

-0.10638297870.3546099291-3080-122220.01063829790.0177304965

(Y INV * X + X INV * Y)(Y INV * X + X INV * Y) -1V = -(Y INV * X + X INV * Y) -1 * X INV

20.1968085106-275.75531914890.18848296780.23410373950.07535224460.0629641309

-11.9893617021222.01773049650.01017842350.0171461730.00358999410.0049973922

U = - YINV * X * V

-0.51620652780.1199976217-0.0753522446-0.0629641309

0.1072482507-0.3538514917-0.0035899941-0.0049973922

Check

0.5162065278-0.1199976217

-0.10724825070.3538514917X * UY * VX*U - Y*V

-0.96806410510.02768434850.03193589490.02768434851-0

0.0120208355-0.98955590190.01202083550.0104440981-01

Y * UX * VY * U + X * V

-0.1528584857-0.12892669710.15285848570.128926697100

-0.055981329-0.05277065510.0559813290.052770655100

Sheet2

Sheet3

Ch2. Matrix Algebra/e.vector.restructure.xlsSheet1

-0.209681.870.42290.9933531142.8230323745-0.00004976092.804268

0.287865-6.511485.71190.097533250100.2753395226

0.0494.30803-7.670480.061124006600.1725550494

0.9999992881

Sheet2

Sheet3

Ch2. Matrix Algebra/eigenvalue.xlsSheet1

42-2.5l

2335.5543592024

-2.534

Determinant

-0.00000028

Sheet2

Sheet3

Ch2. Matrix Algebra/Eigenvector.xlsSheet1

Confirmation of Eigenvector

Au

-5.50981.870.42290.094129

0.287865-11.81165.7119-0.766896

0.0494.30803-12.97060.745248

Eigenvalue

l-17.3975844638(A u - u l)2

0.000000003

A uu l0.0000000001

-1.637562105-1.6376172280

13.342147289413.3421379350.0000000031Objective

-12.9655123627-12.9655150265

Sheet2

Sheet3

Ch3. solution of LP/lpdemo put on solver.xlsSheet1

Choice VariableOBJECTIVE

x10.00000-7.00000=-1*C2-2*C3+-1*C4

x23.00000

x31.00000Constraints

2.000002.00000=2*C2+C3-C4

x1, x2, x3> =0.000006.000002.00000=-2*C2-C3+5*C4

6.000006.00000=4*C2+1.5*C3+1.5*C4

Bound relaxationLimitComputed Value

Sensitivity Report

Adjustable Cells

FinalReducedObjectiveAllowableAllowable

CellNameValueCostCoefficientIncreaseDecrease

$C$2x1 Choice Variable0.000004.00000-110000000000000000000000000000004.0000000001

$C$3x2 Choice Variable3.000000.00000-211000000000000000000000000000000

$C$4x3 Choice Variable1.000000.00000-131

Constraints

FinalShadowConstraintAllowableAllowable

CellNameValuePriceR.H. SideIncreaseDecrease

$G$52.00000-0.50000221.3333333333

$G$62.000000.00000610000000000000000000000000000004

$G$76.00000-1.00000633

Sheet2

Sheet3

Ch3. solution of LP/simplex.xlsSheet1

SIMPLEX METHOD - ILLUSTRATION

BasisDecision VariablesSlack variables

x1x2x3s1s2s3pRHS

s121-110002

s22-1501006

s341.51.500106

-1-2-1000-10

Objective

New Basis

X2s2s3pInverse

10001000

-11001100

1.5010-1.5010

-200-1-200-1

Solution of underspecified system with new basis

x1x2x3s1s2s3pRHS

x221-110002

s24041100=8

s3103-1.50103

-303-2001-4

Objective

New Basis

x2s2x3pInverse

10-10100.33333333330

014001-1.33333333330

0030000.33333333330

003100-11

Solution of underspecified system with new basis

x1x2x3s1s2s3pRHS

2.3333333333100.500.333333333303

2.66666666670031-1.33333333330=4

0.333333333301-0.500.333333333301

-400-0.50-11-7

No Positive numberFinalObjective

s3piinx3inpi

Sheet2

Sheet3

Ch3. solution of LP/Solved Primal-Dual.xlsSensitivity Report 8


Worksheet: [Primal-dual relation.xls]Sheet1


Sensitivity Analysis of Primal Problem

Adjustable Cells

FinalReducedObjectiveAllowableAllowable

CellNameValueCostCoefficientIncreaseDecrease

$A$2x10232.000000007182.00000000221000000000000000000000000000000232.0000000071

$B$2x230099.9999999998299.99999999991000000000000000000000000000000

$C$2x3-57.80200.00000000011000000000000000000000000000000150.0000000002

Constraints

FinalShadowConstraintAllowableAllowable

CellNameValuePriceR.H. SideIncreaseDecrease

$C$52x1+4x2 x3120-75120100000000000000000000000000000015.6

$C$63x1+4x2+6x3 x3-226.80131000000000000000000000000000000239.8

$C$72x1-4x3 x3231.2020031.21000000000000000000000000000000

$C$810x2+5x3 x3114011391000000000000000000000000000000

Sheet1

Primal Problem

x1x2x3

030-57.8Objective

-8560

Constraints

2 x1 + 4 x2120=$A$4Not Binding23.7499992738

$B$3Q2148.749960722$B$3>=$B$4Not Binding48.749960722

$C$3Q3224.9999226673$C$3>=$C$4Not Binding74.9999226673

$D$3Q4302.4998838605$D$3>=$D$4Not Binding102.4998838605

$E$3Q5381.2498512811$E$3>=$E$4Not Binding131.2498512811

$F$3Q6461.2498512811$F$3>=$F$4Not Binding161.2498512811

$B$2Q274.9999614482$B$2>=0Not Binding74.9999614482

$C$2Q376.2499619453$C$2>=0Not Binding76.2499619453

$D$2Q477.4999611932$D$2>=0Not Binding77.4999611932

$E$2Q578.7499674206$E$2>=0Not Binding78.7499674206

$F$2Q680$F$2>=0Not Binding80

$A$2Q173.7499992738$A$2=0Binding0

$F$19P51.320396093$F$19>=0Not Binding1.320396093

$G$19P62$G$19>=0Not Binding2

Adjustable Cells$I$19P83$I$19>=0Not Binding3

CellNameOriginal ValueFinal Value$D$19P33.5$D$19>=0Not Binding3.5

$B$15d10-0.0545011306$B$19P10$B$19=-0.6Not Binding0.5598646362

$B$13V11.0262911993$B$13

optimization principles - electric power

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