New England Reduced Model (39 Bus System) MATLAB Study Report

IEEE PES Task Force on Benchmark Systems for Stability Controls Ian Hiskens November 19, 2013

Abstract

This report summarizes summarizes a study of an IEEE 10-genera 10-generator, tor, 39-bus system. system. Three Three types of analysis analysis were performed: load flow, small disturbance analysis, and dynamic simulation. All analysis was carried out using MATLAB, and this report’s objective is to demonstrate how to use MATLAB to obtain results that are comparable comparable to benchmark benchmark results results from other other analysis analysis methods. Data from other methods may be found on the website website www.sel.eesc.usp.br/ieee. www.sel.eesc.usp.br/ieee.

Contents 1 Syst System em Model Model

1.1 Genera Generator torss . . . . . . . . . . 1.1.1 1.1 .1 State State Variabl ariablee M Model odel . 1.1.2 1.1 .2 Model Model Param Paramete eters rs . . 1.1. 1.1.33 AVR Model Model . . . . . . 1.1.4 1.1 .4 AVR Param Paramete eters rs . . . 1.1. 1.1.55 PSS PSS Model Model . . . . . . 1.1.6 1.1 .6 PSS Param Paramete eters rs . . . 1.1.7 1.1 .7 Gove Governo rnorr Model Model . . . 1.2 1.2 Lo Load adss . . . . . . . . . . . . . 1.2. 1.2.11 Load Load Model Model . . . . . . 1.2.2 1.2 .2 Loa Load d Param Paramete eters rs . . . 1.3 Lines and Transformer ransformerss . . . 1.4 Power Power and Voltag Voltagee Setpoints Setpoints

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2 Load Load Flow Flow Result Resultss

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3 Small Disturban Disturbance ce Analys Analysis is

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4 Dynami Dynamic c Simula Simulatio tion n

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5 Appe Append ndix ix

15 R 

5.1 Accessing Accessing Simu Simulation lation Data with with MATLAB MATLAB . . 5.1.1 5.1 .1 Variabl ariablee Indexi Indexing ng . . . . . . . . . . . . . . 5.1.2 5.1 .2 Exampl Examplee Applic Applicati ation on . . . . . . . . . . . . 5.2 Conten Contents ts of file 39bus.out . . . . . . . . . . . . .

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2 4 6 12 13

List of Figures 1 2 3 4 5

IEEE IEEE 39-b 39-bus us net network ork . . . . . . . . . . . . AVR bloc block k diag diagra ram m . . . . . . . . . . . . PSS PSS bloc block k diag diagra ram m . . . . . . . . . . . . . Graphi Graphical cal depict depiction ion of Eigen Eigenv values alues . . . . δ (in (in radians) versus time for generators .

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Report: 39-bus system (New England Reduced Model)

6 7

ω (in percent per-unit) for generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Con Contents ents of file file 39bus.out (Page 1 of 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 19

List of Tables 1 2 3 4 5 6 7 8 9 10 11

Gene Genera rato torr iner inerti tiaa data data . . . . . . . . . . . . . . . . . . . . . . . Gene Generrato ator data ata . . . . . . . . . . . . . . . . . . . . . . . . . . . Gene Genera rato torr AVR para parame mete ters rs . . . . . . . . . . . . . . . . . . . . Gene Genera rato torr PSS PSS para parame mete ters rs . . . . . . . . . . . . . . . . . . . . . Gene Genera rato torr setpo setpoin intt data data . . . . . . . . . . . . . . . . . . . . . . Activ Activee and reac reactiv tivee power power draw drawss for all all loads loads at init initial ial volt voltage age . Net Network ork data data . . . . . . . . . . . . . . . . . . . . . . . . . . . . Powe Powerr and volta voltage ge setpoin setpointt data data . . . . . . . . . . . . . . . . . Power flow flow resu result ltss . . . . . . . . . . . . . . . . . . . . . . . . . Eigenv Eigenvalues calculated calculated through through small small disturban disturbance ce analysis analysis . . .  Mapp Ma ppin ingg from from MATLAB files to simulation simulation data . . . . . . . . . R

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3 4 5 6 7 8 9 10 11 12 15

The IEEE 39-bus system analyzed in this report is commonly known as ”the 10-machine New-England Power System.” This system’s parameters are specified in a paper by T. Athay et al[ 1] and 1] and are published in a book titled ’Energy Function Analysis for Power System Stability’ [2]. [2]. A diagram of the system system is shown shown in Figure 1, Figure 1, and system models and parameters are introduced in the following section.

G8 <37>

G10 <25>

<26>

<28 <28> >

<29> <29>

<30> <27>

<2>

<18>

<38 38> >

<17>

G9

<24> <24>

<1>

G6

<3> <16>

<35>

<15>

<21>

<22>

G1 <39>

<4>

<14>

<5>

<6>

<12> <19>

<23>

<7> <13>

<20>

<36>

<11>

<8> <31>

G7

<10>

<34>

<33>

<9>

G2

<32>

G5

G4

G3

Figure 1: IEEE 39-bus network


1 1.1 1.1.1

3

System Model Generators State Variable Model

Generator analysis was carried out using a fourth-order model, as defined in the equations below. Equations 1 through 4 model the generator, while the remaining equations relate various parameters. Parameter values for the model are shown in Table 2 on the system base MVA. For more information on this generator model and its parameters, refer to Sauer and Pai pages 101-103 [3]. ˙  = 1 (−E  − (xd − x )I d + E fd ) E q q d  T do ˙  = 1 (−E  + (xq − x )I q ) E d d q  T qo δ ˙ = ω ˙ ω =

(2) (3)

1 (T mech − (φd I q − φq I d ) − Dω ) M

(4)

0 = r a I d + φq + V d 0 = r a I q − φd + V q 0 = −φd − xd I d + E q

(5) (6) (7)

0 = −φq − xq I q − E d

(8)

0 = V d sin δ + V q cos δ − V r 0 = V q sin δ − V d cos δ − V i 0 = I d sin δ + I q cos δ − I r 0 = I q sin δ − I d cos δ − I i

1.1.2

(1)

(9) (10) (11) (12)

Model Parameters

Generator inertia data is given in Table 1. Note that the per unit conversion for M associated with ω is in  radians per second. Also, we changed the value of T qo for Unit 10 from 0.0 to 0.10. Table 1: Generator inertia data Unit No.

M=2*H

1 2 3 4 5 6 7 8 9 10

2 · 500.0/(120π) 2 · 30.3/(120π) 2 · 35.8/(120π) 2 · 28.6/(120π) 2 · 26.0/(120π) 2 · 34.8/(120π) 2 · 26.4/(120π) 2 · 24.3/(120π) 2 · 34.5/(120π) 2 · 42.0/(120π)

4


All other generator parameters are set according to Table 2.

Table 2: Generator data Unit No. 1 2 3 4 5 6 7 8 9 10

1.1.3

H

Ra

xd

xq

xd

xq

 T do

 T qo

xl

500 30.3 35.8 28.6 26 34.8 26.4 24.3 34.5 42

0 0 0 0 0 0 0 0 0 0

0.006 0.0697 0.0531 0.0436 0.132 0.05 0.049 0.057 0.057 0.031

0.008 0.17 0.0876 0.166 0.166 0.0814 0.186 0.0911 0.0587 0.008

0.02 0.295 0.2495 0.262 0.67 0.254 0.295 0.29 0.2106 0.1

0.019 0.282 0.237 0.258 0.62 0.241 0.292 0.28 0.205 0.069

7 6.56 5.7 5.69 5.4 7.3 5.66 6.7 4.79 10.2

0.7 1.5 1.5 1.5 0.44 0.4 1.5 0.41 1.96 0

0.003 0.035 0.0304 0.0295 0.054 0.0224 0.0322 0.028 0.0298 0.0125

AVR Model

All generators in the system are equipped with automatic voltage regulators (AVRs). We chose to use static AVRs with E fd limiters. The model for this controller is shown in Figure 2 below.

Figure 2: AVR block diagram


5

AVR parameters are related according to equations 13 to 24. ˙ f = 1 (V T − V f ) V T R ˙ fd = 1 (K A xtgr − E f d ) E T A x˙ i = xerr − xtgr ˙ REF = V

(13) (14)

0 when t > 0 V T − V setpoint when t < 0

0 = V T 2 − (V r2 + V i2) 0 = T B xtgr − T C xerr − xi 0 = xerr − (V REF + V P SS − V f ) 0 = E f d − E fd Upper Limit Detector < 0 Upper Limit Detector > 0 Lower Limit Detector > 0 Lower Limit Detector < 0

1.1.4

(15)



(16)

(17) (18) (19) (20)

   

0 = Upper Limit Switch − 1 0 = Upper Limit Detector − E f d − E fd,Max

(21)

0 = Upper Limit Switch 0 = Upper Limit Detector − (K A xtgr − E fd )

(22)

0 = Lower Limit Switch − 1 0 = Lower Limit Detector − E fd − E fd,Min

(23)

0 = Lower Limit Switch 0 = Lower Limit Detector − (K A xtgr − E fd )

(24)

AVR Parameters

AVR parameters, as defined in the previous section, are specified in Table 3. Table 3: Generator AVR parameters

1.1.5

Unit No.

T R

K A

T A

T B

T C

V setpoint

E fd,Max

E f d,Min

1 2 3 4 5 6 7 8 9 10

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0

0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015 0.015

10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0300 0.9820 0.9831 0.9972 1.0123 1.0493 1.0635 1.0278 1.0265 1.0475

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

-5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -5.0

PSS Model

Each generator in the system is equipped with a δ and ω type PSS with two phase shift blocks. The model for this PSS is shown in Figure 3 below.

6


Figure 3: PSS block diagram PSS parameters are related according to equations 25 to 33. V W T W x˙ P = V W − V P x˙ Q = V P − V out

x˙ W =

(25)

0 = ωK P SS − V W − xW 0 = V P T 2 − V W T 1 − xP 0 = V out T 4 − V P T 3 − xQ 0 = xPSS,Max − V out − (V P SS,Max − V out ) 0 = V out − xPSS,Min − (V out − V P SS,Min ) 0=

1.1.6

PSS Parameters

 

(26) (27)

V P SS − xPSS,Max V P SS − xPSS,Min V P SS − V out

(28) (29) (30) (31) (32)

when (V P SS,Max − V out ) < 0 when (V out − V P SS,Min ) < 0 otherwise

(33)

PSS parameters defined in the previous section are specified according to Table 4. Table 4: Generator PSS parameters

1.1.7

Unit No.

K

T W

T 1

T 2

T 3

T 4

V P SS,Max

V P SS,Min

1 2 3 4 5 6 7 8 9 10

1.0/(120π) 0.5/(120π) 0.5/(120π) 2.0/(120π) 1.0/(120π) 4.0/(120π) 7.5/(120π) 2.0/(120π) 2.0/(120π) 1.0/(120π)

10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

5.0 5.0 3.0 1.0 1.5 0.5 0.2 1.0 1.0 1.0

0.60 0.40 0.20 0.10 0.20 0.10 0.02 0.20 0.50 0.05

3.0 1.0 2.0 1.0 1.0 0.5 0.5 1.0 2.0 3.0

0.50 0.10 0.20 0.30 0.10 0.05 0.10 0.10 0.10 0.50

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

-0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2 -0.2

Governor Model

No governor dynamics are included in our analysis, and each generator’s mechanical torque is assumed to be constant. Since Unit 2 is the angle reference and resides at the swing node, P set point is determined by the


7

power flow initialization. The value of P set point is given for each generator in Table 5 on the system base of 100 MVA. Table 5: Generator setpoint data

1.2 1.2.1

Unit No.

P set point

1 2 3 4 5 6 7 8 9 10

10.00 6.50 6.32 5.08 6.50 5.60 5.40 8.30 2.50

Loads Load Model

We use a constant impedance load model in our analysis. Loads modeled this way are voltage dependent and behave according to the following equations.

if t < 0

if t > 0

1.2.2

Load Parameters

    

0 = P 0 + V r I r + V i I i 0 = Q0 + V i I r − V r I i V = (V r2 + V i2 )2 0 = P 0 0 = Q0

α

  V V 0

V V 0

β

(34)

+ V r I r + V i I i + V i I r + V r I i

(35)

V = (V r2 + V i2 )2

Table 6 contains load behavior at initial voltage. Due to the voltage dependence discussed above, these values may not be accurate after voltages change.

8


Table 6: Active and reactive power draws for all loads at initial voltage Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 31 39

1.3

Load P [PU] Q [pu] 0.000 0.000 3.220 5.000 0.000 0.000 2.338 5.220 0.000 0.000 0.000 0.075 0.000 0.000 3.200 3.290 0.000 1.580 0.000 6.280 2.740 0.000 2.475 3.086 2.240 1.390 2.810 2.060 2.835 0.092 11.040

0.000 0.000 0.024 1.840 0.000 0.000 0.840 1.760 0.000 0.000 0.000 0.880 0.000 0.000 1.530 0.323 0.000 0.300 0.000 1.030 1.150 0.000 0.846 -0.920 0.472 0.170 0.755 0.276 0.269 0.046 2.500

Lines and Transformers

Network data for the system is shown in Table 7. As with generator data, all values are given on the system base MVA at 60 Hz.


9

Table 7: Network data Line Data From Bus 1 1 2 2 3 3 4 4 5 5 6 6 7 8 9 10 10 13 14 15 16 16 16 16 17 17 21 22 23 25 26 26 26 28 12 12 6 10 19 20 22 23 25 2 29 19

To Bus 2 39 3 25 4 18 5 14 6 8 7 11 8 9 39 11 13 14 15 16 17 19 21 24 18 27 22 23 24 26 27 28 29 29 11 13 31 32 33 34 35 36 37 30 38 20

R 0.0035 0.001 0.0013 0.007 0.0013 0.0011 0.0008 0.0008 0.0002 0.0008 0.0006 0.0007 0.0004 0.0023 0.001 0.0004 0.0004 0.0009 0.0018 0.0009 0.0007 0.0016 0.0008 0.0003 0.0007 0.0013 0.0008 0.0006 0.0022 0.0032 0.0014 0.0043 0.0057 0.0014 0.0016 0.0016 0 0 0.0007 0.0009 0 0.0005 0.0006 0 0.0008 0.0007

Transformer Tap X 0.0411 0.025 0.0151 0.0086 0.0213 0.0133 0.0128 0.0129 0.0026 0.0112 0.0092 0.0082 0.0046 0.0363 0.025 0.0043 0.0043 0.0101 0.0217 0.0094 0.0089 0.0195 0.0135 0.0059 0.0082 0.0173 0.014 0.0096 0.035 0.0323 0.0147 0.0474 0.0625 0.0151 0.0435 0.0435 0.025 0.02 0.0142 0.018 0.0143 0.0272 0.0232 0.0181 0.0156 0.0138

B 0.6987 0.75 0.2572 0.146 0.2214 0.2138 0.1342 0.1382 0.0434 0.1476 0.113 0.1389 0.078 0.3804 1.2 0.0729 0.0729 0.1723 0.366 0.171 0.1342 0.304 0.2548 0.068 0.1319 0.3216 0.2565 0.1846 0.361 0.513 0.2396 0.7802 1.029 0.249 0 0 0 0 0 0 0 0 0 0 0 0

Magnitude 1.006 1.006 1.07 1.07 1.07 1.009 1.025 1 1.025 1.025 1.025 1.06

Angle 0 0 0 0 0 0 0 0 0 0 0 0

10

1.4


Power and Voltage Setpoints

Table 8 contains power and voltage setpoint data, specified on the system MVA base. Note that Generator 2 is the swing node, and Generator 1 represents the aggregation of a large number of generators.

Table 8: Power and voltage setpoint data Bus

Type

Voltage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PQ PV PV PV PV PV PV PV PV PV PV

[PU] 1.0475 0.982 0.9831 0.9972 1.0123 1.0493 1.0635 1.0278 1.0265 1.03

Load MW 0 0 322 500 0 0 233.8 522 0 0 0 7.5 0 0 320 329 0 158 0 628 274 0 247.5 308.6 224 139 281 206 283.5 0 9.2 0 0 0 0 0 0 0 1104

MVar 0 0 2.4 184 0 0 84 176 0 0 0 88 0 0 153 32.3 0 30 0 103 115 0 84.6 -92 47.2 17 75.5 27.6 26.9 0 4.6 0 0 0 0 0 0 0 250

Generator MW 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 250 650 632 508 650 560 540 830 1000

MVar 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -

Unit No.

Gen10 Gen2 Gen3 Gen4 Gen5 Gen6 Gen7 Gen8 Gen9 Gen1

This completes the description of the system and its model. The remainder of this report focuses on the three analyses performed on the system: load flow, small signal stability assessment via Eigenvalue calculation, and numerical simulation. The results of load flow analysis are presented in the next section.


2

11

Load Flow Results

Load flow for the system was calculated using MATLAB . The results are in Table 9. Note that all voltages, active power values, and reactive power values are given in per unit on the system MVA base. For a more complete description of power flow, including flow on each line, see Section 5.2 in the Appendix. Table 9: Power flow results Bus

V

Angle [deg]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

1.0474 1.0487 1.0302 1.0039 1.0053 1.0077 0.997 0.996 1.0282 1.0172 1.0127 1.0002 1.0143 1.0117 1.0154 1.0318 1.0336 1.0309 1.0499 0.9912 1.0318 1.0498 1.0448 1.0373 1.0576 1.0521 1.0377 1.0501 1.0499 1.0475 0.982 0.9831 0.9972 1.0123 1.0493 1.0635 1.0278 1.0265 1.03

-8.44 -5.75 -8.6 -9.61 -8.61 -7.95 -10.12 -10.62 -10.32 -5.43 -6.28 -6.24 -6.1 -7.66 -7.74 -6.19 -7.3 -8.22 -1.02 -2.01 -3.78 0.67 0.47 -6.07 -4.36 -5.53 -7.5 -2.01 0.74 -3.33 0 2.57 4.19 3.17 5.63 8.32 2.42 7.81 -10.05

Bus Total P 0 0 -322 -500 0 0 -233.8 -522 0 0 0 -7.5 0 0 -320 -329 0 -158 0 -628 -274 0 -247.5 -308.6 -224 -139 -281 -206 -283.5 250 511.61 650 632 508 650 560 540 830 -104

Q 0 0 -2.4 -184 0 0 -84 -176 0 0 0 -88 0 0 -153 -32.3 0 -30 0 -103 -115 0 -84.6 92.2 -47.2 -17 -75.5 -27.6 -26.9 146.16 193.65 205.14 109.91 165.76 212.41 101.17 0.44 22.84 -161.7

Load P 0 0 -322 -500 0 0 -233.8 -522 0 0 0 -7.5 0 0 -320 -329 0 -158 0 -628 -274 0 -247.5 -308.6 -224 -139 -281 -206 -283.5

Q 0 0 -2.4 -184 0 0 -84 -176 0 0 0 -88 0 0 -153 -32.3 0 -30 0 -103 -115 0 -84.6 92.2 -47.2 -17 -75.5 -27.6 -26.9

-9.2

-4.6

-1104

-250

Generator P

Q

Unit No.

250 520.81 650 632 508 650 560 540 830 1000

146.16 198.25 205.14 109.91 165.76 212.41 101.17 0.44 22.84 88.28

10 2 3 4 5 6 7 8 9 1

12

3


Small Disturbance Analysis

Small signal stability was assessed by calculating Eigenvalues of the system. To find Eigenvalues, we first calculated reduced A matrices as follows: x = f (x, ˙ y) | 0 = g(x, y) Differential eq.

Linearize:

Algebraic eq.

∆x = f ˙ x ∆x + f y ∆y | 0 = g x ∆x + gy ∆y ∆x = ˙ (f x − f y gy−1 gx ) ∆x

Eliminate algebraic equations:

Reduced A matrices

Eigenvalues, which correspond to machine oscillatory modes, are calculated from the reduced A matrices. Some Eigenvalues of our system are given in Table 10 below. To access all Eigenvalues calculated in our analysis, see Section 5.1 in the Appendix. Table 10: Eigenvalues calculated through small disturbance analysis Index

Real part

Imaginary part

27 29 31 33 35 37 39 41 57

-2.553 -1.8494 -1.5817 -2.5633 -1.8626 -1.3118 -1.8437 -1.523 -2.9563

j10.566 j10.028 j8.5503 j8.6706 j7.4388 j7.1081 j7.0812 j6.3180 j2.5076

The Eigenvalues in Table 10 are plotted in Figure 4. 12

11

27 29

10

9

33

31

8

s i x A y r a 7 n i g a m I 6

35 37

39

41

5

4

3

57

2 −3.5

−3

−2.5

−2

−1.5

Real Axis

Figure 4: Graphical depiction of Eigenvalues

−1


4

13

Dynamic Simulation

Dynamic simulation was performed for a three-phase fault at Bus 16 occurring at t = 0.5s. The fault impedance is 0.001 PU, and it is cleared at t = 0.7s. The simulation method used was numerical trapezoidal integration with a time step of 20 ms. Figure 5 shows the generator state δ as a function of time for the ten generators, with Unit 1 as the angle reference. Figure 6 on the following page shows the generator state ω as a function of time for the ten generators. Gen1

1 0.5 ] d a r [ 0 a t l e D−0.5 −1

] d a r [ a t l e D

0

2

4

] d a r [

a t l e D

a t l e D

0 2

4

a t l e D

2

4

8

1 0 4

6

8

Gen9

1

a t l e D

0 2

8

10

6

8

10

6

8

10

6

8

10

4 6 Time [sec]

8

10

Gen4

0 0

2

4 Gen6

2 1 0 0

2

4 Gen8

4 6 Time [sec]

2 1 0 0

2

4 Gen10

2 ] d a r [

0

6

1

−1

10

2

−1

4

3 ] d a r [ a t l e D

2

2

2

−1

10

2

3

a t l e D

6 Gen7

0

0

3 ] d a r [

0

0

−1

10

0

−1

] d a r [

8

Gen5

3 ] d a r [ a t l e D

6

2

−2

1

3

1

0

2

−1

10

] d a r [

4

a t l e D

8

2

−1

] d a r [

6 Gen3

3

Gen2

3

8

10

1 0 −1

0

2

Figure 5: δ (in radians) versus time for generators

14


Gen1

1.5 ] U 1 P % [ a 0.5 g e 0 m O

−0.5

] U 1 P % [ a 0.5 g e 0 m O

0

2

4

6

8

10

Gen3

1

0

2

4

6

8

10

Gen5

4

6

8

10

6

8

10

6

8

10

6

8

10

4 6 Time [sec]

8

10

Gen4

−1

0

2

4 Gen6

] U P 1 % [ a g e 0 m O

0

2

4

6

8

10

Gen7

2

−1

0

2

4 Gen8

1

] U P 1 % [ a g e 0 m O

] U P 0.5 % [ a g e 0 m O

0

2

4

6

8

10

Gen9

1.5

−0.5

0

2

4 Gen10

1.5

] U 1 P % [ a 0.5 g e 0 m O

−0.5

2

2

] U P 2 % [ a g e 0 m O

−1

0

] U P 1 % [ a g e 0 m O

4

−2

−0.5

2

] U P 0.5 % [ a g e 0 m O

−0.5

Gen2

1.5

] U 1 P % [ a 0.5 g e 0 m O

0

2

4 6 Time [sec]

8

10

−0.5

0

2

Figure 6: ω (in percent per-unit) for generators To generate these and other figures from the MATLAB files available on the website, see ”Accessing Simulation Data with MATLAB” in the appendix.


5

15

Appendix

5.1

Accessing Simulation Data with MATLAB  R

There are seven MATLAB  .mat files available for download. Each .mat file has a csv counterpart for those who prefer to work with Excel or another CSV-friendly program (the procedure described here may be adapted for such programs). Each file is described below. R

• time.mat is a row vector with 677 elements. This vector represents the simulation time period with increments of 20ms. To observe the variation of any model variable with respect to time, simply plot the corresponding row of x.mat or y.mat versus time.mat . • x.mat contains 209 row vectors, each corresponding to a model state variable. See Table 11. • x0.mat contains 209 elements corresponding to initial values of model state variables. It is indexed the same as x.mat. • y.mat contains 617 row vectors, each corresponding to a non-state variable in the model. See Table 11. • y0.mat contains 617 elements corresponding to initial values of non-state variables. It is indexed the same as y.mat. • Aeig.mat contains 209 complex elements corresponding to system Eigenvalues. To obtain any entry of Table 10, one need only extract the element of Aeig corresponding to that entry’s Index. Of course, there are many more Eigenvalues stored in Aeig than those listed in Table 10, and all Eigenvalues may be easily plotted so long as care is taken in isolating the real and imaginary parts. • Ared.mat is a 209-by-209 variable containing reduced A matrix data for the system. (Reduced A matrices are discussed in Section 3.) Model state variables are stored in x.mat while non-state variables are located in y.mat. Table 11 maps from these files to specific model variables by specifying the range of variables corresponding to each component of the model. The order of variables within a component’s range is discussed in the next five subsections. Table 11: Mapping from MATLAB files to simulation data R

5.1.1

Model

Model ID

x range from to

y range from to

Description

Generator

G1 G10

1 37

4 49

1 118

13 130

4 x states and 13 y variables

AVR

G1 G10

110 155

114 159

438 537

448 547


PSS

G1 G10

160 205

164 209

548 611

554 617


Load

Bus1 Bus39

47 107

48 108

303 423

306 426

2 x states and 4 y variables (x states are real and reactive power)

Network

Bus1 Bus39

131 283

134 286

4 y variables

Variable Indexing

This section defines all variables contained in the MATLAB files. When writing code to extract specific variables, refer to this section to determine which indices to use.

16


Generator Variables

(See Section 1.1.1)

x1 x2 x3 x4

AVR Variables

: : : :

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11

V f E f d xi V REF

y1 y2 y3 y4 y5 y6 y7

: xW : xP : xQ

: : : : : : : : : : :

V r V i V T V P SS xerr xtgr E fd Upper Limit Detector Lower Limit Detector Upper Limit Switch Lower Limit Switch

: : : : : : :

ω (input) V W V P V out V P SS V P SS,Max − V out V out − V P SS,Min

(See Section 1.2.1) x1 x2

Network Variables 5.1.2

V r V i I r I i V d V q I d I q ψd ψq E fd ω T mech

(See Section 1.1.5)

x1 x2 x3

Load Variables

E q E d δ ω

: : : : : : : : : : : : :

(See Section 1.1.3)

x1 x2 x3 x4

PSS Variables

: : : :

y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13

: P : Q

y1 y2 y3 y4

: : : :

V r V i I r I i

(See ??) Network variables are indexed in the MATLAB files as follows:

Example Application

The code below will generate Figures 5 and 6. By modifying or adapting this code, the reader may plot or manipulate any variables in x.mat and y.mat . (Note: The placeholders ”TIMEPATH”, ”XPATH”, and ”YPATH” should be replaced with paths to the respective files on your computer.)


%% Import Data % Import time.mat newData = load(’-mat’, ’TIMEPATH\time.mat’); % Create new variables in the base workspace from those fields. vars = fieldnames(newData); for i = 1:length(vars) assignin(’base’, vars{i}, newData.(vars{i})); end % Import x.mat newData = load(’-mat’, ’XPATH\x.mat’); % Create new variables in the base workspace from those fields. vars = fieldnames(newData); for i = 1:length(vars) assignin(’base’, vars{i}, newData.(vars{i})); end % Import y.mat newData = load(’-mat’, ’YPATH\y.mat’); % Create new variables in the base workspace from those fields. vars = fieldnames(newData); for i = 1:length(vars) assignin(’base’, vars{i}, newData.(vars{i})); end clear newData, clear vars %% Plot x or y vs. time % Plot omega vs time figure(’name’,’Generator Omega vs Time’) for i = 1:10 subplot(5,2,i); p = plot(time,x(i*4 - 0,:)./4,’k’); % Note factor of 1/4 set(p,’LineWidth’,1); figtitle = strcat(’Gen’,num2str(i)); title(figtitle) if or(i==9,i==10) xlabel(’Time [sec]’) end ylabel(’Omega [% PU]’) grid on end % Plot delta vs time figure(’name’,’Generator Delta vs Time’) for i = 1:10 subplot(5,2,i); p = plot(time,x(i*4 - 1,:)-x(3,:),’k’); % Subtract Unit 1 (angle reference) set(p,’LineWidth’,1);

17

18


figtitle = strcat(’Gen’,num2str(i)); title(figtitle) if or(i==9,i==10) xlabel(’Time [sec]’) end ylabel(’Delta [rad]’) grid on end

5.2

Contents of file 39bus.out

The next four pages show the contents of the file 39bus.out . This file contains power flow data with enough granularity to observe flows on all lines.

2 2 GENERATION LOAD LOSSES BUS SHUNTS LINE SHUNTS SWITCHED SHUNTS MISMATCH

MW 6140.81 6097.10 43.71 0.00 0.00

MVAR 1250.37 1408.90 -158.52 0.00 0.00 0.00

0.00

0.01

39bus.out

BASE MVA : 100.0 TOLERANCE: 0.000100 PU SLACK BUSES ----- ----Gen31 100.0

Page 4

Contents of file 39bus.out (Page 4 of 4)

R e p o r t : 3 9 b u s s y s t e m ( N e w E n g l a n d R e d u c e d M o d e l )


23

References [1] T. Athay, R. Podmore, and S. Virmani. “A Practical Method for the Direct Analysis of Transient Stability”. In: IEEE Transactions on Power Apparatus and Systems PAS-98 (2 Mar. 1979), pp. 573– 584. [2] M. A. Pai. Energy function analysis for power system stability . The Kluwer international series in engineering and computer science. Power electronics and power systems. Boston: Kluwer Academic Publishers, 1989. [3] Peter W Sauer and M. A Pai. Power system dynamics and stability . English. Champaign, IL.: Stipes Publishing L.L.C., 2006.

New England Reduced Model (39 Bus System) MATLAB Study Report

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