Chapter 9- Fault Calculation

CHAPTER 9: FAULT CALCULATIONS
9.1 Zbus Matrix Modification
9.2 Manual Fault Calculations
9.3 Unified Solution for 1LG,2LG,3P,LL
9.4 Flows in Lines with Mutual Coupling
9.5 Computer Implementation of Branch flows
9.6 Computer Program, Full Matrix (FOLT.CPP)
9.7 Computer Program, Sparse Matrix (FAULT.CPP)

Three-Phase Fault Calculation, Manual Method
Manual method requires us to reduce the network into a single impedance as seen by an ohm meter on node 1. Get the series impedance on lines 1-2 and 2-0.
Z(L1)=z(12) + z(20) = j0.15 + j0.2= j0.35.
Then get the series of lines 1-4, 4-3, and 3-0.
Z(L2)=z(14) +z(43) +z(30) = j0.1 + j0.1 +j0.2 = j0.4.
Let Z(L3) be the parallel result of Z(L1) and Z(L2).
=(j0.35*j0.4)/(j0.35+j0.4)=j0.18667.
Then get Z(t), the parallel result for line 0-1 and Z(L3)
=(j0.05*j0.18667)/(j0.05+j0.18667)
=j0.03944
I(3 phase) = 1/j0.03944 = -j25.3572 pu.
The current distribution will be,
I(1)=-j25.3572*(0.18667/0.23667)=-j20.0 pu
I(2)=-j25.3572*(0.05/0.23667)=-j5.3572
Also I(3)=-j5.3572*(0.35/0.75) =-j2.5
and I(4)=-j5.3572*(0.4/0.75)= -j2.857

Check By Using the elements of Zbus
k=1, P=2, Q=1

I(2,1)=[(0.03944 - 0.02254)/0.15]*(-j25.3572)
= -j2.857 pu
Opening a Line During a Study

A line may be opened by adding a line in parallel with the existing one with a negative impedance of the original. A line conversion can also be implemented by adding a line such that the equivalent impedance of the original one and the added line has the value of the desired conversion. Use Types-2,-4 or -6. During a fault study it is desirable to modify the Zbus for sliding fault, line-end fault and line outage. The best method is to extract a small matrix whose elements are the one needed in branch contributions for a desired NBACK.

Opening a non-coupled line
Opening line 1-2 will result to z(1,2)=infinity. What should be added to line 1-2 to make z(1,2) = infinity? Answer: negative z(1,2).

Use type 5 Equations,

with line 1-2 added and x= -0.15. The resulting Zbus is

Z(L,i=1)=Z(P=1,i=1)-Z(Q=2,i=1)=0.0394346-0.0225341=0.0169
...up to
Z(L,4)
Z(L,L)=Z(P=1,P=1)+Z(Q=2,Q=2)-2Z(P=1,Q=2)+z(P=1,Q=2)
=0.03943+0.09859-2(0.022534)+(-0.15)= -0.05704
After Kron Reduction,

This Zbus will be the same if it is formed from an input data without z(1,2).

Opening a coupled line
Opening line 1-4 will require a coupling matrix as shown below. Line 1'- 4' is not coupled to line 1-4. Its self impedance is the negative of line 1-4.

Here P=1', Q=4'. There are 3RS (RS= 1-4, 1-2, 4-3). The use of prime is for convenience only. It is not shown in a program code. Applying Type-6,
Z(L,i=1)=Z(P=1,i=1)-Z(Q=4,i=1)+ [y(PQ,RS)...][Z(R,i) - Z(S,i)...]/y(PQ,PQ)
Z(L,1)=Z(1,1)-Z(4,1)+[y(1'-4',1-4)*[Z(1,1)-Z(4,1)]+y(1'-4',1-2)*[Z(1,1)-Z(2,1)]+
+y(1'-4',4-3)*[Z(4,1)-Z(3,1)]]/y(1'-4',1'-4')

Z(L,1)=0.08617-0.06596+[(-0.9375)*[0.08617-0.0596] + 1.875*[0.0861-0.03723]
+(-1.25)*[0.06596-0.03191]]/(-4.0625) = 0.01276
...Z(L,4)
Z(L,L)=Z(P,L)-Z(Q,L) +(1+[y(PQ,RS)...][Z(R,L)-Z(S,L)...])/y(PQ,PQ) = -0.04255 and after applying Kron Reduction results to,

which will be the same result if we rebuild the matrix from start without line 1-4.

Flows in the Zero Sequence Network with Mutual Coupling

The line flow in the positive sequence is

This is also the equation for the zero sequence, if line P-Q is uncoupled.Q

From the coupling equation as shown on the figure,

Where R-S can be more than one item. Note that

Example Fault Calculation SLG with Coupling
From Fig_1, Table_1 of section 4.2, the positive/zero Zbus, x-only is,

For a single-line-to-ground fault on bus=1,
= -j605.877 Mva.

= 1 - (-j6.05877)(j0.03944)= 0.76104 pu,
= 0 - (-j6.05877)(j0.08617)= -0.52208 pu,
, = 0 - (-j6.05877)(j0.03944)= -0.2389 pu
These voltages are in real numbers because (-j)*(j)= +1.

=solved using coupling formula.

The desired flow is from P=2, to Q=1, but what is available in [z] as shown on the coupling matrix above is P=1, Q=2. Solve instead then just reverse direction. See the figure at the right side for easy interpretation.
= (-0.22587 +0.046638 +0.104769)*(-j605.87) = j45.119 Mva
From the right side figure, the result of (0.03723- 0.08617)*4.61538) is -0.22587 as shown above. Also (-j)*(j)= +1. The (-j) from the Figure is for the admittance, the (+j) is for the impedance.

Output Node information from JPFault2.exe for a fault at node=1

Branch Contributions from JPFault1.exe for a fault in node=1.

Output Node information from JPFault1.exe for a fault at node=2

Branch Contributions for a fault in node=2. You can compare these last screen dumps to the scrollable output. Source code from this scrollable output is shown at Chapter 8. Sparz

Output from sparz.exe

Fault Calculation                                                       
Faulted Bus:1                                                           
3P  Fault Level Mva=   2535.71                                          
1LG Fault Level Mva=   1817.70                                          
From (1LG)E+  (1LG)Eo   (3P)E+--->To  (1LG)Io  (1LG)Ia  (3P)Ia          
-------------------------------------------------------------------     
  1  0.7611 -0.5221  0.0000                                             
                               ---> 0  -522.11 -1477.89 -2000.00        
primitive [z]                                                           
         1         1         4                                          
         2         4         3                                          
         1         1         1                                          
  0.400000  0.100000  0.200000                                          
  0.100000  0.200000  0.000000                                          
  0.200000  0.000000  0.300000                                          
                                                                       
Enter to continue                                                       
After Inversion [y]                                                 
         1         1         4                                      
         2         4         3                                      
         1         1         1                                      
  4.615385 -2.307692 -3.076923                                      
 -2.307692  6.153846  1.538461                                      
 -3.076923  1.538461  5.384615                                      
                                                                   
Enter to continue                                                   
                                                                    
                               -*-> 2   -45.12  -181.66  -285.71    
                               -*-> 4   -38.67  -158.15  -250.00    
  2  0.8635 -0.2256  0.4286                                         
                               ---> 0   -45.12  -181.66  -285.71    
                               -*-> 1    45.12   181.66   285.71    
  3  0.8805 -0.1934  0.5000                                         
                               ---> 0   -38.67  -158.15  -250.00    
                               -*-> 4    38.67   158.15   250.00    
  4  0.8208 -0.3996  0.2500                                         
                               -*-> 1    38.67   158.15   250.00    
                               -*-> 3   -38.67  -158.15  -250.00    
Enter to continue                                                   
                                                                     
Faulted Bus:2                                                        
3P  Fault Level Mva=   1014.29                                       
1LG Fault Level Mva=    741.42                                       
From (1LG)E+  (1LG)Eo   (3P)E+--->To  (1LG)Io  (1LG)Ia  (3P)Ia       
-------------------------------------------------------------------  
  1  0.9443 -0.0920  0.7714                                          
                               ---> 0   -92.02  -314.79  -457.14     
                               -*-> 2   144.60   395.22   514.29     
                               -*-> 4   -52.58   -80.43   -57.14     
  2  0.7563 -0.5127  0.0000                                          
                               ---> 0  -102.54  -346.20  -500.00     
                               -*-> 1  -144.60  -395.22  -514.29     
  3  0.9722 -0.2629  0.8857                                          
                               ---> 0   -52.58   -80.43   -57.14     
                               -*-> 4    52.58    80.43    57.14     
  4  0.9582 -0.1315  0.8286                                          
                               -*-> 1    52.58    80.43    57.14     
                               -*-> 3   -52.58   -80.43   -57.14     
Enter to continue                                                    
3P  Fault Level Mva=    910.26                                      
1LG Fault Level Mva=    661.14                                      
From (1LG)E+  (1LG)Eo   (3P)E+--->To  (1LG)Io  (1LG)Ia  (3P)Ia      
------------------------------------------------------------------- 
  1  0.9565 -0.0703  0.8205                                         
                               ---> 0   -70.33  -244.16  -358.97    
                               -*-> 2   -46.89   -71.72   -51.28    
                               -*-> 4   117.22   315.88   410.26    
  2  0.9752 -0.2344  0.8974                                         
                               ---> 0   -46.89   -71.72   -51.28    
                               -*-> 1    46.89    71.72    51.28    
  3  0.7579 -0.5158 -0.0000                                         
                               ---> 0  -103.16  -345.26  -500.00    
                               -*-> 4  -117.22  -315.88  -410.26    
  4  0.8572 -0.2579  0.4103                                         
                               -*-> 1  -117.22  -315.88  -410.26    
                               -*-> 3   117.22   315.88   410.26    
Enter to continue                                                   
Faulted Bus:4                                                        
3P  Fault Level Mva=   1028.99                                       
1LG Fault Level Mva=    744.64                                       
From (1LG)E+  (1LG)Eo   (3P)E+--->To  (1LG)Io  (1LG)Ia  (3P)Ia       
-------------------------------------------------------------------  
  1  0.9266 -0.1637  0.6957                                          
                               ---> 0  -163.72  -457.38  -608.70     
                               -*-> 2   -26.41   -68.36   -86.96     
                               -*-> 4   190.12   525.74   695.65     
  2  0.9580 -0.1320  0.8261                                          
                               ---> 0   -26.41   -68.36   -86.96     
                               -*-> 1    26.41    68.36    86.96     
  3  0.8392 -0.2905  0.3333                                          
                               ---> 0   -58.09  -218.91  -333.33     
                               -*-> 4    58.09   218.91   333.33     
  4  0.7588 -0.5176 -0.0000                                          
                               -*-> 1  -190.12  -525.74  -695.65     
                               -*-> 3   -58.09  -218.91  -333.33     
Enter to continue
                                             
Portion from sparz.c Chapter 8. This snippet performed the output above.
------------------------------
 296: //Discussion about fault calculation is not in this Chapter.
 297: void FaultCalc(void)
 298: {int lp,lq,lr,ls,i,j,k,LastGroup=0,pq,pqq,rs;               
 299:  float ZPqk,ZPpk, ZZqk,ZZpk,  Zsk,Zrk;
 300:  float If_3,If_1, Ep_1,Eo_1,Ep_3,  Ip_1,Io_1,Ia_3,Ia_1,Io;
 301:  printf("Fault Calculation\n");//print(ZP);print(ZZ);                           
 302:  for (k=1;k<=nb;k++)    //loop for all nodes
 303:   {If_3=1/ZP[k][k]; If_1=1/(2*ZP[k][k]+ZZ[k][k]);
 304:    printf("Faulted Bus:%d\n",k);
 305:    printf("3P  Fault Level Mva=%10.2f\n",If_3*100);
 306:    printf("1LG Fault Level Mva=%10.2f\n",3*If_1*100);
 307:    printf("From (1LG)E+  (1LG)Eo   (3P)E+");
 308:    printf("--->To  (1LG)Io  (1LG)Ia  (3P)Ia\n");
 309:    for (i=1;i<=67;i++)printf("-");printf("\n"); //straight line
 310:    
 311:    for (lp=1;lp<=nb;lp++)    //loop for all nodes wrt k
 312:     {if (lp<k) 
 313:        {ZPpk=ZP[lp][k]; ZZpk=ZZ[lp][k];}
 314:      else 
 315:        {ZPpk=ZP[k][lp]; ZZpk=ZZ[k][lp];}
 316:       
 317:      Ep_1=1-If_1*ZPpk; 
 318:      Eo_1=0-If_1*ZZpk; 
 319:      Ep_3=1-If_3*ZPpk;
 320:      printf("%3d%8.4f%8.4f%8.4f\n",lp,Ep_1,Eo_1,Ep_3);
 321:      for (j=1;j<=nl;j++)     //loop for all connections to lp
 322:       {if (kp[j]==lp || kq[j]==lp)
 323:         {if (kp[j]==lp) lq=kq[j]; else lq=kp[j];
 324:          ZPqk=GetZij(ZP,lq,k); ZZqk=GetZij(ZZ,lq,k);
 325:          ZPpk=GetZij(ZP,lp,k); ZZpk=GetZij(ZZ,lp,k);
 326:          Ip_1=100*If_1*(ZPqk-ZPpk)/x1[j];
 327:          Ia_3=100*If_3*(ZPqk-ZPpk)/x1[j];
 328:          if (mut_yn[j])     //for coupled Io
 329:           {if (LastGroup!=mut_yn[j])
 330:             {primitive(mut_yn[j]); LastGroup=mut_yn[j];}
 331: //consider direction of coupling
 332:            pqq=Find_pq(lp,lq,ckt[j]); pq=abso(pqq);
 333:            Io=(float)0.0;   //perform [y pq-rs]*[Zsk-Zrk]*If_1
 334:            for (rs=1;rs<=z_size;rs++)
 335:             {lr=zP[rs]; ls=zQ[rs];
 336:              Zsk=GetZij(ZZ,ls,k); Zrk=GetZij(ZZ,lr,k);
 337:              Io+=cmat[pq][rs]*(Zsk-Zrk);
 338:             }
 339:            if (pqq<0) Io_1= -Io*100*If_1; else Io_1=Io*100*If_1;
 340:           }  //if (mut_yn[j]) loop
 341:          else
 342:           Io_1=100*If_1*(ZZqk-ZZpk)/x0[j];  //uncoupled flow
 343:          
 344:          Ia_1=2*Ip_1 + Io_1;
 345:          for (i=1;i<=31;i++)printf(" "); //31 spaces
 346:          if (mut_yn[j]) 
 347:            printf("-*->%2d%9.2f%9.2f%9.2f\n",lq,Io_1,Ia_1,Ia_3);
 348:          else
 349:            printf("--->%2d%9.2f%9.2f%9.2f\n",lq,Io_1,Ia_1,Ia_3);                   
 350:         } //if
 351:        } //for j
 352:      } //for lp
 353:      Enter();
 354:    } //for k
 355: }

HOME
CHAPTER ONE: INTRODUCTION
CHAPTER TWO: PROGRAMMING TOPICS
CHAPTER THREE: MATRIX INVERSION ROUTINE
CHAPTER FOUR: STEP-BY-STEP ALGORITHMS
CHAPTER FIVE: DIRECT METHODS IN Ybus FORMATION
CHAPTER SIX: SPARSITY
CHAPTER SEVEN: MATRIX FACTORIZATION AND OPTIMAL ORDERING
CHAPTER EIGHT: SPARSE Zbus FROM FACTORED SPARSE Ybus
CHAPTER NINE: FAULT CALCULATIONS
REFERENCES