# A Robust and Efficient Power Series Method for Tracing PV Curves

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A Robust and Efficient Power Series Method for Tracing PV Curves
```A Robust and Efficient
Power Series Method for
Tracing PV Curves
Xiaoming Chen, David Bromberg, Xin Li,
Lawrence Pileggi, and Gabriela Hug
Electrical and Computer Engineering Department
Carnegie Mellon University
{xchen3, dbromber, xinli, pileggi, ghug}@ece.cmu.edu
Outline
 Background
 Power Series Method
 Tracing PV Curve
 Results
 Conclusion
2
Background
 Power flow analysis
 Fundamental problem in power system analysis
 Voltage stability
 Critical for secure and stable operating
 Power-voltage curve & voltage collapse
point
 Margin of current operating point
3
Conventional Methods
 Newton-Raphson (NR) iterative method
 Cons: singular Jacobian matrix at collapse point
 Continuation power flow (CPF) [Ajjarapu 1992]
 Extend Jacobian matrix by one row & column
 Pros: trace whole PV curve including collapse
point in most cases
 Cons: extended Jacobian matrix may still be
singular at collapse or non-collapse point [Zhao 2006]
[Ajjarapu 1992] V. Ajjarapu and C. Christy, “The continuation power flow: a tool for
steady state voltage stability analysis,” Power Systems, IEEE Transactions on, vol. 7,
no. 1, pp. 416–423, Feb 1992
[Zhao 2006] J. Zhao and B. Zhang, “Reasons and countermeasures for computation
failures of continuation power ﬂow,” in Power Engineering Society General Meeting,
2006. IEEE, 2006, pp. 1-6.
4
State-of-the-art Methods
 Holomorphic embedding load flow method
(HELM) [Trias 2012]
 Direct method, no divergence problem
 Machine learning based methods [Zhang 2013]
 Expensive for training
 Only get collapse point, no PV curve
 AQ-bus method [Ghiocel 2014]
 A PQ bus  AQ bus
 Similar to CPF
[Trias 2012] A. Trias, “The holomorphic embedding load ﬂow method,” in Power and
Energy Society General Meeting, 2012 IEEE, July 2012, pp. 1–8
[Zhang 2013] R. Zhang, Y. Xu, Z. Y. Dong, P. Zhang, and K. P. Wong, “Voltage
stability margin prediction by ensemble based extreme learning machine,” in Power
and Energy Society General Meeting (PES), 2013 IEEE, July 2013, pp. 1–5.
[Ghiocel 2014] S. Ghiocel and J. Chow, "A power flow method using a new bus type
for computing steady-state voltage stability margins," Power Systems, IEEE
Transactions on, vol. 29, no. 2, pp. 958–965, March 2014.
5
Power Series Method (PSM)
 A known operating point f (x(old) )  y (old)
 RHS changes to y (new)
 Write RHS as a new form y  y (old)  u y (new)  y (old) 

 Break x into power series x  x(old)   u k x[k ]
 New power flow equation
k 1
 (old)  k


f x   u x[k ]  y (old)  u y (new )  y (old) 


k 1
 Physical meaning
(old)
(old)
 u=0, f (x )  y
 u=1, f (x (new ) )  y (new )
6
Power Series Method (PSM)
 New power flow equation
 (old)  k


f x   u x[k ]  y (old)  u y (new )  y (old) 


k 1
 If f is polynomial, both sides are polynomials
of u
 x[k] (k=1,2,…) can be solved by equating kth-order
coefficients of both sides
 If using rectangular coordinate in DC power
flow, f is polynomial
 If f is not polynomial, use Taylor expansion
of f
7
Power Series Method (PSM)
 Example of active power equation
(rectangular coordinate)
 First-order
N
(old)
i
e
 G e [1]  B
ij
j 1
N
fi
(old)
 G
ij
j 1
j
ij
N
f j [1]  ei [1] Gij e(old)
 Bij f j(old)  
j
j 1
N
f j [1]  Bij e j [1]  fi [1] Gij f j(old)  Bij e(old)
  Pi (new )  Pi (old)
j
j 1
Jx[1]  y (new)  y (old)
8
Power Series Method (PSM)
 Higher-order (n=2,3,…)
N
(old)
i
e
 G e [n]  B
ij
j 1
N
fi
(old)
 G
ij
j 1
j
ij
N
 Bij f j(old)  
f j [n]  ei [n] Gij e(old)
j
j 1
N
f j [n]  Bij e j [n]  fi [n] Gij f j(old)  Bij e(old)
  Pi (n)
j
j 1
N


 ei [k ] Gij e j [n  k ]  Bij f j [n  k ] 
n1 

j 1


Pi (n)   

N
k 1 

 f i [k ] Gij f j [n  k ]  Bij e j [n  k ]
j 1


Jx[n]  y (n)
RHS only depends on lower-order solutions
9
Tracing PV Curve
f (x ( m ) )  y ( m )  y base   ( m ) y ch
 Increase  until power series diverges
(collapse point reached)
f (x (0) )  y (0)
f (x (1) )  y (1)
f (x (2) )  y (2)
 Step size control by estimating truncation
error of power series
 Details are provided in paper
10
Experimental Setup
 Eight cases provided by MATPOWER
 Methods for comparison
 Our implementation of HELM (embedding)
 CPF by MATPOWER (golden solution, if no failure)
 PSM & embedding are implemented by C++
 i7 3.6GHz CPU & 16GB memory
11
Comparison on Robustness
 MATPOWER (CPF) fails for a 3012-bus case
 Inaccurate results near collapse point
12
Comparison on Accuracy
 Max and root-mean-square (RMS) error of
upper-half PV curve
 Embedding has much larger errors
 Embedding solves a new point by extrapolating
from a trivial point; our PSM extrapolates from
previous point
13
Comparison on Accuracy
 Accurate collapse points compared with
MATPOWER (CPF)
14
Comparison on Runtime
 Faster (& more accurate) than embedding
15
Conclusion
 Power series method for power flow
analysis
 Extrapolating from a known point
 Stepping scheme
 Trace PV curve & obtain collapse point
 Robust & efficient
16
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