A Unified and Efficient Approach to Power Flow Analysis
Abstract
:1. Introduction
2. United Framework
2.1. Power Flow Problems
2.2. Optimal Power Flow Problems
2.3. Probabilistic Power Flow Problems
2.4. State Estimation Problems
2.5. Summary
3. Solution Algorithm
3.1. Alternating Least Square Approach
3.2. Newton-Raphson Method
3.3. Proposed Method
Algorithm |
Set l = 0 (l: iteration index) and εth (tolerance in error) |
(1) If , terminate the process; |
(2) Otherwise compute using Equation (15); |
(3) Update , l ← l+1, go to Equation (1). |
3.4. Convergence of the Proposed Method
3.5. Determination of αl for the Interior Point Method
3.6. Illustrative Examples
4. Numerical Examples
4.1. Power Flow Studies
4.2. Optimal Power Flow Studies
4.3. Probabilistic Power Flow Studies
4.4. State Estimation
5. Conclusions
Funding
Acknowledgement
Conflicts of Interest
Nomenclature
Aj | set of buses that are directly connected to Bus j |
Ej | voltage magnitude squares at j |
Emin, Emax | min. and max. of voltage magnitude squares |
Fk | matrix associated with power flow at k in the cardinality of 2N-by-2N |
G | number of generators |
Ij | complex current at Bus j |
L | number of branches |
Ldem | matrices indicating the locations of loads in the cardinality of N-by-G |
Lgen | matrices indicating the locations of generators in the cardinality of N-by-N |
N | number of buses |
NPMU | number of PMU measurements |
NSCADA | number of SCADA measurements |
P | perfect permutation matrix |
Sj | matrix associated with power injections at j in the cardinality of 2N-by-2N |
T | Transpose |
real and imaginary part of X | |
Ybus | nodal admittance matrix |
diag(·) | diagonal matrix of the vector inside () transformed into a column vector |
a2 | quadratic cost coefficient |
a1 | linear cost coefficient |
d | vector of real and reactive power demand in R2N |
ej | jth column vector of an identity matrix |
f | real and reactive plow vector in R4L |
flowk | maximum power flow on Branch k |
g | real and reactive power generation in R2G |
gmin, gmax | min. and max. limits of power generation |
j, k, m | bus index, branch index, and generation index |
ref | angle reference bus where the imaginary part of the voltages is set to zero |
u | measurements |
vec(·) | vectorization of the matrix inside () |
v | real and imaginary voltage vector in R2N |
w | weight factors associated with the measurements u |
|v|, θ | voltage magnitude and angle |
zf | slack variable corresponding to maximum flow inequality constraints |
zg | slack variable corresponding to max/min generation limit constraints |
zmax | slack variable corresponding to max voltage magnitude constraints |
zmi | slack variable corresponding to min voltage magnitude constraints |
Ψ | Lagrange function |
ρ | shadow price corresponding to an equality constraint |
μf | shadow price corresponding to maximum flow inequality constraints |
μg | shadow price corresponding to max/min generation limit constraints |
μmax | shadow price corresponding to max voltage magnitude constraints |
μmin | shadow price corresponding to min voltage magnitude constraints |
* | complex conjugate |
equivalent |
Appendix A. Power Flow Problem
Appendix B. Optimal Power Flow Problem
Appendix B.1. Equality Constraints
Appendix B.2. Inequality Constraints
Appendix C. State Estimation Problem
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Oh, H. A Unified and Efficient Approach to Power Flow Analysis. Energies 2019, 12, 2425. https://doi.org/10.3390/en12122425
Oh H. A Unified and Efficient Approach to Power Flow Analysis. Energies. 2019; 12(12):2425. https://doi.org/10.3390/en12122425
Chicago/Turabian StyleOh, HyungSeon. 2019. "A Unified and Efficient Approach to Power Flow Analysis" Energies 12, no. 12: 2425. https://doi.org/10.3390/en12122425