A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables
Abstract
:1. Introduction
2. Traditional CM for P-OPF
- Take the mean values of wind power outputs and loads, and solve the aforementioned deterministic OPF model using LBIPM.
- Obtain the KKT first-order conditions, when the optimization is converged:
- Treat loads and wind power outputs as random input variables, and formulate new KKT first-order conditions:
- Take the full derivative of (5), and find a linear relationship between input variables and output variables:From Equation (7), an unknown variable can be formulated as a linear combination of known input variables (loads and wind power outputs):
- If the known input variables (loads and wind power outputs) are independent of each other, the cumulants of unknown output variables can be computed by a linear combination of cumulants of known input variables based on the property of cumulants (see Appendix A):
3. The Proposed Method for P-OPF
3.1. The Overall Procedure of the Proposed Method
- Input the basic data, including network data, the distribution functions of wind speeds and loads, and the correlation matrix.
- Group the samples of wind power outputs and loads into a number of clusters using the K-means algorithm.
- In each cluster, the CM for P-OPF considering correlations among input variables is applied. Firstly, the samples for each cluster are converted to uncorrelated samples. Then, the cumulants of uncorrelated input variables are calculated based on uncorrelated samples [33]. Finally, the CM for P-OPF is applied to compute the cumulants of system variables for each cluster.
- Compute the final cumulants of system variables for the total samples.
- PDFs of system variables are produced by Cornish–Fisher series expansion [42].
- Output the cumulants and PDFs of system variables.
3.2. Generating Samples of Correlated Wind Power Outputs and Loads
3.3. Application of the K-Means Algorithm to Group Samples into Clusters
- (1).
- Pick initial mean values of all clusters, which are defined as the following equations:
- (2).
- Calculate Euclidean distances from each point to each cluster mean according to the following equation:
- (3).
- Assign every point to the nearest cluster according to the Euclidean distances, and update the mean values of all clusters.
- (4).
- Repeat steps 2 and 3 until points in each cluster are no longer changed.
3.4. The Method of Handling Correlations among Input Variables
3.5. Computation of the Cumulants of System Variables
- (1).
- In each cluster, cumulants of a system variable for each cluster can be obtained based on the algorithms introduced in Section 3.2, Section 3.3 and Section 3.4. The moments of the system variable for each cluster are computed using the following equation:
- (2).
- The property that moments meet the total probability formula is applied to combine the moments of the system variable for all clusters. The moments of the system variable for the total samples are calculated using the following equation:
- (3).
- The final cumulants of the system variable for the total samples are obtained using the following equation:
3.6. Computation of PDFs of System Variables
4. Case Studies
4.1. The Modified IEEE 9-Bus Test System
4.1.1. Application of the K-Means Algorithm to Group Samples into Clusters
4.1.2. Cumulants of System Variables and Comparison
4.1.3. PDFs of System Variables and Comparison
4.2. The Modified IEEE 118-Bus Test System
4.3. Discussions about Number of Clusters
5. Conclusions
- When input variables have large fluctuations, P-OPF results obtained using the traditional CM have large APE values and significant errors at the tails of PDFs, which indicates that the traditional CM is not suitable to solve P-OPF problems with large fluctuations of stochastic variables.
- The proposed method can handle correlations and large fluctuations of input variables. Case studies indicate that the proposed method has more accurate results than traditional CM and is more efficient than MCS.
- The performance of the proposed method is influenced by the number of clusters. Generally, the proposed method with more clusters has more accurate results, but will require more computation time. The appropriate number of clusters can be determined by the weighted average radius.
Acknowledgments
Author Contributions
Conflicts of Interest
Nomenclature
CM | cumulant method |
OPF | optimal power flow |
PLF | probabilistic load flow |
P-OPF | probabilistic optimal power flow |
KKT | Karush–Kuhn–Tucker |
CDFs | cumulative distribution functions |
PDFs | probability density functions |
IS | importance sampling |
LHS | Latin hypercube sampling |
LSS | Latin supercube sampling |
NPNT | ninth-order polynomial normal transformation |
PEM | point estimation method |
UTM | unscented transformation method |
LT | Laplace transform |
FFT | fast Fourier transform |
FOSMM | First-Order Second-Moment Method |
GMM | Gaussian mixture model |
LBIPM | Logarithmic Barrier Interior Point Method |
APE | absolute percent error |
, | the active and reactive power generation of conventional generator at bus |
, , | cost coefficients of generator at bus |
the active power output of wind farm at bus | |
the reactive power injected by compensation device at bus | |
, | the active and reactive loads at bus |
the magnitude of voltage at bus | |
the phase angle difference between bus and | |
, | the real and imaginary parts of the element in bus admittance matrix |
the complex power flow of branch | |
, | the lower and upper bounds of |
, | the lower and upper bounds of |
, | the lower and upper bounds of |
the line rating of branch | |
the number of buses | |
the number of branches | |
the set of equations defining the KKT first-order conditions | |
a vector consisting of magnitude and angle of voltage at each bus, active and reactive generation of each conventional generator, slack variables and Lagrange multipliers | |
the vector of load at each bus | |
the vector of wind power output at each bus | |
the Hessian of the Lagrangian function with respect to when the optimization is completed | |
, , | vectors of the changes in , and |
, | matrixes obtained by taking the partial derivatives with respect to and |
an unknown variable | |
the value of evaluated by the deterministic OPF | |
the change of | |
, | the values at row and column of and |
, | the j-th variables in and |
, | the mean values of and |
, | the number of load variables and wind power variables |
the v-th order cumulant of | |
, | the v-th order cumulants of and |
the wind speed of wind farm at bus | |
the rated power of a wind farm | |
the cut-in speed of a wind farm | |
the rated speed of a wind farm | |
the cut-out speed of a wind farm | |
, , | coefficients of wind power output model |
a multi-dimensional vector comprised of initial mean values of all clusters | |
the pre-set number of clusters | |
the mean value vector of cluster | |
the mean value of variable in cluster | |
the Euclidean distance from point to the mean of clusters | |
a variable in | |
the standardized vector corresponding to | |
the standardized variable corresponding to | |
the mean of | |
the standard deviation of | |
the correlation matrix of | |
D | a diagonal matrix |
a lower triangular matrix | |
an uncorrelated vector | |
the v-th order moment of a variable for cluster | |
the v-th order cumulant of a variable for cluster | |
a combination of elements from v − 1 different elements | |
the v-th order moment of a system variable for the total samples | |
the proportion of the number of elements in cluster to the total samples | |
the v-th order cumulant of a system variable for the total samples | |
the weighted average radius | |
the radius of cluster | |
a random variable | |
the probability density function of | |
the moment generating function of | |
the cumulant generating function of | |
the v-th order cumulant of | |
a linear combination of independent variables | |
the v-th order cumulant of |
Appendix A
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Systems | Buses Connected with Wind Farms | Rated Capacities (MW) |
---|---|---|
IEEE 9-Bus Test System | 1, 3 | 60 |
IEEE 118-Bus Test System | 59, 80, 90 | 250 |
Wind Speed Distribution | Shape Parameter | Scale Parameter |
---|---|---|
W1 (Connected to Bus 1) | 1.732 | 6.611 |
W2 (Connected to Bus 3) | 2.036 | 7.933 |
Input Variable | |||
---|---|---|---|
Wind Power 1 (Connected to Bus 1) | 7.0850 | 10.9982 | 2.8596 |
Wind Power 2 (Connected to Bus 3) | 7.2758 | 11.6619 | 3.6242 |
Total System Load | 8.1510 | 14.3702 | 3.4837 |
Algorithm | Mean | Mean APE | Standard Deviation | Standard Deviation APE |
---|---|---|---|---|
MCS | 4769.75 | \ | 992.97 | \ |
UCCM | 4697.13 | 1.52% | 910.72 | 8.28% |
CCCM | 4697.13 | 1.52% | 1022.13 | 2.94% |
Proposed Method | 4765.08 | 0.10% | 994.90 | 0.19% |
Algorithm | Time (s) |
---|---|
MCS | 1912.72 |
UCCM | 4.21 |
CCCM | 4.58 |
Proposed Method | 6.27 |
Wind Speed Distribution | Shape Parameter | Scale Parameter |
---|---|---|
W1 (Connected to Bus 59) | 1.732 | 6.611 |
W2 (Connected to Bus 80) | 2.036 | 7.933 |
W3 (Connected to Bus 90) | 1.350 | 5.774 |
W1 | W2 | W3 | |
---|---|---|---|
W1 | 1.00 | 0.76 | 0.64 |
W2 | 0.76 | 1.00 | 0.36 |
W3 | 0.64 | 0.36 | 1.00 |
Area | Bus Number |
---|---|
Area 1 | 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, 113, 114, 115, 117 |
Area 2 | 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 116 |
Area 3 | 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 118 |
Algorithm | Mean | Mean APE | Standard Deviation | Standard Deviation APE |
---|---|---|---|---|
MCS | 124,281.33 | \ | 11,864.39 | \ |
UCCM | 124,093.19 | 0.15% | 10,969.98 | 7.54% |
CCCM | 124,093.19 | 0.15% | 11,961.51 | 0.82% |
Proposed Method | 124,244.89 | 0.03% | 11,884.53 | 0.17% |
Algorithm | Time (s) |
---|---|
MCS | 6195.28 |
UCCM | 13.24 |
CCCM | 13.46 |
Proposed Method | 171.48 |
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Deng, X.; He, J.; Zhang, P. A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables. Energies 2017, 10, 1623. https://doi.org/10.3390/en10101623
Deng X, He J, Zhang P. A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables. Energies. 2017; 10(10):1623. https://doi.org/10.3390/en10101623
Chicago/Turabian StyleDeng, Xiaoyang, Jinghan He, and Pei Zhang. 2017. "A Novel Probabilistic Optimal Power Flow Method to Handle Large Fluctuations of Stochastic Variables" Energies 10, no. 10: 1623. https://doi.org/10.3390/en10101623