3. POWER SYSTEM TOPOLOGY CONTROL FOR GRID ECONOMIC EFFICIENCY: PROBABILISTIC FORMULATIONS AND DECISION MAKING *
3.2 Probabilistic Optimal Topology Control Framework
3.2.3 Probabilistic Topology Control Optimization
As the deterministic OPF evaluations cannot fully reveal the state of the system, probabilistic analysis is becoming of considerable importance and interest due to the increasing trend of facing many random distortions or uncertainties arisen from measurement errors, forecasting errors, variation of system variables due to adoption of renewable generation resources and load uncertainties. Performing OPF analysis for every possible or probable combination of loads, generation, and network topology is impractical or at least computationally cumbersome. As an analytical tool with tractable computation burden and acceptable level of accuracy, the PEM is suggested to be used for probabilistic formulation of the problem. Using the PEM method for probabilistic OPF analysis, the impact of uncertain input variables and the propagation of such uncertainties over the output parameters would be well captured. The PEM method is selected over the other probabilistic techniques as it is easier to implement and imposes less computational complexities for large-scale scenarios [133]-[138].
Vectors of input and output random variables as well as corresponding nonlinear functions are presented in (3.4)-(3.6), respectively.
, ,
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The probabilistic DCOPF-based optimization for transmission topology control problem is formulated below, where the objective function is introduced in (3.7) subject to system and security constraints in (3.8)-(3.13)* [139].
min GC
The output power of generator g at node n is limited to its physical capacities in (3.8). Constraint (3.9) limits the power flow across transmission line k within the minimum and maximum line capacities. Power balance at each node is enforced by (3.10) and Kirchhoff’s laws are incorporated in (3.11) and (3.12). The status of any transmission line k of the system is identified via an integer variable in (3.13). Parameter Mk is a user-specified large number greater than or equal to Bk
maxmin
which is selected to make the constraints nonbinding and relax those associated with Kirchhoff’s laws when a line
* The bar notation over variables show the probabilistic (expected) values.
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is removed from service regardless of the difference in the bus angles [68], [69]. Parameter
introduced in (3.14) limits the number of open transmission lines in the new optimal network reconfiguration.
1 k
Lk
k
(3.14)The optimization engine is able to provide several sets of optimal solutions for any selection of. In doing so, the probabilistic optimization algorithm is first simulated to suggest the best optimal solution for the topology control problem. A Not-To-Switch (NTS) list is designed where the obtained best optimal solution is stored. The optimization engine is simulated again neglecting the solutions previously stored in the NTS box and the process will continue to obtain the second best, third best, etc. optimal switching solution. Such implementation design would not only increase the chance that at least one set of the solutions would survive all the subsequent AC feasibility/stability tests and other operational concerns, but also would provide the operator with more flexibility in final decision making [139].
The two point estimation method (2-PEM) decomposes (3.5) into several sub problems by taking only two deterministic values of each uncertain variable located on the two sides of its mean value. Figure 4 demonstrates a general idea of the 2-PEM application to capture the uncertainties in the probabilistic optimization model. The deterministic topology control optimization (3.7)-(3.13) is then simulated twice for each uncertain variable, one for the value below and the other for the value above the mean,
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Figure 4. Illustration of the 2-PEM application for capturing the uncertainties.
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while keeping the other variables at their mean values. These two pints may or may not be selected symmetrically around the mean of a given variable [138]. As each set of the selected sample points undergoes the optimization problem to obtain the transformed samples, the mean and standard deviation of output variables (e.g., the generation dispatch cost) as well as the status of each line would be calculated at each scenario. The probabilistic optimal topology control formulation would eventually result in the probability distribution function (PDF) of the generation dispatch cost as well as the final status for each transmission line.
Regarding the selection of the final optimal lines to switch, the following procedure is pursued: in each studied scenario, the optimization engine is simulated and the optimal dispatch cost and the optimal switching status of the lines (0 or 1) would be obtained. Having conducted the same process for all the studied probabilistic scenarios (i.e., 2X scenarios for X uncertain factors) for a given generation and load profile at an hour, the final status for each transmission lines would be selected as the “most repeated status” in all the simulated scenarios. For instance, if the status for a given transmission line is 1 in many (more than a threshold) of the studied probabilistic scenarios, the final status of that transmission line would be considered 1.
Note that the transmission line switching embedded in the probabilistic ACOPF formulation can be also approached, if the computational facilities allow, by adding the voltage magnitude and reactive power constraints. Recent advancement in parallel processing and new heuristics to handle such complexities are necessary to make an AC model of the suggested framework work.
42 3.2.4 The 2-PEM Core Algorithm
The algorithm of the 2-PEM procedure for the above optimization formulation is presented as follows [133], [139]; the requisite variables of the 2-PEM algorithm are initialized in Step 1 using (3.15) and (3.16):
( )(1) 0
E Y (3.15)
2 (1)
( ) 0
E Y (3.16)
In Step 2, the locations and probability of concentrations are calculated through (3.17)-(3.20) as follows:
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In Step 4, the deterministic topology control optimization is solved for both concentrations xz i, with respect to vector X presented in (3.23).
,1 ,2 , ,
[ z , z ,...,xz i,...,z r] i 1,2
X (3.23)
Equations (3.24) and (3.25) are updated in Step 5 as follows:
2 deviation would be estimated in (3.26) and (3.27), respectively.
Y E Y( )
3.3 Probabilistic Reliability Cost/Value Framework for Optimal Topology Control