5.6 Computational Effort
6.1.6 Numerical Results using FDA
This section is dedicated to the numerical results using the Fully Distributed Algo-rithm FDA). We use the 47-bus network described in Appendix A and adjust the DER presence and capabilities to induce or relieve congestion. We start with results on non-congested conditions and continue with congested instances.
FDA Results on Uncongested Networks
We begin by comparing the iterations needed for convergence with the use of:
1. Constant penalty
2. Centrally adapted penalty 3. Bus-specific penalty
4. Bus-and-quantity specific penalty
ρ1 Constant Centrally Adapted Bus Specific Bus& Quantity
5 1205 409 196 205
10 2298 514 261 241
20 1595 519 355 331
50 >> 5000 442 417 342
Table 6.1: Number of iterations required for convergence using Fully Distributed Algorithms with various penalty updates rules.
We conclude that a large constant penalty can indeed result in no convergence.
On the other hand, local penalties result not only in lower communication burden but also in decreased number of iterations needed for convergence.
The following figures show the evolution of a centrally adapted penalty and the evolution of bus-specific penalties, respectively, starting from much different initial values.
Figure 6·1: Evolution of centrally adapted penalty across iterations.
Figure 6·2: Evolution of bus-specific penalty across iterations at a specific distribution bus, starting from different penalty values.
In addition, to underline the importance of locational penalties, we show the
updates of bus-specific penalty for two different distribution buses starting from the same penalty value.
Figure 6·3: Evolution of bus-specific penalty across iterations at two distribution buses, starting from the same initial penalty value.
The two figures below show the updates of the bus-and-quantity specific penalties for two different buses.
Figure 6·4: Updates of bus-specific penalty of real power, reactive power and voltage consistency across iterations at distribution bus 30.
Figure 6·5: Updates of bus-specific penalty of real power, reactive power and voltage consistency across iterations at distribution bus 2.
We reach the following conclusions regarding penalties:
1. In agreement with (Kraning et al., 2014), the choice of the starting value of the penalty is not important, if the penalty is allowed to change for many iterations.
2. Our choice of penalty update rules, based on imbalances and the imbalances’
changes, indeed leaves the penalties unchanged for many iterations before con-vergence, as is desired for ADMM’s theoretical convergence guarantees.
3. Bus specific penalties can vary significantly across buses, as in Figure 6·3.
4. Starting from the same initial value and specifying quantity and bus specific penalties, can result in penalties for each quantity varying greatly in the space domain, as in Figures 6·5 and 6·4.
The following figures show the convergence of the prices of real and reactive power, by means of the maximum and average deviation across iterations, with the use of bus and quantity specific penalties that are initialized at 50.
Figure 6·6: Percent deviation of real power price estimates from op-timal reactive power price across iterations.
Figure 6·7: Percent deviation of reactive power price estimates from optimal reactive power price across iterations.
For completeness, the tables below show the deviation from the optimal real and reactive DLMPs, for bus specific penalties and for all tested starting values of the penalty (Ntakou and Caramanis, 2014).
ρ1 Average Deviation Minimum Deviation Maximum Deviation
5 0.0513 0.0001 0.2821
10 0.0585 0.0001 0.5445
20 0.0895 0.0003 0.5553
50 0.0290 0.0000 0.1706
Table 6.2: Average, minimum and maximum deviation of real power prices from P-DLMPs (%) at convergence, with the use of bus specific penalties.
ρ1 Average Deviation Minimum Deviation Maximum Deviation
5 0.6435 0.4048 0.8442
10 1.3242 0.8299 1.5025
20 1.3455 0.8738 1.5254
50 1.2026 0.9728 1.3571
Table 6.3: Average, minimum and maximum deviation of reactive power prices from Q-DLMPs (%) at convergence, with the use of bus specific penalties.
The tables below show the same metrics for bus-and-quantity specific penalties and for all tested starting values of the penalty.
ρ1 Average Deviation Minimum Deviation Maximum Deviation
5 0.0476 0.0005 0.2016
10 0.0171 0.0001 0.0523
20 0.0176 0.0032 0.0808
50 0.0255 0.0003 0.1413
Table 6.4: Average, minimum and maximum deviation of real power prices from P-DLMPs (%) at convergence, with the use of bus-and-quantity specific penalties.
ρ1 Average Deviation Minimum Deviation Maximum Deviation
5 0.1547 0.0497 0.2162
10 1.6385 1.2788 1.7887
20 1.3719 1.1010 1.4991
50 1.1715 0.9868 1.3566
Table 6.5: Average, minimum and maximum deviation of reactive power prices from Q-DLMPs (%) at convergence, with the use of bus-and-quantity specific penalties.
(Peng and Low, 2015) examines how the network size and the network diam-eter (distance, measumed in number of lines, of the two furthest buses) affect the number of iterations required for convergence of ADMM to solve an OPF problem over a tree network. Through simulations, (Peng and Low, 2015) concludes that the network distance is the most important factor affecting the convergence speed. We contribute to this result by comparing the convergence speed of FDA for the 47-bus network, whose results are shown above, to the convergence speed of FDA for a 253-bus network, that on top of higher network size, also has a larger diameter. This results in slower convergence (Ntakou and Caramanis, 2014). The specifications of both networks can be found in the Appendix.
Figure 6·8: Average percent deviation of real power price estimates from optimal real power price across iterations, 47-bus network versus 253-bus network.
Figure 6·9: Maximum percent deviation of real power price estimates from optimal real power price across iterations, 47-bus network versus 253-bus network.
FDA Results on Congested Networks
We simulate a congested instance of the 47-bus network A, where hard voltage bound inequality constraints bind (Ntakou and Caramanis, 2016).
Motivated by the superiority of results with the use of bus-and-quantity specific penalties, we proceed in the analysis of FDA for congested conditions using bus-and-quantity specific penalties.
The figure below shows that the sum of voltage shadow prices over all distribution buses, P
bµib(h) = P
b,b0µib,b0(h) converges to the optimal sum in a small number of iterations.
Figure 6·10: Percent deviation of the sum of the voltage magnitude shadow prices across all buses from the optimal value across iterations.
However, individual values µib(h) take much longer to converge. The following two figures show the percent deviation of real and reactive power price estimates across iterations, respectively, by means of the minimum, average and maximum deviation.
Figure 6·11: Minimum, average and maximum percent deviation of real power price estimates across all buses from the optimal value across iterations.
Figure 6·12: Minimum, average and maximum percent deviation of reactive power price estimates across all buses from the optimal value across iterations.
Figure 6·13 below, zooms into Figures 6·11 and 6·12 above, to show the percent deviation of real and reactive power price estimates many iterations after the tolerance
criterion is met.
Figure 6·13: Average percent deviation of the real and reactive power price estimates across all buses from the optimal value long after con-vergence.
We conclude that the slow convergence of the individual µib(h) values propagates to a decreased rate of convergence of the real and reactive power marginal costs.
To proceed, we show the discrepancy in satisfying the first order optimality constraints 6.23. In specific, the figure below shows the maximum and average values of the ratio 100·Pb(i)(h)
πbP,i(h) across iterations and over all distribution buses.
Figure 6·14: Deviation in satisfying first order optimality conditions of real power DLMPs as a percentage of the benchmark real power DLMP.
The results of the proposed price estimate correction process follow below. First, voltage bound inequality constraints’ shadow prices are corrected through 6.25 and 6.27 as shown in Figure 6·15 below. Figure 6·15 only shows buses that correspond to non-zero voltage magnitude shadow prices for at least one of the methods that we compare.
Figure 6·15: Shadow prices of voltage bound constraints, before and after the correction process.
Compared to the optimal solution, obtained by the centralized solution, that we use as a benchmark, the correction process results in 2 additional buses appearing to be binding, versus 4 additional buses in the unassisted FDA results. For the 2 additional buses, the shadow prices assigned are very small. Interestingly, the sum of the voltage magnitude shadow prices after the correction process is still equal to the optimal sum, even if we do not constrain it explicitly in 6.25 and 6.27.
We conclude the reporting of results of the correction process by showing the resulting corrected price estimates. Specifically, as described above, we feed the cor-rected voltage magnitude shadow prices in 5.26 and 5.27 and get updated price es-timates, that are closer to the optimal DLMPs. Figure 6·16 overlays the average deviation of the updated price estimates after the correction process to the average deviation of the unassisted price estimates shown in Figure 6·13 above.
Figure 6·16: Average percent deviation of real and reactive power price estimates from the optimal values before and after the correction process.
Lastly, we present results when using FDA with soft voltage constraints for congested instances. Note that our benchmark results are now different, since they are based on the centralized formulation with soft voltage constraints. We choose the parameters in the voltage barrier function to be such that the voltage magnitudes in the new optimal solution will be as close to the voltages in the original optimal solution with the hard voltage bound inequality constraints.
The two figures below show the superior convergence performance of FDA with soft voltage constraints by means of the maximum and average percent deviation of real power price estimates as compared to the benchmark values obtained from the centralized algorithm with soft voltage constraints.
Figure 6·17: Fully Distributed Algorithms, Average Error in Real power DLMPs across iterations (%)
Figure 6·18: Fully Distributed Algorithms, Maximum Error in Real power DLMPs across iterations (%)
The FDA with soft voltage constraints is able to converge fully (accuracy of 0.1%
error in the prices) to the centralized benchmark after 7500 iterations. FDA-OPT (FDA with hard voltage bound constraints) is unable to reach this level of accuracy
even after 50000 iterations. Therefore, the computational effort improvement of using soft voltage constraints is at least 6 times.
We repeat our clarification that the deviations of FDA-OPT are based on the benchmark values of C-OPT and the deviations of FDA-SVC are based on the bench-mark vales of C-SVC.