Figure 4. Incremental marginal cost of electricity. B. Experimental Results For the parameter setting of the PSO algorithm, is set to be 0.6 and updated as (0.5 + unif(0,1))/2 in every iteration , 1 = 2 is set to be 1.49 and updated by following the nonlinear acceleration strategy [10]. The maximum allowed iteration number is 100, swarm size is 30, total run time is set to 50. If the best fitness is not improved in consecutive 10 iterations, a random disturbance between [-0.5, 0.5] is added to the current position of each particle to mutate current particles. If the best fitness is not improved in successive 30 iterations, stop the algorithm. The obtained price signals from PSO in 50 runs are shown in Fig. 5, and the corresponding DSO objective values are shown in Fig. 6, where the red line is the optimal objective value $8316.39 obtained from the BI approach; see results summary in Table I. Figure 5. Prices for 50 runs of PSO. Figure 6. DSO objective values for 50 runs of PSO. Since PSA is a direct search method and doesn’t rely on derivative information, its performance greatly depends on the maximum allowable function evaluation number (here, one function evaluation means one complete negotiation process between a DSO and LAs). For instance, the objective iteration is shown in Fig. 7 with function evaluation number of 2000. The final objective value approaches the optimal value after about 40 iterations. The final objective value and solving time are shown in Fig. 8 along with a function evaluation number from 100 to 2000. For comparison purposes, the BI approach is implemented using a non-commercial nonlinear solver SCIP [16]. Both fixed price (FIX) and Time-of-Use (TOU) price structures are tested. The average price in multiple runs are calculated and the best price is picked based on the best objective value for PSO. Overall, as indicated by the price comparison in Fig. 9, the TOU prices are blow the fixed price before 11am and after 23pm, which will motivate LAs to shift demand from relatively peak hours 11am-23pm (see Fig. 2) or high generation cost hours (see Fig. 4) to off-peak hours before 11am. This is supported by the aggregated demand response outcome shown in Fig. 10. Note that this peak-load shifting effect is greatly determined by users’ preference on satisfaction (see Fig. 3 and Fig. 1); small value of before 11am means more load or regulation flexibility, and large value of means that users are reluctant to the price response in that period. Figure 9. Resulted prices from the different solution approaches. Figure 7. Iteration process of PSA with function evaluation number of 2000. Figure 8. DSO objective and computational time of PSA with different function evaluation numbers. Figure 10. Resulted aggregated loads from the different solution approaches. To better compare the different solution approaches, the detailed experimental results are summarized in Table I. The best price for PSO is used, and the function evaluation number is fixed as 2000 for PSA. Compared with fixed price, TOU prices from BI, PSO, and PSA have a lower PAR value and higher overall social welfare (objective sum of DSO and LAs) for one-day operation. The average price for each aggregator ∑∗∙∗,. ∑, provide customers with a lower average price which means is calculated by The BI (TOU) and PSO (Best) they could consume more energy with a similar bill payment and satisfaction level. The average price of PSA is higher than BI and PSO mainly due to the limitation of the function evaluation number. As a centralized approach, BI will need to collect all sensitive information from customers, such as their load profile, satisfaction preference, etc. And its computational cost (solver SCIP) is higher than the distributed algorithms (PSO and PSA), for instance, it takes 300 seconds to reach a relative gap of 2.88% and 600 seconds to reach a relative gap of 2.69% for this small case study. Each iteration in PSO takes about 0.3 seconds for parallel execution and it will need around 30 seconds for max 100 iterations limit. Seen from Fig. 6, it starts to converge around 50 iterations. The solving time for PSA is shown in Fig. 8. Although PSO performs slightly better than PSA in Table I, its performance is not stable due to its stochastic nature and it requires a parameter tuning process and a large number of repetitions. On the other hand, PSA is a deterministic method and its convergence property is guaranteed given enough function evaluations. TABLE I. PROFIT AND COST IN SUMMARY (PRICE: CENT, COST/REVENUE/SATISFACTION: $), (CUSTERMOR SATISFACTION: PEAK PENALTY = 3:1). DSO Level BI (FIX) BI (TOU) PSO (Best) PSA (Final) Peak Load 255.43 243.69 243.72 244.30 PAR Value 1.17 1.09 1.09 1.10 Total Satisfaction 2593.66 2600.74 2600.79 2597.71 DSO Revenue 790.33 780.87 780.70 786.89 DSO Gen. Cost 193.32 193.60 193.67 191.85 DSO Profit 597.01 587.27 587.03 595.03 DSO Objective 8301.39 8316.39 8316.31 8314.90 Aggregator Level BI (FIX) BI (TOU) PSO (Best) PSA (Final) n1 Average Price 15.11 14.63 14.61 14.84 n2 Average Price 15.11 14.57 14.56 14.79 n3 Average Price 15.11 14.76 14.76 15.02 n1 Bill Payment 206.15 203.57 203.48 205.05 n2 Bill Payment 293.70 289.45 289.37 291.59 n3 Bill Payment 290.46 287.85 287.84 290.24 n1 Satisfaction 682.23 684.70 684.71 684.14 n2 Satisfaction 955.92 959.30 959.41 958.35 n3 Satisfaction 955.50 956.73 956.67 955.21 V. CONCLUSION In this paper, a Stackelberg pricing model is formulated for a DSO and LAs, where in the upper level, the DSO maximizes its profit as well as its social obligation to customers, while in the lower level, LAs minimize their electricity bill payment and discomfort. To accelerate the solving process and protect sensitive information, two distributed algorithms, PSO and PSA, are implemented and compared. Numerical results show that the performance of the proposed distributed approaches is in close proximity to the optimal solution. The problem of how the committed aggregated demand response can be allocated to each individual customer will be explored in a future paper. ACKNOWLEDGMENT This material is based upon work supported by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Building Technology Office under contract DE-AC05-00OR22725. REFERENCES [1] H. Yang, J. zhang, J. Qiu, S. Zhang, M. Lai, and Z. Y. Dong, “A practical pricing approach to smart grid demand response based on load classification,” IEEE Transactions on Smart Grid, vol. 9, no. 1, pp. 179–190, 2018. [2] P. Yang, G. Tang, and A. Nehorai, “A game-theoretic approach for optimal time-of-use electricity pricing,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 884–892, 2013. [3] M. Yu and S. H. Hong, “Supply–demand balancing for power management in smart grid: A Stackelberg game approach,” Applied Energy, vol. 164, pp. 702–710, 2016. [4] A. Paudel, K. Chaudhari, C. Long, and H. Gooi, “Peer-to-peer energy trading in a prosumer based community microgrid: A game-theoretic model,” IEEE Transactions on Industrial Electronics, vol. 66, no. 8, pp. 6087–6097, 2018. [5] S. Fan, Q. Ai, and L. Piao, “Bargaining-based cooperative energy trading for distribution company and demand response,” Applied Energy, vol. 226, pp. 469–482, 2018. [6] R. Gibbons, “Game theory for applied economists,” Princeton University Press, 1992. [7] H. Tarish, O. Hang, and W. Elmenreich, “Residential demand response scheme based on adaptive consumption level pricing,” Energy, vol. 113, pp. 301–308, 2016. [8] M. Motalleb and R. Ghorbani, “Non-cooperative game-theoretic model of demand response aggregator competition for selling stored energy in storage devices,” Applied Energy, vol. 202, pp. 581–596, 2017. [9] Z. Fadlullah, D. M. Quan, S. Member, N. Kato, and I. Stojmenovic, “GTES: An optimized game-theoretic demand-side management scheme for smart grid,” IEEE Systems Journal, vol. 8, no. 2, pp. 588– 597, 2014. [10] Y. Chen and M. Hu, “A swarm intelligence based distributed decision approach for transactive operation of networked building clusters,” Energy and Buildings, vol. 169, pp. 172–184, 2018. [11] Y. Chen and M. Hu, “Swarm intelligence-based distributed stochastic model predictive control for transactive operation of networked building clusters,” Energy and Buildings, vol. 198, pp. 207–215, 2019. [12] Y . Chen and M. Hu, “A guided particle swarm optimizer for distributed operation of electric vehicle to building integration,” Proc. of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 2A, pp. 1–9, 2017. [13] C. Audet, “Convergence results for generalized pattern search algorithms are tight,” Optimization and Engineering, vol. 5, no. 2, pp. 101–122, 2004. [14] C. Bogani, M. G. Gasparo, and A. Papini, “Generalized pattern search methods for a class of nonsmooth optimization problems with structure,” Journal of Computational and Applied Mathematics, vol. 229, no.1, pp. 283–293, 2009. [15] M. A. Diniz-Ehrhardt, D. G. Ferreira, and S. A. Santos, “A pattern search and implicit filtering algorithm for solving linearly constrained minimization problems with noisy objective functions,” Optimization Methods and Software, vol. 34, no. 4, pp. 827–852, 2019. [16] “SCIP Optimization Suite,” 2019. [Online]. Available at: https://scip.zib.de/. [Accessed: 20-Apr-2019].
ACKNOWLEDGMENT This material is based upon work supported by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Building Technology Office under contract DE-AC05-00OR22725.