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SCIENTIA SINICA Informationis, Volume 48, Issue 10: 1450-1466(2018) https://doi.org/10.1360/N112018-00132

Investigating the market-based operation mechanism of DR resources using the equilibrium model

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  • ReceivedJul 31, 2018
  • AcceptedOct 4, 2018
  • PublishedNov 22, 2018

Abstract

The bidding of DR resources in the day-ahead market has become a trend with the reformation of the electricity market and the development of smart grids. Therefore, a market-based operation mechanism with DR resources must be investigated. First, a market mechanism in an auction-based framework for the participation of DR resources in the electricity market was designed. In this mechanism, a DR market was built to manage the flexibility of DR resources for a DRA, and the DRA was considered to bid in the day-ahead market. A compensation mechanism was established to encourage the DRA to participate in the day-ahead market. Second, a joint equilibrium model of the day-ahead and DR markets was proposed. Herein, the supply function bid form was applied by the DRA and generators in the day-ahead market. Third, in the DRX market, the supply function bid mode was also applied by DR customers, and the price in the DR market was determined on the basis of the bids of DR customers. Then, the existence and uniqueness of the Nash equilibrium were theoretically proven. Considering information asymmetry in practical application, a distributed algorithm was further proposed to determine equilibrium outcomes. Finally, numerical examples were presented to verify the reasonableness and effectiveness of the proposed model and algorithms.


Supplement

Appendix

Proof. 对于传统发电商而言, 令${B_{~-~i}}~=~\sum\nolimits_{i~=~1}^n~{{B_i}}~+~{B_{\rm~DRA}}~-~{B_i}$, $i~=~1,\ldots,n$, 代入式(2)和(5)得到 \begin{equation}{p_1} = \frac{D}{{{B_i} + {B_{ - i}}}}, {Q_i} = \frac{{D{B_i}}}{{{B_i} + {B_{ - i}}}}, \tag{A1}\end{equation} 将式(35)代入式(11), 并对${B_i}$求导, 有 \begin{equation}\frac{{\partial {\pi _i}}}{{\partial {B_i}}} = \frac{{{D^2}}}{{{{({B_{ - i}} + {B_i})}^2}}}\left[ {\frac{{{B_{ - i}} - {B_i}}}{{{B_{ - i}} + {B_i}}} - \frac{{{B_{ - i}}}}{D}C_{1i}^\prime \left(\frac{{D{B_i}}}{{{B_{ - i}} + {B_i}}}\right)} \right]. \tag{A2}\end{equation}

由于$\frac{{{B_{~-~i}}~-~{B_i}}}{{{B_{~-~i}}~+~{B_i}}}~\le~1$, 当$\frac{{{B_{~-~i}}}}{D}C_{1i}^\prime~(0)~>~1$时, $\frac{{\partial~{\pi~_i}}}{{\partial~{B_i}}}~<~0$, ${\pi~_i}$关于${B_i}$单调递减, 此时$B_i^*~=~0$, 即$Q_i^*~=~0$, 发电商$i$不参与日前市场投标.

当$\frac{{{B_{~-~i}}}}{{{D_t}}}C_{1i}^\prime~(0)~\le~1$时, 令$\frac{{\partial~{\pi~_i}}}{{\partial~{B_i}}}~=~0$, 即 \begin{equation}\frac{{B_{ - i}^* - B_i^*}}{{B_{ - i}^* + B_i^*}} - \frac{{B_{ - i}^*}}{D}C_{1i}^\prime \left(\frac{{DB_i^*}}{{B_{ - i}^* + B_i^*}}\right) = 0. \tag{A3}\end{equation}

由于$\frac{{B_{~-~i}^*}}{D}C_{1i}^\prime~\big(\frac{{DB_i^*}}{{B_{~-~i}^*~+~B_i^*}}\big)~>~0$, 因此可得$B_{~-~i}^*~-~B_i^*~>~0$, 即 \begin{equation}Q_i^* = \frac{{DB_i^*}}{{B_i^* + B_{ - i}^*}} < \frac{D}{2}. \tag{A4}\end{equation}

对于DRA而言, 有 \begin{equation}{p_1}{\rm{ = }}\frac{D}{{\sum\nolimits_{i = 1}^n {{B_i}} + {B_{\rm DRA}}}}, {Q_{\rm DRA}} = \frac{{D{B_{\rm DRA}}}}{{\sum\nolimits_{i = 1}^n {{B_i}} + {B_{\rm DRA}}}}. \tag{A5}\end{equation}

将式(39)代入式(14), 并对${B_{\rm~DRA}}$求导, 有 \begin{equation}\frac{{\partial {\pi _{\rm DRA}}}}{{\partial {B_{\rm DRA}}}} = \frac{{{D^2}}}{{{{(\sum\nolimits_{i = 1}^n {{B_i}} + {B_{\rm DRA}})}^2}}}\left[ {\frac{{(1 + \lambda )(\sum\nolimits_{i = 1}^n {{B_i}} - {B_{\rm DRA}})}}{{\sum\nolimits_{i = 1}^n {{B_i}} + {B_{\rm DRA}}}} - ({p_2} + \lambda {p_3})\frac{{\sum\nolimits_{i = 1}^n {{B_i}} }}{D} + \lambda } \right]. \tag{A6}\end{equation}

由于$\frac{{(1~+~\lambda~)(\sum\nolimits_{i~=~1}^n~{{B_i}}~-~{B_{\rm~DRA}})}}{{\sum\nolimits_{i~=~1}^n~{{B_i}}~+~{B_{\rm~DRA}}}}~+~\lambda~\le~1~+~2\lambda$, 当$\frac{{\sum\nolimits_{i~=~1}^n~{{B_i}}~}}{D}({p_2}~+~\lambda~{p_3})~>~1~+~2\lambda~$时, ${\pi~_{\rm~DRA}}$关于${B_{\rm~DRA}}$单调递减, 此时$B_{\rm~DRA}^*~=~0$, 即$Q_{\rm~DRA}^*~=~0$, DRA不参与日前市场投标.

当$\frac{{\sum\nolimits_{i~=~1}^n~{{B_i}}~}}{D}({p_2}~+~\lambda~{p_3})~\le~1~+~2\lambda~$时, 令$\frac{{\partial~{\pi~_{\rm~DRA}}}}{{\partial~{B_{\rm~DRA}}}}~=~0$, 即 \begin{equation}\frac{{(1 + \lambda )(\sum\nolimits_{i = 1}^n {B_i^*} - B_{\rm DRA}^*)}}{{\sum\nolimits_{i = 1}^n {B_i^*} + B_{\rm DRA}^*}} - ({p_2} + \lambda {p_3})\frac{{\sum\nolimits_{i = 1}^n {B_i^*} }}{D} + \lambda = 0. \tag{A7}\end{equation}

由于$({p_2}~+~\lambda~{p_3})\frac{{\sum\nolimits_{i~=~1}^n~{B_i^*}~}}{D}~>~0$, 因此可得$(1~+~2\lambda~)\sum\nolimits_{i~=~1}^n~{B_i^*}~-~B_{\rm~DRA}^*~>~0$, 即 \begin{equation}Q_{\rm DRA}^* = \frac{{DB_{\rm DRA}^*}}{{\sum\nolimits_{i = 1}^n {B_i^* + B_{\rm DRA}^*} }} < \frac{{1 + 2\lambda }}{{2 + 2\lambda }}D. \tag{A8}\end{equation}

Proof. 首先, 通过推导可以得出$f_{1,i}^{\prime~\prime~}({Q_i})~>~0$, $f_{1,{\rm~DRA}}^{\prime~\prime~}({Q_{\rm~DRA}})~>~0$, 即优化问题(23)$\sim$(26)是一个严格的凸优化问题, 并且存在唯一的最优解. 该凸优化问题的最优性条件为

当$1~\le~i~\le~n$, \begin{equation}\left[ {\left(1 + \frac{{Q_i^*}}{{D - 2Q_i^*}}\right)C_{1i}^\prime (Q_i^*) - {\omega _1}} \right]({Q_i} - Q_i^*) \ge 0; \tag{B1}\end{equation}

当$i~=~{\rm~DRA}$, \begin{equation}\left[ {\frac{{({p_2} + \lambda {p_3})(D - Q_{\rm DRA}^*)}}{{(1 + 2\lambda )D - (2 + 2\lambda )Q_{\rm DRA}^*}} - {\omega _1}} \right]({Q_{\rm DRA}} - Q_{\rm DRA}^*) \ge 0. \tag{B2}\end{equation}

原均衡问题的最优性条件:

当$1~\le~i~\le~n$, \begin{equation}\nabla {\pi _i}(B_i^*)({B_i} - B_i^*) = \left[ {\frac{D}{{B_{ - i}^* + B_i^*}} - \frac{{B_{ - i}^*}}{{B_{ - i}^* - B_i^*}}C_{1i}^\prime \left(\frac{{DB_i^*}}{{B_i^* + B_{ - i}^*}}\right)} \right]({B_i} - B_i^*) \le 0; \tag{B3}\end{equation}

当$i~=~{\rm~DRA}$, \begin{equation}\nabla {\pi _{\rm DRA}}(B_{\rm DRA}^*)({B_{\rm DRA}} - B_{\rm DRA}^*) = \left[ {\frac{D}{{\sum\nolimits_{i = 1}^n {B_i^*} + B_{\rm DRA}^*}} - ({p_2} + \lambda {p_3})\frac{{\sum\nolimits_{i = 1}^n {B_i^*} }}{{(1 + 2\lambda )\sum\nolimits_{i = 1}^n {B_i^*} - B_{\rm DRA}^*}}} \right]({B_{\rm DRA}} - B_{\rm DRA}^*) \le 0, \tag{B4}\end{equation} \begin{equation}\left[ {p_1^* - \frac{{({p_2} + \lambda {p_3})(D - Q_{\rm DRA}^*)}}{{(1 + 2\lambda )D - (2 + 2\lambda )Q_{\rm DRA}^*}}} \right]({Q_{\rm DRA}} - Q_{\rm DRA}^*) \le 0. \tag{B5}\end{equation}

可以看出式(B1)和(B3)、式(B2)和(B5)分别等价, 即原均衡问题的最优性条件与该凸优化问题的最优性条件等价, 因此可将原问题转化为该凸优化问题进行求解, 该凸优化问题的最优解即为原问题的均衡解. 由于该凸优化问题的最优解存在且唯一, 可得原均衡问题的均衡解存在且唯一.


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  • Figure 1

    Trading framework of the electricity market

  • Figure 2

    Solution algorithm flow chart of the equilibrium model

  • Figure 3

    (Color online) Evolution of price in the day-ahead market

  • Figure 4

    (Color online) Evolution of DRA and generators' bidding strategies in the day-ahead market

  • Figure 5

    (Color online) Evolution of price in the DRX market

  • Figure 6

    (Color online) Evolution of 4 DR customers' bidding strategies in the DRX market

  • Figure 7

    (Color online) The impacts of compensation coefficient on retailers' revenues and compensation amounts

  • Figure 8

    (Color online) The impacts of DR resources and day-ahead market demand on the price of day-ahead market

  • Table 1   The impacts of the compensation coefficient on equilibrium results
    Item Compensation coefficient
    0 (without DR) 0.1 0.3 0.5 0.7 0.9
    3*G1 Supply function (MW$^{2}$h/) 0.293 0.293 0.277 0.284 0.289 0.293
    Bid output (MW) 24.00 24.00 20.04 19.51 19.08 18.71
    Profit (/h) 1256 1256 887 799 733 683
    3*G2 Supply function (MW$^{2}$h/) 0.285 0.285 0.269 0.275 0.280 0.283
    Bid output (MW) 23.33 23.33 19.48 18.94 18.49 18.12
    Profit (/h) 1200 1200 848 762 698 649
    3*G3 Supply function (MW$^{2}$h/) 0.277 0.277 0.262 0.267 0.271 0.274
    Bid output (MW) 22.68 22.68 18.94 18.38 17.92 17.54
    Profit (/h) 1148 1148 811 727 665 617
    3*DRA Supply function (MW$^{2}$h/) 0 0.021 0.046 0.068 0.088
    Bid output (MW) 0 1.54 3.17 4.51 5.63
    Profit (/h) 0 36 120 231 360
    Price of day-ahead market (/MWh) 81.94 81.94 72.29 68.74 66.07 63.97
    Price of DRX market (/MWh) 86.76 88.65 90.20 91.49
    Total profit of DR customers (/h) 0 8 20 33 45
    Increase in retailers' revenue (/h) 0 193 366 485 573
    Net increase in retailers' revenue (/h) 0 135 183 146 57
  • Table 2   The impacts of the day-ahead market demand on equilibrium results
    Item Demand of day-ahead market (MW)
    40 50 60 70 80
    3*G1 Supply function (MW$^{2}$h/) 0.213 0.250 0.284 0.314 0.341
    Bid output (MW) 13.63 16.66 19.51 22.28 24.99
    Profit (/h) 525 666 799 937 1082
    3*G2 Supply function (MW$^{2}$h/) 0.208 0.244 0.275 0.304 0.329
    Bid output (MW) 13.33 16.23 18.94 21.55 24.10
    Profit (/h) 506 638 762 890 1023
    3*G3 Supply function (MW$^{2}$h/) 0.204 0.237 0.267 0.294 0.317
    Bid output (MW) 13.04 15.81 18.38 20.85 23.26
    Profit (/h) 488 612 727 846 969
    3*DRA Supply function (MW$^{2}$h/) 0 0.019 0.046 0.075 0.104
    Bid output (MW) 0 1.30 3.17 5.31 7.65
    Profit (/h) 0 43 120 227 365
    3*Price of day-ahead market (/MWh) With DR (/MWh) 63.98 66.63 68.74 70.97 73.29
    Without DR (/MWh) 63.98 69.97 75.96 81.94 87.93
    Price drop (%) 0 5.0 10.5 15.5 20.0
    Price of DRX market (/MWh) 86.48 88.65 91.12 93.83
    Total profit of DR customers (/h) 0 7 20 41 70
    Net increase in retailers' revenue (/h) 0 68 183 334 522
  • Table 3   The impacts of the retail price on equilibrium results
    Item Retail price (/MWh)
    70 90 110 130 150
    3*G1 Supply function (MW$^{2}$h/) 0.290 0.284 0.278 0.272 0.293
    Bid output (MW) 19.00 19.51 20.00 20.45 24.00
    Profit (/h) 721 799 880 964 1256
    3*G2 Supply function (MW$^{2}$h/) 0.281 0.275 0.270 0.264 0.285
    Bid output (MW) 18.40 18.94 19.44 19.91 23.33
    Profit (/h) 686 762 840 923 1200
    3*G3 Supply function (MW$^{2}$h/) 0.272 0.267 0.262 0.257 0.277
    Bid output (MW) 17.84 18.38 18.90 19.38 22.68
    Profit (/h) 654 727 804 884 1148
    3*DRA Supply function (MW$^{2}$h/) 0.073 0.046 0.023 0.003 0
    Bid output (MW) 4.76 3.17 1.67 0.26 0
    Profit (/h) 182 120 62 9.74 0
    Price of day-ahead market (/MWh) 65.58 68.74 71.99 75.30 81.94
    Price of DRX market (/MWh) 90.49 88.65 86.91 85.28
    Total profit of DR customers (/h) 35 20 9 1 0
    Net increase in retailers' revenue (/h) 301 183 87 12 0
  • Table 4   The impacts of the number of DR customers on equilibrium results
    Item Number of DR customers
    5 10 20 40 80 800
    3*G1 Supply function (MW$^{2}$h/) 0.293 0.279 0.284 0.287 0.288 0.290
    Bid output (MW) 24.00 19.91 19.51 19.28 19.14 19.01
    Profit (/h) 1256 864 799 762 742 723
    3*G2 Supply function (MW$^{2}$h/) 0.285 0.271 0.275 0.278 0.279 0.281
    Bid output (MW) 23.33 19.35 18.94 18.69 18.55 18.42
    Profit (/h) 1200 826 762 726 707 688
    3*G3 Supply function (MW$^{2}$h/) 0.277 0.263 0.267 0.270 0.271 0.272
    Bid output (MW) 22.68 18.80 18.38 18.13 17.99 17.85
    Profit (/h) 1148 789 727 692 674 655
    3*DRA Supply function (MW$^{2}$h/) 0 0.027 0.046 0.058 0.065 0.072
    Bid output (MW) 0 1.94 3.17 3.90 4.32 4.73
    Profit (/h) 0 73 120 148 164 181
    Price of day-ahead market (/MWh) 81.94 71.38 68.74 67.25 66.44 65.64
    Price of DRX market (/MWh) 95.37 88.65 84.82 82.73 80.67
    Total profit of DR customers (/h) 0 25 20 10 7 2
    Net increase in retailers' revenue (/h) 0 119 183 217 235 252

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