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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


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