工程科学与技术   2021, Vol. 53 Issue (6): 235-243

1. 智慧能源技术湖北省工程研究中心（三峡大学），湖北 宜昌 443002;
2. 贵州电网有限责任公司电力调度控制中心，贵州 贵阳 550002

Hybrid Decentralized Optimization of Dispatching Electrical Units with Consideration of Demand-side Response
CHENG Shan1, SHANG Dongdong1, DAI Jiang2, ZHONG Shiling1
1. Hubei Provincial Eng. Center for Intelligent Energy Technol. (CTGU), Yichang 443002, China;
2. Electric Power Dispatching and Control Center of Guizhou Power Grid Co., Ltd., Guiyang 550002, China
Abstract: In order to alleviate the burden of continuous increasing energy consumption falling on the power system and solve the complex calculation problem in the joint dispatching of large-scale electrical equipment, a hybrid decentralized optimization of dispatching the large-scale controllable appliances and energy storage equipment considering demand side response was proposed in this paper. Firstly, two mathematical models of controllable electrical equipment load and energy storage equipment were established. On this basis, a mixed integer non-linear centralized optimization model was mathematically formulated under the constraints of the operation characteristics of the system and equipment, with the objective of minimizing the sum of electricity purchase cost, users’ dissatisfaction cost and energy storage equipment loss cost. Secondly, for tackling the difficult nonlinear centralized optimization problems of high dimensionality, multi objectives and multiple constraints, the Lagrange relaxation method was used to decompose the problem into two sub-problems, namely, optimally scheduling the controllable electrical equipment load and optimizing the dispatch of the energy storage equipment. Then, the former was further decomposed into optimizing dispatch of each controllable electrical equipment and solved by the interior point method, while the latter was decomposed into a set of mixed integer linear optimization sub-problems of scheduling each energy storage equipment and solved in parallel by the Benders decomposition method. Thirdly, a series of numerical simulations together with comparison analysis were performed to verify the effectiveness and superiority of the proposed dispatch optimization method. For example, the optimization objective value and the optimal dispatch solution corresponding to the proposed method were illustrated and compared with those of the centralized method to demonstrate the effectiveness of the hybrid decentralized optimization method. And the influence of different numbers of dispatching equipment on the computation efficiency was investigated on the centralized and decentralized optimization method to show the superiority of the proposed hybrid decentralized optimization method. According to the numerical simulation results, the optimization objective value of the proposed method is basically consistent with that of the centralized. Moreover, the identified dispatch solution enables to efficiently respond to the time-of-use and results in good effect of peak-shaving and valley-filly. Besides, the calculation efficiency of the proposed hybrid decentralized optimization method is of high computation efficiency and not affected by the increasing number of the schedulable electrical equipments.
Key words: demand response    mixed integer nonlinear programming    decentralized optimization    Lagrangian relaxation method    interior point method    Benders decomposition

1 用户侧集中式优化模型 1.1 集中式调度框架

 图1 集中式调度框架图 Fig. 1 Centralized scheduling architecture

1.2 可调控负荷和储能设备模型

1.2.1 可调控负荷模型

1） 设备的能量约束[20]

 $\sum\limits_{h \in H} {p_a^h} \Delta T \ge {E_a}{\text{ ,}}\forall a \in A$ (1)

2） 可控时间段约束

 $\left\{ {\begin{array}{*{20}{c}} {{J_{{\text{j}},a}} = \left\lfloor {\dfrac{{{T_{{\text{j}},a}}}}{{\Delta T}}} \right\rfloor {\text{ }}} \\ {{J_{{\text{d}},a}} = \left\lceil {\dfrac{{{T_{{\text{d}},a}}}}{{\Delta T}}} \right\rceil {\text{ }}} \end{array}} \right.,\forall a \in A$ (2)

3） 负荷功率上下限约束

$a$ 台可调控负荷的可调度时间段 ${H_a} = \left[ {{J_{{\text{j}},a}},{J_{{\text{d}},a}}} \right]$ 。调度时段必须在可控时间段内，在可控时间段外都无法进行调度。即满足：

 ${\;\;\;\;\;\;\;\;\;\;\;\;\left\{\begin{array}{l}{p}_{a}^{\mathrm{min}}\le {p}_{a}^{h}\le {p}_{a}^{\mathrm{max}}, \forall h\in {H}_{a}\\ {p}_{a}^{h}=0, 其他\end{array},\forall a\in A \right.}$ (3)

1.2.2 储能模型

1）状态约束

 {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}[b] &\partial _{b,{\text{c}}}^h + \partial _{b,{\text{d}}}^h \le 1,{\text{ }}\partial _{b,{\text{c}}}^h,\partial _{b,{\text{d}}}^h \in \left\{ {0,1} \right\}{\text{, }} \\ &\quad\quad \forall b \in B,h \in H{\text{ }} \end{aligned}} (4)

2） 功率上下限约束

 ${\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left\{ {\begin{array}{*{20}{c}} {0 \le p_{b,{\text{c}}}^h \le \partial _{b,{\text{c}}}^hp_{b,{\text{c}}}^{\max }} \\ {0 \le p_{b,{\text{d}}}^h \le \partial _{b,{\text{d}}}^hp_{b,{\text{d}}}^{\max }} \end{array}} \right.,\forall b \in B,h \in H}$ (5)

3）能量及其上下限约束[22]

 {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}[b] &e_b^{\min } \le e_b^0 + \sum\limits_{t = 1}^h \left( n_b^{\text{c}}p_{b,{\text{c}}}^t - \frac{{p_{b,{\text{d}}}^t}}{{n_b^{\text{d}}}}\right) \le e_b^{\max },{\text{ }} \\ &\quad\quad \forall b \in B,h \in H \end{aligned}} (6)

1.2.3 系统功率约束

1） 用户侧功率供需平衡。即：

 ${\;\;\;\;\;\;\;p_0^h} + \sum\limits_{a \in A} {p_a^h + \sum\limits_{b \in B} {p_{b,{\text{c}}}^h = \sum\limits_{b \in B} {p_{b,{\text{d}}}^h + {g^h}} } } ,\forall h \in H$ (7)

2） 购电功率上下限约束

 $0 \le {g^h} \le {g^{\max }},\forall h \in H$ (8)

 ${\;\;\;\;\;0} \le p_0^h + \sum\limits_{a \in A} {p_a^h + \sum\limits_{b \in B} {p_{b,{\text{c}}}^h - \sum\limits_{b \in B} {p_{b,{\text{d}}}^h \le {g^{\max }}} } } ,h \in H$ (9)
1.3 目标函数

 {\;\;\;\begin{aligned}[b]&\underset{{p}_{\text{l}},\partial ,{p}_{\text{s}}}{\mathrm{min}}\;F({p}_{\text{l}},\partial ,{p}_{\text{s}})={\displaystyle \sum _{h\in H}{c}^{h}}{g}^{h}+\\ &\;\;{{\displaystyle \sum _{h\in {H}_{a}}{\displaystyle \sum _{a\in A}{\omega }_{a}^{h}({p}_{a}^{h}-{\overline{p}}_{a}^{h})}}}^{2}+{\displaystyle \sum _{h\in H}{\displaystyle \sum _{b\in B}({r}_{b,\text{c}}{p}_{b,\text{c}}^{h}+{r}_{b,\text{d}}{p}_{b,\text{d}}^{h})}}\text{ }\\ &{\rm{s}}.{\rm{t}}.\;式（1）、（3）～（7）、（9）\end{aligned} } (10)

2 分散式优化模型

2.1 分散式优化框架

 图2 混合分散式调度框架图 Fig. 2 Hybrid decentralized scheduling framework

2.2 上层分解协调过程

 \begin{aligned}[b] &L({p_{\text{l}}},\partial ,{p_{\text{s}}}) = F({p_{\text{l}}},\partial ,{p_{\text{s}}}) + \hfill \\ &\;\;\;\;\sum\limits_{h \in H} {{\lambda ^h}\left(p_0^h + \sum\limits_{a \in A} {p_a^h + \sum\limits_{b \in B} {p_{b,{\text{c}}}^h - \sum\limits_{b \in B} {p_{b,{\text{d}}}^h - {g^{\max }}} } } \right)} + \hfill \\ &\;\;\;\;\sum\limits_{h \in H} {{\mu ^h}\left( - p_0^h - \sum\limits_{a \in A} {p_a^h - \sum\limits_{b \in B} {p_{b,{\text{c}}}^h + \sum\limits_{b \in B} {p_{b,{\text{d}}}^h} } } \right)} = \hfill \\ &\;\;\;\;\sum\limits_{h \in H} {\sum\limits_{a \in A} {[({c^h} + {\lambda ^h} - {\mu ^h})p_a^h + \omega _a^h{{(p_a^h - \overline p _a^h)}^2}]} } + \hfill \\ &\;\;\;\;\;\;\;\;\;\;\sum\limits_{h \in H} {\sum\limits_{b \in B} {[m_b^hp_{b,{\text{c}}}^h + t_b^hp_{b,{\text{d}}}^h]} } + \tau \hfill \\[-15pt] \end{aligned} (11)

1）可调控负荷调度子问题，如式（12）所示：

 {\;\;\;\;\;\;\begin{aligned}[b] &\mathop {\min }\limits_{{p_{\text{l}}}} {\text{ }}{L_1}({p_{\text{l}}},\lambda ,\mu ) = \sum\limits_{h \in H} {\sum\limits_{a \in A} {[({c^h} + {\lambda ^h} - {\mu ^h})} } p_a^h + \\ &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\omega _a^h{(p_a^h - \overline p _a^h)^2}] \end{aligned} } (12)

2）储能调度子问题，如式（13）所示：

 ${\;\;\;\;\;\;\;\mathop {\min }\limits_{\partial ,{p_{\text{s}}}} {\text{ }}{L_2}(\partial ,{p_{\text{s}}},\lambda ,\mu ) = \sum\limits_{h \in H} {\sum\limits_{b \in B} {[m_b^hp_{b,{\text{c}}}^h + t_b^hp_{b,{\text{d}}}^h]} }}$ (13)

 ${\;\;\;\;\;\;\;\;\tau {\text{(}}\lambda {\text{,}}\;\mu {\text{) = }}\sum\limits_{h \in H} {[({c^h} + {\lambda ^h} - {\mu ^h})p_0^h - } {\lambda ^h}{g^{\max }}] }$ (14)

$\lambda$ $\mu$ 确定时，式（14）作为常数项返回到式（11）的对偶问题迭代求解。将松弛函数中的 $\lambda$ $\; \mu$ 看成变量，式（11）的对偶问题的表达式为：

 {\;\;\;\;\;\;\;\begin{aligned}[b] &D{\text{(}}\lambda {\text{,}}\;\mu )=\mathop {{\text{max}}}\limits_{\lambda ,\mu } [\mathop{\min }\limits_{{p_{\text{l}}}}\;{L_1}({p_{\text{l}}},\lambda ,\mu ) + \\ &\;\;\;\;\;\;\;\;\mathop {\min }\limits_{\partial ,{p_{\text{s}}}} \;{L_2}(\partial ,{p_{\text{s}}},\lambda ,\mu ) + \tau ] \end{aligned}} (15)

 ${\;\;\;\;\;\;\;\;\;\;\left\{ {\begin{array}{*{20}{c}} {\lambda (k + 1) = \lambda (k) + {y_\lambda }(k)\dfrac{{{s_\lambda }(k)}}{{||{s_\lambda }(k)|{|_1}}},} \\ {\mu (k + 1) = \mu (k) + {y_\mu }(k)\dfrac{{{s_\mu }(k)}}{{||{s_\mu }(k)|{|_1}}}} \end{array}} \right.}$ (16)

${s_\lambda }\left( k \right)$ ${s_\mu }\left( k \right)$ 的更新公式为：

 ${\;\;\;\;\;\;\;\;\left\{ \begin{gathered} s_\lambda ^h(k + 1) = p_0^h + \sum\limits_{a \in A} {p_a^h(k) + \sum\limits_{b \in B} {p_{b,{\text{c}}}^h(k) - } } \hfill \\ \quad\quad\quad\quad\;\;\;\sum\limits_{b \in B} {p_{b,{\text{d}}}^h(k) - {g^{\max }}} , \hfill \\ s_\mu ^h(k + 1) = - p_0^h - \sum\limits_{a \in A} {p_a^h(k) - \sum\limits_{b \in B} {p_{b,{\text{c}}}^h(k) + } } \hfill \\ \quad\quad\quad\quad\;\;\;\sum\limits_{b \in B} {p_{b,{\text{d}}}^h(k)} \hfill \\ \end{gathered} \right. }$ (17)

2.3 下层分散优化过程 2.3.1 各可调控负荷的优化调度

 ${\;\;\;\;\;\;\;\begin{array}{l}\underset{{p}_{a}}{\mathrm{min}}\;{L}_{1a}({p}_{a},\lambda ,\mu )={\displaystyle \sum _{h\in H}[({c}^{h}+{\lambda }^{h}-{\mu }^{h})}{p}_{a}^{h}+\\ \;\;\;\;\;\;\;\;\;{\omega }_{a}^{h}{(p_a^h - \overline p _a^h)^2}],\;a\in A\\ \text{s}\text{.t}.\;式（1）、（3）\end{array}}$ (18)

 \begin{aligned}[b] &\psi ({p_{a1}},\lambda ,\mu ,{r^{(i)}}) = {L_{1a}}({p_{a1}},\lambda ,\mu ) - \hfill \\ &\;\;{r^{(i)}}\left[\frac{1}{{{E_a} - \displaystyle\sum\limits_{h \in H} {p_a^h} {\text{ }}}} + \frac{1}{{p_a^h - p_a^{\max }}} + \frac{1}{{p_a^{\min } - p_a^h}}\right]{\text{ , }} \\ &\;\;\;\;\;\;\;\;\;\forall a \in A,h \in {H_a} \end{aligned} (19)

 图3 内点法流程图 Fig. 3 Internal point method flowchart

2.3.2 各储能的优化调度

 ${\;\;\;\;\;\;\begin{array}{l}\underset{{\partial }_{b},{p}_{b}}{\mathrm{min}}\text{ }{L}_{2b}({\partial }_{b},{p}_{b},\lambda ,\mu )={\displaystyle \sum _{h\in H}({m}_{b}^{h}{p}_{b,\text{c}}^{h}+{t}_{b}^{h}{p}_{b,\text{d}}^{h})}\\ \text{s}\text{.t}\text{. }\;式（4）、（6）{\text{，}}b\in B\end{array}}$ (20)

1）主问题数学形式为：

 ${\;\;\;\begin{array}{l}{L}_{2b,\text{lower}}=\underset{{\partial }_{b}}{\mathrm{min}}\;{L}_{2b}\\ \text{s}\text{.t}\text{.}\;式（4）\text{、}可行割约束、不可行割约束\end{array}}$ (21)

2）子问题数学形式为：

 $\begin{array}{l}{L}_{2b,\text{upper}}=\underset{{p}_{b}}{\mathrm{min}}\text{ }{L}_{2b}\\ \text{s}\text{.t}\text{.}\;式（\text{5}）、（\text{6}）\end{array}$ (22)

 \begin{aligned}[b] {L_{2b}}(\alpha ,\beta ,\chi ,\delta ) = &\sum\limits_{h \in H} {(m_b^hp_{b,{\text{c}}}^h + t_b^hp_{b,{\text{c}}}^h)} + \hfill \\ &{\alpha ^h}[p_{b,{\text{c}}}^h - \partial _{b,{\text{c}}}^hp_{b,{\text{c}}}^{\max }] + {\beta ^h}[p_{b,{\text{d}}}^h - \partial _{b,{\text{d}}}^hp_{b,{\text{d}}}^{\max }] + \hfill \\ &{\chi ^h}\left[e_b^0 + \sum\limits_{t=1}^h\left(n_b^{\text{c}}p_{b,{\text{c}}}^t-\frac{{p_{b,{\text{d}}}^t}}{{n_b^{\text{d}}}}\right) - e_b^{\max }\right] + \hfill \\ &{\delta ^h}\left[e_b^{\min } - e_b^0 - \sum\limits_{t = 1}^h \left( n_b^{\text{c}}p_{b,{\text{c}}}^t - \frac{{p_{b,{\text{d}}}^t}}{{n_b^{\text{d}}}}\right)\right]\\[-20pt] \end{aligned} (23)

 {\;\;\;\;\;\;\;\;\;\;\;\begin{aligned}[b] &\sum\limits_{h \in H} \left[({\chi ^h} - {\delta ^h})\sum\limits_{t = 1}^h \left( n_b^{\text{c}}p_{b,{\text{c}}}^t - \frac{{p_{b,{\text{d}}}^t}}{{n_b^{\text{d}}}}\right)\right] = \hfill \\ &\;\;\;\;\;\;\sum\limits_{h \in H} \left[\left(n_b^{\text{c}}p_{b,{\text{c}}}^t - \frac{{p_{b,{\text{d}}}^t}}{{n_b^{\text{d}}}}\right)\sum\limits_{t = h}^H ( {\chi ^h} - {\delta ^h})\right] \end{aligned}} (24)

 \begin{aligned}[b] {L_{2b}}(\alpha ,\beta ,\chi ,\delta ) =& \sum\limits_{h \in H} \left[(m_b^h + {\alpha ^h} + n_b^{\text{c}}\sum\limits_{t = h}^H ( {\chi ^h} - {\delta ^h}) \right]p_{b,{\text{c}}}^h + \hfill \\ &\sum\limits_{h \in H} \left[(t_b^h + {\beta ^h} + \frac{1}{{n_b^{\text{d}}}}\sum\limits_{t = h}^H ( {\chi ^h} - {\delta ^h}) \right]p_{b,{\text{d}}}^h + \hfill \\ &\sum\limits_{h \in H} [ - \partial _{b,{\text{c}}}^hp_{b,{\text{c}}}^{\max }{\alpha ^h} - \partial _{b,{\text{d}}}^hp_{b,{\text{d}}}^{\max }{\beta ^h} + \hfill \\ & {\chi ^h}(e_b^0 - e_b^{\max }) + {\delta ^h}(e_b^{\min } - e_b^0)]\\[-10pt] \end{aligned} (25)

 $\begin{gathered} \mathop {\max }\limits_{\alpha ,\beta ,\chi ,\delta } \;{Y_{2b}} = \sum\limits_{h \in H} {[ - \partial _{b,{\text{c}}}^hp_{b,{\text{c}}}^{\max }{\alpha ^h} - \partial _{b,{\text{d}}}^hp_{b,{\text{d}}}^{\max }{\beta ^h} + } \hfill \\ \;\;\;\;\;\;\;\;(e_b^0 - e_b^{\max }){\chi ^h} + (e_b^{\max } - e_b^0){\delta ^h}] \hfill \\ {\text{s}}{\text{.t}}{\text{. }}m_b^h + {\alpha ^h} + n_b^{\text{c}}\sum\limits_{t = h}^H {({\chi ^h} - {\delta ^h})} \ge 0, \hfill \\ \;\;\;\;\;\;{\text{ }}t_b^h + {\beta ^h} - \frac{1}{{n_b^{\text{d}}}}\sum\limits_{t = h}^H {({\chi ^h} - {\delta ^h})} \ge 0, \hfill \\ \;\;\;\;\;\;\;{\text{ }}{\alpha ^h},{\beta ^h},{\chi ^h},{\delta ^h} \ge 0,\forall b \in B \hfill \\ \end{gathered}$ (26)

1）对偶问题无可行解，则原问题无最优解，即该模型可行域为空集。

2）对偶问题有最优解，求解得到各拉格朗日乘子的值，则返回可行割约束式（27）到主问题求解，并更新上界。

 \begin{aligned}[b] {L_{2b}} \ge & \sum\limits_{h \in H} {[ - \partial _{b,{\rm c}}^hp_{b,{\rm c}}^{\max }{\alpha ^h} - \partial _{b,{\rm d}}^hp_{b,{\rm d}}^{\max }{\beta ^h} + } {\text{ }} \hfill \\ &(e_b^0 - e_b^{\max }){\chi ^h} + (e_b^{\max } - e_b^0){\delta ^h}],\;\forall b \in B \end{aligned} (27)

3）对偶问题存在无界最优解，则返回不可行割约束式（28）到主问题求解，并更新上界。

 \begin{aligned}[b] 0 \ge & \sum\limits_{h \in H} {[ - \partial _{b,{\rm c}}^hp_{b,{\rm c}}^{\max }{\alpha ^h} - \partial _{b,{\rm d}}^hp_{b,{\rm d}}^{\max }{\beta ^h} + } {\text{ }} \hfill \\ &(e_b^0 - e_b^{\max }){\chi ^h} + (e_b^{\max } - e_b^0){\delta ^h}],\;\forall b \in B \end{aligned} (28)

2.4 算法流程

 图4 算法流程图 Fig. 4 Algorithm flowchart

3 算例分析 3.1 基础数据

 图5 分时电价和不可转移负荷功率 Fig. 5 Time-of-use electricity price and nontransferable load power

3.2 仿真结果 3.2.1 不同调度策略对比

3.2.2 联合调度分散式优化结果

 图6 用电设备调度结果 Fig. 6 Power equipment scheduling results

 图7 储能充放电功率和存储能量 Fig. 7 Storage charge and discharge power and stored energy

3.2.3 算法计算效率对比

 图8 算法优化时间和迭代次数 Fig. 8 Algorithm optimization time and number of iterations

4 结　论

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