基于鞍点方程的分布式经济调度算法
Saddle dynamic based distributed algorithm for economic dispatch problem
通讯作者:
收稿日期: 2019-01-16
Received: 2019-01-16
作者简介 About authors
时侠圣(1992—),男,博士生,从事分布式资源分配研究.orcid.org/0000-0001-9079-5705.E-mail:
为了实现智能电网的安全稳定经济运行,针对电力系统中广泛研究的经济调度问题,受到一致性模型和鞍点动态法的启发,提出基于一阶连续系统的分布式算法. 该算法考虑了迭代过程中节点生产能力和网络总负荷需求的约束,且每个节点只知道自身的代价函数. 为了解决上述约束,该算法设计3种对应的拉格朗日乘子. 为了实现控制参数的常量化,该算法添加了一个变量,用于平衡局部梯度差值. 由于有向网络的权矩阵是非对称的,该算法引入一变量用于平衡各有向边的权增益. 通过节点局部梯度与拉格朗日乘子,获取节点输出功率. 实验结果表明,该算法针对经济调度问题是可行且有效的.
关键词:
A distributed algorithm based on the first-order continuous-time multi-agent system was proposed for the widely studied economic dispatch problem in the power system inspired by the consensus model and saddle point dynamic method in order to realize the safe, stable, and economical operation of the smart grid. The total demand and generating capacity of each generators during its iteration were considered, in which each agent only knew its own cost function. Three Lagrange multipliers were designed in order to solve above constraints. The control parameters in the above proposed algorithm were constants by adding one variable to balance the difference of the local subgradient. Since the adjoint matrix of the directed network was asymmetrical, one variable was introduced to balance the weight gain of each edge. The output power of each agent was obtained by using the local subgradient and the corresponding Lagrange multipliers. The simulation results show that the proposed algorithm is effective and useful for the economic dispatch problem.
Keywords:
本文引用格式
时侠圣, 郑荣濠, 林志赟, 颜钢锋.
SHI Xia-sheng, ZHENG Rong-hao, LIN Zhi-yun, YAN Gang-feng.
为了克服上述集中式算法所带来的缺陷,许多学者提出大量基于离散系统的分布式优化算法[3-18],例如梯度算法[3-6]、残差法[7-9]、push-sum算法[10-11]、二分法[12]、一致性算法[13-16]、交替乘子法[17]、粒子群算法[18]等. 近年来,更多的学者寻找更合适的连续系统算法来解决分布式优化问题[19-25]. 从连续系统角度来看,通过微分包含理论等分析方法,可以得到更强的最优协调控制理论条件. 因为连续系统对应目标函数最陡的梯度方向. 通过罚函数法消掉不等式约束,Cherukuri等[26]提出针对一阶连续多智能体系统的分布式协调算法. 受到上述文献的激发,He等[27]设计集中式二阶算法.Wang等[28]研究分布式2阶系统的动态方程. 与上述方法不同,Hoang等[29]以鞍点动态方程出发,提出分布式控制算法用于解决最优资源分配问题。Zheng等[30]以多智能体一致性为基础,提出分布式算法.
上述算法都是从无向图或有向平衡网络出发,无向网络算法中都需要构建一系列双随机矩阵,阻碍了这些算法的开发和实际应用,特别是在随时间变化的一般非平衡有向实际网络环境中,因为双随机矩阵的条件难以以分布式方式满足. 在现代电力系统的应用中,无向时变网络已经不能完全满足智能电网中复杂度更高以及不对等通信条件下的发展要求[9].
本文针对有向网络下的电力系统经济分配问题,通过建立鞍点动态方程,设计基于连续系统的分布式1阶连续系统算法,且所设计算法只需每个节点各自的入度.
1. 预备知识
1.1. 图论
1.2. 问题描述
考虑带
s.t.
式中:
式中:
引理 1[29] 假设
式(3c)表明:
假设 1 为了解决有向网络下的经济调度问题,作出如下假设.
1) 网络节点图
2) 对于任意的节点
上述假设在分布式优化算法中是合理的,更多的相关知识介绍可以参考文献[12].
1.3. 无向图下的一阶分布式算法
为了实现以分布式的方式获取最优解,对于每个节点
式中:
2. 有向图下的一阶分布式算法
1.3节中,针对无向网络下的经济调度问题,Hoang等[29]设计基于多智能体连续系统和不动点理论的分布式算法。但该算法无法解决有向网络下的经济调度问题,因为有向图的拉普拉斯矩阵
其中所添加的平衡变量
算法的运行流程如下。
1)初始化控制参数
2)通过式(5a)计算平衡变量
3)通过式(5b)计算辅助变量
4)通过式(5c)~(5e)获取拉格朗日乘子
5)通过映射(5f)获取功率分配值
6)判断算法是否收敛至最优值。若是,则结束;若否,则转至2)步继续计算.
在给出收敛性证明之前,先给出以下引理.
引理2[23] 在假设1满足的条件下,存在一个正交矩阵
式中:
引理3[25] 假设本文讨论有向网络
1) 矩阵
2)
3)
定理1 令假设1成立且控制参数满足
则算法(5)收敛到最优解,即
3. 数值仿真
图 1
表 1 代价函数参数及生产能力约束范围
Tab.1
节点编号 | | | xi/MW |
1 | 0.04 | 2.0 | [0,80] |
2 | 0.03 | 3.0 | [0, 70] |
3 | 0.035 | 4.0 | [0, 70] |
6 | 0.03 | 4.0 | [0, 70] |
8 | 0.04 | 2.5 | [0, 80] |
在该仿真案例中,设置系统参数如下:
图 2
图 3
图 3
案例中拉格朗日乘子
Fig.3
Trajectories of Lagrange multiplier
图 4
图 4
案例中拉格朗日乘子
Fig.4
Trajectories of Lagrange multiplier
图 5
图 5
案例中各节点平衡变量
Fig.5
Trajectories of balance variable
图 6
图 6
案例中的辅助变量
Fig.6
Trajectories of auxiliary variable
图 7
图 7
Fig.7
Error trajectories of balance variable
4. 结 语
针对智能电网中的经济调度问题,基于鞍点动态方程理论,本文研究有向网络下的基于连续系统的分布式优化算法. 在带约束的分布式优化问题中,最优点的局部梯度不一定都是零,本文通过增加一辅助变量yi来平衡各节点的局部最优梯度,可以实现控制参数的常量化. 针对非平衡有向网络的经济调度问题,由于此时网络的权值是不平衡的,本文通过引入变量
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