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JXYXY01-026@高空作业车工作臂结构设计及有限元分析设计
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Optimal design of signal-controlled road network Abstract： An optimal design of signalized road network is considered where total travelers delay is minimized subject to user equilibrium flow. This problem can be formulated as an optimization problem by taking the user equilibrium traffic assignment as a constraint. In this paper, a projected conjugate gradient method is presented to solve the signalized road network problem with global convergence. Numerical examples are investigated on simple grid road network. As it shows, the proposed method achieved substantially better performance than did traditional approaches when solving the signalized road network design problem with equilibrium flows. Keywords: Conjugate gradient; Signalized road network; Equilibrium constraints; Optimization 1. Introduction An optimal design of signalized road network is considered while the route choice of users is taken into account. This problem can be formulated as an optimization problem by taking the user equilibrium traffic assignment as a constraint. In the past decades, many researchers via the techniques of optimization have investigated this problem 1,6,9,2. In this paper, a bilevel programming program is considered to formulate an optimal design of signal-controlled road network. In the upper level program, the performance index is defined as the sum of a weighted linear combination of rate of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model from TRANSYT 7. The corresponding mathematical approximations of the performance index and the average delay to a vehicle at the downstream junction in the TRANSYT model have been obtained. At the lower level the user equilibrium traffic assignment obeying Wardrops first principle can be formulated as a minimization problem. Because the user equilibrium assignment constraint is nonlinear, which leads the signalized road network design problem to be a non-convex problem, only locally optimal solutions can therefore be found. In this paper, a projected conjugate gradient (PCG) approach is proposed to determine the optimal signal settings and network flows with global convergence. Numerical computations are conducted on a grid network with signalized junctions where the proposed PCG method outperforms traditional methods on varioussets of initials and sets of demand scalars. The rest of the paper is organized as follows. In the next section, a signalized road network is formulated where users route choice is taken into account. In Section 3, a projected conjugate gradient method is developed with global convergence. In Section 4, a grid network with signal-controlled junctions under various sets of initials is considered where numerical computations for the proposed PCG and traditional methods are conducted. Conclusions and discussions for this paper are remarked in Section5. 2. Problem formulation 2.1. Notation Notation used throughout this paper is summarized below. G(N, l) denotes a directed road network, where N is the set of signal controlled junctions ntsand L is the set of links denotes the set of signal setting variables, respectively for the reciprocal of cycle time, start and duration of greens, where and represents the vector of starts and durations of green for signal group j at junction m as proportions of common cycle time represents the duration of effective green for link a represents the minimum green for signal group j at junction m represents the clearance time between the end of green for group j and the start of green for incompatible group l at junction m represents saturation flow rate on link a represents a collection of numbers 0 and 1 for each pair of incompatible signal groups at junction m; where if the start of green for signal group j proceeds that of l and otherwise represents the rate of delay on link a represents the number of stops per unit time on link a denotes the set of OD pairs denotes the travel demands for OD pairs denotes the set of paths between OD pair w denotes vector of path flow denotes vector of link flow denotes the link-path incidence matrix, where if path p between OD pair w uses link a and otherwise C denotes the link travel times 2.2. Signalized road network problem The signalized road network design problem can be formulated as to subject to ntswhere and are respectively link-specific weighting factors for the rate of delay and the number of stops per unit time used in TRANSYT. The first constraint is for the common cycle time and the constraints (3)(5) are for the green phase, link capacity and clearance time at each junction. Also the equilibrium flows are found by solving the following traffic assignment problem. 3. A solution method for signalized road network design problem In this section, an effective search for solving problems (1)(9) is developed, for which a search direction of descent is generated and a new iterate is created. The search process will be terminated at a KKT point or a new search direction can be generated. In the following, a projected conjugate gradient method is proposed to obtain a descent search direction. 3.1. A projected conjugate gradient method Lemma 1 (Fletcher & Reeves conjugate gradient method). Consider a continuously differentiable function to generate a sequence of iterates according to Then for the sequence of points generated by the conjugate gradient method whenever The direction generated by (11) for an unconstrained nonlinear problem is a descent direction which strictly decreases the objective function value provided that the corresponding gradient value is not zero. Suppose that the first-order partial derivatives for the objective function in problems (1)(9) evaluated at exist, which can be expressed as where the first-order derivatives with respect to signal settings and flows are derived from Chiou 3 and the second item are from the sensitivity analysis for network flows in Patriksson 5. Let A denote the coefficient matrix and B the constant vector in constrains (2)(5) the problems (1)(9) can be rewritten as nts In the followings, we apply Fletcher & Reeves conjugate gradient method to a linear constraint set as given in (14) and (15) by introducing a matrix in projecting the gradient of the objective function onto a null space of active constraints as in (2)(5) with equality in order to effectively search for implementable points. Theorem 1 (Projected conjugate gradient (PCG) method). Consider the problem in (14) and (15) a sequence of feasible iterates can be generated according to where is the conjugate gradient direction determined by (11) and is the step length minimizing along for which is within the feasible region defined by (2)(5). Suppose that has full rank at which is the gradient of active constraints with equalities in (2)(5) and the projection matrix is of the following form: A modified search direction can be determined in the following form: Then the sequence of feasible points generated by the projected conjugate gradient method monotonically decreases the performance value, whenever Proof. Following the results of Lemma 1, we have Multiply Eq. (20) by projection matrix it becomes Thus for sufficiently small we have ntsBecause by definition is the step length which minimize Z1 along from it implies which completes this proof. Theorem 2 (PCG method as In Theorem 1, when if all the Lagrange multipliers corresponding to the active constraint gradients with equalities in (2)(5) are positive or zeros, it implies the current is a KKT point. Otherwise choose one negative Lagrange multiplier, say and construct a new of the active constraint gradients by deleting the jth row of which corresponds to the negative component and make the projection matrix of the following form: The search direction then is determined by (18) and the results of Theorem 1 hold. Proof. Let be a negative component of the Lagrange multiplier and defined in (17), we show By contradiction, suppose and let then (25) can be rewritten as For any there exists a corresponding jth row, , of the active constraint in (2)(5) and such that We subtract (27) from (26) and it follows: since which contradicts the assumption that has full rank. Thus Corollary 1 (Stopping condition). If is a KKT point for problem in (14) and (15) then the search process may stop; otherwise a new descent direction at can be generated according to Theorems 1 and 2. 3.2. PCG solution scheme Consider the signalized road network optimal design problem in (1)(9), a PCG ntssolution scheme is given below: Step 1: Start with set index k=0. Step 2: Solve a traffic assignment problem with signal settings find the first-order derivatives by (13). Step 3: Use the projected conjugate gradient method to determine a search direction by (18). Go to Step 4. Step 4: If find a new in (16) and let Go to Step 2. If and all the Lagrange multipliers corresponding to the active constraint gradients are non-negative, is the KKT point and stop. Otherwise, find the most negative Lagrange multiplier and cancel the corresponding constraint and find a new projection matrix and go to Step 3. nts4. Numerical example and computational comparisons In this section, numerical experiments are conducted twofold. Firstly, numerical computations are carried out for showing the effectiveness and robustness of the proposed PCG as compared to those of traditional methods, e.g. LCA 4 and SAB 8 in a signal-controlled grid network with distinct sets of initials. Secondly, numerical comparisons are made continuously with various congestions by monotonically increasing travel demand scalars. 4.1. Grid network with signal-controlled junctions A 5* 5 grid-size network shown in Fig. 1 is considered for illustrating the effectiveness of the proposed PCG method in optimal design of signalized network where 4-pair OD trips and 9 signalized junctions are taken into account. Trip rates are set at 100 veh/h and link capacity is set 1800 veh/h. Numerical results with two sets of distinct initials are summarized in Table 1. As it seen in Table 1, the proposed PCG method achieved near global optimum of 40 veh within 2.6% of SO which is short for system optimization. The proposed PCG method significantly outperforms traditional method LCA in performance index values (PI) approximately by 39% and 31% and by 11% and 9% over the SAB method. Regarding the computational times taken by the proposed PCG method as it obviously seen from Table 1, it requires less computational efforts either in CPU times or in solving the corresponding traffic assignments as it does for LCA or SAB. 4.2. Grid signalized network with increasing congestions The second investigation is to test robustness of the proposed PCG method on increasingly congested grid signalized networks when compared to conventional methods. The increasing traffic congestion is caused by continuously increasing scalars to base travel demands. Computational results for the proposed PCG method are summarized in Table 2, where results are expressed in the relative difference percentage to SO at two sets of nts nts distinct initials. As it shown in Table 2, the proposed PCG method achieved the system optimum within 3% by consistently yielding the least relative difference percentages at 20 sets of travel demands. On the other hand, the conventional methods, like LCA and SAB, achieved the system optimum with much higher values of the relative difference percentages than those did the proposed PCG method. Furthermore, taking the LCA method for example, the values of the relative difference percentages were larger as traffic congestion becamesevere and the values on average were as high as 72 and 63 for two sets of initials. Consider the performance of SAB, it achieved the system optimum with 19%, which seems relatively better than those did LCA but still far less robust than those did the proposed PCG method. The implementations for carrying out the computational efforts on LCA, SAB and PCG methods have been conducted on SUN SPARC Ultra II workstation under operating system Unix SunOS 5.5.1 using C+ compiler. The stopping criterion for these solutions is set when the relative difference in the performance index value between the consecutive iterations less than 0.05%. 5. Conclusions and discussions In this paper, we presented a newly projected conjugate projection method (PCG) with global convergence to effectively solve the signalized road network problem. A 5 * nts5 grid road network with signal-controlled junctions is used to numerically demonstrate the robustness and superiority of the proposed method to conventional approaches in solving optimal design of signalized road network. As it reported from numerical comparisons under traffic congestion at various sets of initials, the proposed PCG method consistently outperformed other conventional methods with obvious significance and less computational efforts have been taken. It is envisaged to test the proposed PCG method on a wide range of general road signalized networks. Further investigations will be made for the signalized road network design problem with link capacity expansions and elastic travel demands. Acknowledgement The author is grateful for support from Taiwan National Science Council via grant NSC 95-2416-H-259-014. References 1 M. Abdulaal, L.J. LeBlanc, Continuous equilibrium network design models, Transportation Research B 13 (1979) 1932. 2 H. Ceylan, M.G.H. Bell, Traffic signal timing optimisation based on genetic algorithm approach, including drivers routing, Transportation Research B 38 (2004) 329342. 3 S-W. Chiou, TRANSYT derivatives for area traffic control optimization with network equilibrium flows, Transportation Research B 37 (2003) 263290. 4 B.G. Heydecker, T.K. Khoo, The equilibrium network design problem, in: Proceedings of AIRO90 Conference on Models and Methods for Decision Support, Sorrento, 1990, pp. 587602. 5 M. Patriksson, Sensitivity analysis of traffic equilibria, Transportation Science 38 (2004) 258281. 6 S. Suh, T.J. Kim, Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review, Annals of Operations Research 34 (1992) 203218. 7 R.A. Vincent, A.I. Mitchell, D.I. Robertson, User guide to TRANSYT, version 8. TRRL Report, LR888, Transport and Road Research Laboratory, Crowthorne, 1980. 8 H. Yang, S. Yagar, Traffic assignment and signal control in saturated road networks, Transportation Research A 29 (1995) 125139. 9 H. Yang, M.G.H. Bell, Models and algorithms for road network design: a review and some new developments, Transport Reviews 18(1998) 257278. nts网 路 控制信号的 优 化 设计 摘要 ： 考虑到整个 使用 信号的迟滞 ,网路信号的 优选设计 使平衡流的最小化 。这个问题可 以 由 用户平衡交通任务作为限制 ,对优化设计进行公式化 。在本文 中 , 一个突出 的共轭梯度方法被提出 ,解决全球 集中 的网 路优化设计 问题。数例 用于研究简单栅格 网 路 。 据 显示 , 在解决网路优化设计的平衡流 ,教传统的方法 ,被提出的方法更能出色的完成。 关键词 : 共轭梯度 ;网 路信号 ; 平衡 约束 ; 优化 1. 介绍 当考虑到使用者的路线选择， 网 路信号 的 优 化 设计 是需要 考虑 的 。 这个问题可 以 由 用户平衡交通任务作为限制 ,对优化设计进行 公式化 。 在过去 的几 十年 中 , 许多研究 者 通过优化技术 研究 了这个问题 1,6,9,2 。在本文里 , 一个 双层规划设计可以用 网 路控制信号 来阐述 。在上层 规划中 , 性能指标可定义为迟滞率的有利的线性综合，以及在整个交通流中每段时间停止的数量 ,可以用 TRANSYT 7 交通模拟来评估 。 工作性能相应的数学近似值 和 在下游汇合处的 媒介物的 平均延迟 ， 在 TRANSYT 模型 中已经 获得。在底层 规划中， 用户 的 平衡交通任务服从Wardrop 第一原则 ，可以作为最小化问题来阐明 。由于用户 的均 衡 分配约束 是非线性的 , 导致 网路 信号 化的 设计问题 是不突出的 , 因此 创建了 唯一最佳方案。 在本文里 , 一个计划的共轭梯度 (PCG)提议确定优选的信号设置和 全球 集中的 网络流程。 栅格网络 的数值 计算在 明显的交接处 提出的 PCG 方法胜过传统方法在各种各样 的初始 需求量。本文的 其他章节安排如下 。在下 一章 ,表明网路的信号化考虑到使用者的路线选择 。 在第三章中， 提出 的共轭梯度方法 在全球集中开发。 在第四章 , 一个栅格网络以信号控制的连接点在各种 初 始环节 之下被考虑提出的 PCG的数值 计算和传统方法被 提出 。 第五章是本文的 结论 与 讨论。 2. 问题公式化 2.1. 符号 这 篇论文的符号概述如下： G(N, l) 表示一个 指定的 网 路 , N 是信号 控制 连接并且 L 是套链接 表示套信号设置可变物 , 各自地为相互绿色周期、开始和期间 , nts那里 和 代表开始传染媒介 绿色的期间 为信号小组 j 在连接点 m 如同共同的周期的比例 代表有效的绿色的期间为链接 a 代表极小值绿色为信号小组 j 在连接点 m 代表清除时间在绿色的结尾为小组 j 和绿色之间开始为不相容的小组 l 在连接点 m 代表饱和流速在链接 a 代表第号 0 和 1 的一件收藏品为各对不相容的信号小组在连接点 m; 如果绿色开始为信号小组 j 进行那 l 和 否则 代表延迟的率在链接 a 代表中止的数量每单位时间在链接 a 表示套 OD 对 表示对 OD 对的旅行需求 表示套道路在 OD 对 w 之间 表示道路流程传染媒介 表示链接流程传染媒介 表示链接道路发生矩阵 , 如果道路 p 在 OD 对 w 之间使用链接 a 和 c 表示链接旅行时间 2.2. 信号化的公路网问题 信号化的公路网设计问题可能被公式化至于 依于 nts 和 是各自链接具体衡量的因素为延迟的率和数字中止每单位时间 被使用在 TRANSYT 。第一限制在在共同的周期并且 constraints(3)-(5) 在在绿色阶段、链接容量和清除时间在各个连接点。并且平衡流程 由解决发现以下交通分配问题。 分钟 依于 3. 一个解答方法为信号化的公路网设计问题 在这个部分 , 一次有效的查寻解决问题 (1)-(9) 被开发 , 为哪些下降的查寻方向引起并且新重复被创造。查寻过程将被终止在 KKT 点或 a 新查寻方向可能引起。在以下 , 一个计划的共轭梯度方法提议获得下降查寻方向。 3.1. 一计划的共轭梯度 methodIn 以下 , 一 个计划的共轭梯度方法提议获得下降查寻方向。 题词 1 (Fletcher & 穿过了共轭梯度方法 ) 。考虑一个连续能区分的作用 引起序列重复 根据 nts 每当 然后为点 xk 序列由共轭梯度方法引起 方向引起由 (11) 为一个跌宕的非线性问题是严密地的下降方向减少目标函数价值在对应的梯度价值不是零条件下。 那里优先处理的衍生物谈到信号设置和流程从 Chiou 3 被获得并且第二个项目是从灵敏度分析为网络流程在 Patriksson 5 。让 A 表示系数恒定的传染媒介压抑的矩阵和 B (2)-(5) 问题 (1)-(9) 可能被重写 在追随者 , 我们应用 Fletcher & 依照被给作为穿过了共轭梯度方法对一个线性限制被设置在 (14) 和 (15) 由介绍一个矩阵在射出目标函数的梯度活跃限制空空间 (2)-(5) 以平等为了有效地寻找 implementable 点。 定理 1 (计划的共轭梯度 (PCG) 方法 ) 。考虑问题在 (14) 和 (15) 序列可行重复星期 能引起根据 那里 是共轭梯度方向由 (11) 确定和 是步长度使 减到最小 是在可行的区域之内的 定义了 (2)-(5) 。 假设 , 有充分的等级在星期 , 是活跃限制梯度以平等 (2)-(5) 并且投射矩阵 是以下形式 : 一个修改过的查寻方向 sk+1 可能被确定以以下形式 : nts然后可行的点 序列由计划的共轭梯度方法单调地引起减少表现价值 , 每当 和 是从 (13) 。 证明。在题词以后 1 的结果 , 我们有 倍增 Eq 。 (20) 由投射矩阵 它成为 因而为充足地小 我们有 由于由定义使 减到最小沿 从 的 是步长度 , 它暗示 哪些完成这证明。 定理 2 ( ) 。在定理 1, 当 , 如果所有拉格朗日乘算器对应于活跃限制梯度以平等 (2)-(5) 是正面或零 ,它暗示当前的 是 KKT 点。否则选择一个消极拉格朗日乘算器 , 说 , 和修建新活跃限制梯度由删除 列 , 对应于消极组分 , 并且做投射矩阵以下形式 : 查寻方向由 (18) 和定理 1 举行的结果然后确定。 nts证明。让 是拉格朗日乘算器的一个消极组分并且 被定义 (17), 我们显示 。由矛盾 , 假设 并且让 然后 (25) 可能被重写 为任何 , 那里存在对应的 列 , , 活跃限制 (2)-(5) 并且 这样 我们减去 (27) 从 (26) 并且它随后而来 : 自从抗辩假定的 有充分的等级。因而 。 推论 1 (停止状态 ) 。 如果 是 KKT 点为问题在 (14) 并且 (15) 查寻过程然后可以中止 ; 否则一个新下降方向在 能引起根据定理 1 和 2 。 3.2. PCG 解答计划 认为信号化的公路网优选设计问题 (1)-(9), PCG 解答计划被给如下 : 步骤 1: 开始时 ,设置索引 k =0 。 步骤 2: 解决有信号设置的交通分配问题 ,发现优先处理的衍生物 (13) 。 步骤 3: 使用计划的共轭梯度方法确定查寻方向 (18) 。进入步骤 4 。 步骤 4: 如果 发现一新 在 (16) 和让进入步骤 2 。如果 和 所有拉格朗日乘算器对应于活跃限nts制梯度 non-negative, 是 KKT 点和中止。否则 , 发现最消极的拉格朗日乘算器和取消对应限制和发现一个新投射矩阵和进入步骤 3。 4. 数字例子和计算比较 在这个部分 , 数字试验做两重。首先 , 数字计算被执行为显示提出的 PCG 的有效率和强壮与那些传统方法 , 即 LCA 4 并且 SAB 8 比较 在一个信号控制的栅格网络以分明套最初。第二 ,数字比较用各种各样的 壅塞连续做由
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