序率最佳控制理論應用於水庫系統之優選操作

Abstract

本研究是以序率最佳控制理論求得水庫在考慮不確定因素下之較佳 (suboptimal)操作策略，並將水庫模擬成線性、序率之動態系統。傳統序 率動態規劃雖能考慮系統之不確定性，但是卻有隨變數增加而產生計算量 大、計算時間長之維度的困擾。而定率型之動態規劃理論雖已有方法能部 份解決計算量之限制，但不能考慮系統之不確定因素，因此本研究乃結合 線性二次高斯法（Linear Quadratic Gaussian ，LQG）及限制型微分動 態規劃理論（Constrainted Differential Dynamic Progra- mming ， CDDP）繼續發展限制型序率微分動態規劃理論（Constrainted Stotchastic Differential Dynamic Programming，CSDDP）。 LQG法， 乃利用控制學上之分離理論（Separation theorem）將問題分解成估測器 （Estimator）及調節控制器（Actuator）兩部份。估測器是用來估測狀 態變數之條件期望值，調節控制器則是將此狀態變數估測值乘上增益矩陣 （Gain matrix）而得到最佳控制值。由於LQG法只能考慮二次目標函數且 未含有不等號限制條件。因此乃結合LQG方法及CDDP理論使其能考慮二次 以上之目標函數及不等號之機率型限制條件（Chance con- strains）， 使本研究所提之方法更趨完備。本研究採用簡化之淡水河流域翡翠、石門 水庫作為演算範例，以驗證本研究所提之方法能適用於序率過程之水庫優 選操作，並提供水庫管理當局在操作上之參考。 This research apply the stochastic optimal control theorem to obtain a suboptimal policy for the operation of a reservoir system with uncertainty. The reservoir system will be formulated as a dynamic linear stochastic system. Although the traditional discrete type of stochastic dynamic programming can consider the system uncertainty, it demands large amount of computational power as the number of state variables increase, which is the curse of dimensionality. The deterministic differential dynamic programming can overcome the computational limitation but it cannot consider the system uncertainty. Therefore, to ease the computational limitation and consider the system uncertainty, this study develops a Constrained Stochastic Differential Dynamic Programming（CSDDP） algorithm by integrating the Linear Quadratic Gaussian（LQG） method and Constrained Differential Dynamic Pro- gramming （ CDDP）. The LQG method use the separation theorem to decompose the problem into an estimator and an actuator. The estimator is to estimate the expected value of the state variables and the actuator compute the optimal control by multiply the estimated expected states with the system gain matrix. The traditional LQG method can only solve a problem with quadratic objective function and without unequal sign constraints. The study combine the LQG method with the CDDP scheme to develop a CSDDP algorithm. The CSDDP algorithm can consider the system uncertainty and solve a optimal control problem with high order objective function and unequal sign constraints.This research will select the simplified Feitsui- Shihmen reservoir system on the Tanshui river basin as a test example to demonstrate the model capacity.