Flexibility Analysis Of Dynamic Systems With Delays

Chemical process systems often have to operate successfully throughout time in the presence of uncertainties.These uncertainties may be external factors such as feed compositions,product requirements and utilities which vary with market conditions.They can also be internal factors such as chemical reaction rate constants that are affected by catalysts,and heat transfer coefficients which change due to fouling in heat exchangers.The existence of uncertainties brings disturbance to the system and makes it unstable,which may result in quality problems,even accidents.Flexibility analysis associated with the system is an effective way to reduce this risk.However,an important factor,time delay,is usually ignored in the previous literature on this aspect.The delay phenomenon is common in chemical processes.For example,intermediate storage tanks and cascade systems may cause mass transfer lag.In order to deal with dynamic systems with uncertainties and delays,the delay factor is added in this paper to the flexibility analysis framework of dynamic systems proposed by Dimitriadis and Pistikopoulos,and the flexibility analysis framework of dynamic systems with fixed delays is thus established.Through the analysis of the influence of time delays on state variables,this paper presents a method to deal with the fixed time delays by using the method of orthogonal collocation on finite elements(OCFEM).On this basis,an approach for solving the flexibility problems with fixed time delays is proposed.In this approach,the proposed method for dealing with fixed time delays and an active constraint strategy are embedded into OCFEM,so that the time factor can be removed by discretization,and the delay factor can also be removed by the corresponding processing method.In addition,the active constraint strategy can serve to transform two-stage dynamic optimization problems into mixed integer nonlinear programming problems,which can be solved using MATLAB.In order to illustrate the applicability of the proposed framework and method,this paper takes a two stage chemical reactor train with delayed recycle as an example.Feasibility and flexibility analysis of the system with a fixed time delay has been taken.The results show that the proposed framework and method are feasible.Finally,based on the analysis of the uncertainty of time delays,it is concluded that the OCFEM based method fails to deal withproblems with uncertain delays.For this reason,an accurate method is proposed for estimating the flexibility index of systems with uncertain delays.