An α-Model Parametrization Algorithm for Optimization with Differential-Algebraic Equations

Abstract

An optimization task with nonlinear differential-algebraic equations (DAEs) was approached. In special cases in heat and mass transfer engineering, a classical direct shooting approach cannot provide a solution of the DAE system, even in a relatively small range. Moreover, available computational procedures for numerical optimization, as well as differential- algebraic systems solvers are characterized by their limitations, such as the problem scale, for which the algorithms can work efficiently, and requirements for appropriate initial conditions. Therefore, an $\alpha$DAE model optimization algorithm based on an $\alpha$-model parametrization approach was designed and implemented. The main steps of the proposed methodology are: (1) task discretization by a multiple-shooting approach, (2) the design of an $\alpha$-parametrized system of the differential-algebraic model, and (3) the numerical optimization of the $\alpha$-parametrized system. The computations can be performed by a chosen iterative optimization algorithm, which can cooperate with an outer numerical procedure for solving DAE systems. The implemented algorithm was applied to solve a counter-flow exchanger design task, which was modeled by the highly nonlinear differential-algebraic equations. Finally, the new approach enabled the numerical simulations for the higher values of parameters denoting the rate of changes in the state variables of the system. The new approach can carry out accurate simulation tests for systems operating in a wide range of configurations and created from new materials.

1. Introduction

The advanced environments of computational optimization, which can take into account different types of constraints, are the tools that can be used by science and industry to solve the most pressing engineering problems. There are some branches of engineering and technology for which defining complex optimization tasks seems to be a natural issue, such as, e.g., chemical engineering [1,2], biotechnology [3], power engineering [4], or aviation [5] and astronautics [6]. It is worth emphasizing that there is a wide group of defined finitely dimensional optimization tasks with a classical scalar objective function and equality and/or inequality constraints that can arise in various branches of industry. This means that not only the specific scope of the problems belongs to the optimization area, but also well-known formulations reflect a common desire in process design, such as increasing efficiency, the reduction of losses, or reaching a compromise among the chosen goals.

Mathematical modeling, as well as a model-based optimization are important subjects of the heat and mass transfer area. Recently, Singh and Ghoshdastidar considered a new approach for computational simulation of heat transfer in a special case of alumina and cement rotary kilns [7]. The model relations take into account such items and phenomena as: (a) radiation exchange among solids, walls, and gas, (b) convective heat transfer from the gas to the wall and the solids, (c) contact heat transfer between the covered wall and solids, and (d) heat loss to the surrounding, as well as chemical reactions. In particular, the energy equation for the wall was computed by the finite-difference method. Finally, the performed simulations resulted in new insights into axial solids and gas temperature distributions, as well as the axial variation of chemical composition.

Najim and Krishnan designed a new mathematical model to obtain a similarity solution, which can be further applied for heat transfer analysis in progressive freeze-concentration-based desalination [8]. Then, the calculated similarity solution can predict the temperature distributions in the ice, the thickness of the ice, as well as heat flux. An important part of the mathematical model is the Scheil equation, useful in an analysis of simultaneous heat and mass transfer phenomena. The calculated similarity solution was used in a further analysis to investigate, e.g., the effect of the ice–liquid interface speed on heat transfer.

Li et al. designed and analyzed a new mathematical model of a novel evaporator [9]. In the presented new approach, the main rules of the vapor–liquid adjustment evaporator, as well as an appropriated configuration were considered. The authors indicated that the vapor–liquid adjustment evaporator can be further enhanced by an optimization procedure.

Najib and co-workers considered a new approach to calculate the transfer functions (g-functions) for computer simulation for the thermal performance of large-diameter, shallow bore, helical Ground heat exchangers [10]. It was indicated that the g-functions are generated using a validated numerical capacitance resistance model—helical ground heat exchangers for different bore diameters, bore depths, and helical pipe pitches. Moreover, a simplified resistance-based model, for the calculation of traditional borewell temperature-based g-functions, has also been presented.

Ghrissi et al. investigated a new mathematical approach based on the darcien model [11]. The prepared solution enabled the analysis of the influence of effective coupled parameters, which were heat, moisture, and air, on the evaporation performance of a porous layer. Then, to solve the designed system of equations, a combination of the Boltzmann method and the finite-volume method was proposed. Finally, the obtained results indicated that blowing air in winter conditions markedly affects the heat and mass exchanges at the interface between the porous layer and the channel.

The mentioned technological applications resulted in the problem functions expressed by the specified mathematical formulas. There are some important aspects that should be reflected in the problem formulation. In particular, the optimization task can be defined by considered the limitations and additional conditions:

• The single- and multi-objective optimization is related to the form of the objective function [12];
• Nonlinear formulas indicate whether local solutions can be detected [13];
• Optimization with complementarity constraints (mathematical programs with complementarity constraints (MPCC)) considers special pairs of restrictions [14,15,16];
• Mixed-integer nonlinear programming (MINLP) combines constraints with combinatorial problems [17].

Usually, in the optimization tasks related to the technological problems, a mathematical model of the considered process is included. Then, the mathematical model can be considered as a system of constraints. In particular, there are purely dynamical constraints in ordinary differential equation (ODE) form, dynamic constraints with distributed variables (partial differential equations (PDEs)), as well as dynamical constraints with discrete–continuous algebraic path constraints in the differential-algebraic equation (DAE) formulation. The optimization subject to the dynamical constraints is commonly known as dynamical optimization.

Moreover, solving the model equation system is a difficult task, because the obtained solution trajectories depend on many particular situations that may arise. The initial conditions for the performed numerical computations, as well as the course of the input variables can have a decisive influence on the numerical simulations and their final result. In the other words, the features, such as, e.g., model instability and its stiffness, can vary during an iterative optimization procedure. Therefore, in each computation step, there is a possibility that the computations can fail [18]. This is especially true when the initial solution is far from the final result.

In the literature, there are two main perspectives on the optimized equation system treatment. The first one comes from an obvious observation that in the actual process, all physical relations and laws must be always fulfilled. On this basis, the processes can be optimized in such a way that the model equations are always satisfied during the performed numerical calculations [19]. In this case, according to the parametrization methodology used, the dynamical optimization tasks result often in small- or medium-scale nonlinear optimization problems [20]. The second view is definitely different in this aspect. That is, the mathematical equations of a process may be violated during the optimization procedure. However, they should be met at the end of the last iteration to define a useful final solution. This methodology can be observed in the direct transcription method [21]. Therefore, there are two main ways for optimization with dynamical constraints: the sequential, as well as the simultaneous approaches.

The numerical sequential approaches for dynamic optimization are strongly based on the assumption that there is an available numerical procedure that is able to obtain solution trajectories satisfying the model equations. This assumption seems to be difficult to accept, because the observed increasing precision and complexity of technological processes are reflected in mathematical models, which are often highly nonlinear, unstable, numerically ill-conditioned, and possibly multi-stage [22,23,24]. Currently, simulation studies concerning the new solutions in the field of heat and mass transfer engineering indicate that an exact numerical solution of the differential-algebraic model of the system may not be possible [25]. This is especially true for new advanced mathematical models that are built according to the white-box approach. Moreover, by the appropriate selection of the model parameters, one can freely influence the numerical conditioning of the considered model equation system [26].

To improve the required stability condition, the multiple-shooting approach can be applied. This method is often used in numerical simulations of the nonlinear, as well as multi-stage production systems [27]. The multiple-shooting method can be combined together with other modifications, to ensure a failure-free operation of the designed computational algorithms [28]. Recently, to parametrize the trajectory described by the dynamical equations, variability constraints have been proposed [29]. Therefore, the appropriate model of the parametrization procedures can be applied to rewrite the optimization task in a well-tractable form.

Some of the most-often-used parametrization approaches result in large-scale nonlinear optimization tasks. Moreover, methodologies such as the direct transcription method [5] and collocation-based approaches [20] require experience in the numerical treatment of the differential-algebraic constraints. On the other hand, they are independent of the external computational procedures for the DAE system solution. Although the fully parametrization methods are able to efficiently co-operate with large-scale numerical optimization procedures, the computational experience indicates that the external DAE solvers can be applied for some classes of constraints and thus reduce the dimension of the optimization task significantly. It is worth noticing that medium-scale optimization problems can be successfully solved by personal computers. Therefore, the use of specialized computing stations seems not to be necessary.

The problem considered in this article is to overcome the difficulties associated with solving a system of the nonlinear differential-algebraic equations in a sequential optimization approach. One of the most important issues discussed in this work is an improvement of the optimization task’s tractability. This problem was considered in the context of a homotopy method, which was adjusted for solving optimization tasks with the DAE constraints [30].

The main contributions of this article are:

• The design of the $\alpha$-model parametrization procedure as a combination of homotopy and the multiple shooting method;
• The implementation of the $\alpha$DAE model optimization algorithm;
• The application of the $\alpha$DAE model optimization algorithm for the design of a counter-flow exchanger.

In the presented solution method, a well-known homotopy-based approach was used. Additionally, the designed method has never been used before to solve systems of differential- algebraic equations. Therefore, it is a significant extension of the applicability area of homotopy in solving optimization tasks subject to highly nonlinear differential-algebraic constraints.

Based on the possibilities given by the homotopy and multiple-shooting method, an $\alpha$DAE model optimization algorithm was designed and implemented. The detailed discussion of the considered optimization task, a formula of the model constraints, as well as the features of the new methodology are discussed in the next sections. The optimization problem taken into account is formulated in Section 2. The $\alpha$-model parametrization procedure is introduced in Section 3. Then, the main steps of the $\alpha$DAE model optimization algorithm are presented in Section 4. An application of the new approach to the design task of a heat and mass transfer process is described in Section 5. Finally, the presented considerations are concluded in Section 6.

2. The Problem Formulation

Let us consider a classical optimization task subject to the differential-algebraic model equations:

$$\underset{u\left(t\right)}{min}\mathcal{F}(y\left(t\right),z\left(t\right),u\left(t\right),p,t)$$

(1)

s.t

$$\begin{array}{ccc}\hfill \left(DAE\right)& & \left\{\begin{array}{ccc}\hfill \dot{y}\left(t\right)& =& f\left(y\right(t),z(t),u(t),p,t)\hfill \\ \hfill 0& =& g\left(y\right(t),z(t),u(t),p,t)\hfill \\ \hfill t& \in & \left[t_0{t}_f\right]\hfill \\ \hfill y\left(t_0\right)& =& y_0\hfill \\ \hfill z\left(t_0\right)& =& z_0\hfill \end{array}\right.\hfill \end{array}$$

(2)

where:

$$\mathcal{F}:{\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to \mathcal{R}$$

(3)

is an optimized performance index, $\mathcal{R}$ denotes a set of real numbers, $y\left(t\right)\in {\mathcal{R}}^{n_y}$ is a vector of state variables modeled by differential equations with $\dot{y}\left(t\right)=\frac{dy\left(t\right)}{dt}$, and $z\left(t\right)\in {\mathcal{R}}^{n_z}$ represents a vector of state variables described by algebraic constraints. Moreover, $u\left(t\right)\in {\mathcal{R}}^{n_u}$ denotes a vector of input functions; $p\in {\mathcal{R}}^{n_p}$ is a vector of global constant parameters; $t\in \mathcal{R}$ is an independent variable with a known a priori range $t\in \left[t_0{t}_f\right]$. The considered relations take the form of the differential-algebraic equations in a semi-explicit form with:

$$\begin{array}{cc}\hfill f:& {\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to {\mathcal{R}}^{n_y}\\ \hfill g:& {\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to {\mathcal{R}}^{n_z}\end{array}$$

(4)

To solve the model equations (2), consistent initial conditions:

$${\left[y\left(t_0\right)z\left(t_0\right)\right]}^T={\left[y_0{z}_0\right]}^T$$

(5)

need to be provided. For a given initial value of the input function $u\left(t_0\right)$, the consistent initial conditions must fulfill the relation:

$$g(y_0,z_0,u\left(t_0\right),p,t_0)=0,$$

(6)

which is equivalent to the algebraic constraints at the point $t_0$ of the independent variable domain.

It is worth explaining what the main difference between the differential-algebraic equations and differential-algebraic constraints is. The DAEs in their classical understanding can be solved by specialized single- or multi-step numerical schemes [31]. For the given consistent initial conditions (6), the DAEs (2) can be solved by Gear’s method or modified Runge–Kutta procedures [32]. Finally, starting from the consistent initial conditions, a set of feasible pointwise solutions can be found. This set can be further used to construct the solution trajectories $y\left(t\right)$ and $z\left(t\right)$. On the other hand, there is a group of methods that treat the DAE relations as the constraint system. While searching for a solution, the imposed restrictions do not have to be met. Even the starting solution may be infeasible. Finally, only the last result of the searching procedure should meet all of the model equations in the form of the differential-algebraic constraints. Therefore, it is possible to find a feasible solution without the knowledge about an explicit formulation of the constraint functions. The effectiveness of this approach in the context of simultaneous optimization has been described in the literature [33,34].

Although some solution procedures for the differential-algebraic equations have been just listed, it is worth presenting the main features of the direct shooting method—an approach designed for solving the highly nonlinear DAE systems. To solve the system of DAEs (2) characterized by a highly nonlinear dynamics, according to the multiple-shooting approach, the range of the independent variable is divided into an assumed number N of subintervals. Then, in each subinterval $t^i\in \left[t_0^i{t}_f^i\right]$, for $i=1,\dots ,N$ and:

$$t_0=t_0^1<t_f^1=t_0^2<\cdots <t_f^{N-1}=t_0^N<t_f^N=t_f,$$

(7)

the DAE system can be considered independently in each subinterval:

$$\begin{array}{ccc}\hfill \left(DA{E}^i\right)& & \left\{\begin{array}{ccc}\hfill {\dot{y}}^i\left(t^i\right)& =& f(y^i\left(t^i\right),z^i\left(t^i\right),u^i\left(t^i\right),p,t^i)\hfill \\ \hfill 0& =& g(y^i\left(t^i\right),z^i\left(t^i\right),u^i\left(t^i\right),p,t^i)\hfill \\ \hfill y^i\left(t_0^i\right)& =& y_0^i\hfill \\ \hfill z^i\left(t_0^i\right)& =& z_0^i\hfill \\ \hfill u^i\left(t^i\right)& =& u^i=\mathrm{const}\hfill \\ \hfill t^i& \in & \left[t_0^i{t}_f^i\right]\hfill \\ \hfill i& =& 1,\cdots ,N\hfill \end{array}\right.\hfill \end{array}$$

(8)

where $y_0^i\in {\mathcal{R}}^{n_{y^i}}$ and $z_0^i\in {\mathcal{R}}^{n_{z^i}}$ denote the vectors of the initial conditions for the differential and algebraic state trajectories, respectively. It was assumed that the input function $u^i\left(t^i\right)$ is constant in each subinterval.

The presented procedure, known as the multiple-shooting method, is a basis for computational optimization algorithms for problems with a highly nonlinear dynamics. This is because the dynamical relations can be solved more accurately on shorter subintervals than over one long range. Moreover, the initial conditions vectors $y_0^i$ and $z_0^i$, with the state trajectories’ continuity requirements:

$$\begin{array}{ccc}\hfill y^i\left(t_f^i\right)-y^{i+1}\left(t_0^{i+1}\right)& =& 0\hfill \\ \hfill z^i\left(t_f^i\right)-z^{i+1}\left(t_0^{i+1}\right)& =& 0\hfill \\ \hfill i& =& 1,\cdots ,N-1\hfill \end{array}$$

(9)

can be used to introduce additional discrete process constraints. Therefore, inequality constraints on the differential state trajectory:

$$\begin{array}{ccccc}\hfill y^L& \le & y\left(t\right)\hfill & \le & y^U\\ \hfill & & t\hfill & \in & \left[t_0{t}_f\right]\end{array}$$

(10)

with $y^L,y^U\in \mathcal{R}$, can be effectively considered in the following pointwise form:

$$\begin{array}{ccccc}\hfill y^L& \le & y^i\left(t_0^i\right)\hfill & \le & y^U\\ \hfill & & i\hfill & =& 1,\cdots ,N.\end{array}$$

(11)

The presented multiple-shooting approach enables us to approximate an infinite-dimensional optimization task by a finite-dimensional formulation. The further considerations concern a new approach for optimization according to the sequential approach rules, where the differential-algebraic Equation (8) is fulfilled in each step of a computational procedure.

Finally, the application of the multiple-shooting approach can be used to transform the classical task (1) and (2) to an optimization problem with pointwise-continuous constraints:

$$\underset{\mathbf{X}}{min}F\left(\mathbf{X}\right)$$

(12)

where $\mathbf{X}$ is a matrix of decision variables:

$$\begin{array}{ccc}\hfill \mathbf{X}& =& \left[\begin{array}{ccc}\hfill {\left({\mathbf{x}}_{y_0}^1\right)}^T& {\left({\mathbf{x}}_{z_0}^1\right)}^T& {\left({\mathbf{x}}_u^1\right)}^T\hfill \\ \\ \hfill {\left({\mathbf{x}}_{y_0}^2\right)}^T& {\left({\mathbf{x}}_{z_0}^2\right)}^T& {\left({\mathbf{x}}_u^2\right)}^T\hfill \\ \\ ⋮& ⋮& ⋮\\ \\ \hfill {\left({\mathbf{x}}_{y_0}^N\right)}^T& {\left({\mathbf{x}}_{z_0}^N\right)}^T& {\left({\mathbf{x}}_u^N\right)}^T\hfill \end{array}\right]\hfill \end{array}\in {\mathcal{R}}^{N\times (n_y+n_z+n_u)}$$

(13)

and the objective function:

$$F:{\mathcal{R}}^{N\times (n_y+n_z+n_u)}\to \mathcal{R}$$

(14)

should be optimized subject to the parametrized DAEs:

$$\begin{array}{ccc}\hfill \left(DA{E}^i\left(\mathbf{X}\right)\right)& & \left\{\begin{array}{ccc}\hfill {\dot{y}}^i\left(t^i\right)& =& f(y^i\left(t^i\right),z^i\left(t^i\right),u^i\left(t^i\right),p,t^i)\hfill \\ \hfill 0& =& g(y^i\left(t^i\right),z^i\left(t^i\right),u^i\left(t^i\right),p,t^i)\hfill \\ \hfill y^i\left(t_0^i\right)& =& {\mathbf{x}}_{y_0}^i\hfill \\ \hfill z^i\left(t_0^i\right)& =& {\mathbf{x}}_{z_0}^i\hfill \\ \hfill u^i\left(t^i\right)& =& {\mathbf{x}}_u^i\hfill \\ \hfill t^i& \in & \left[t_0^i{t}_f^i\right]\hfill \\ \hfill i& =& 1,\cdots ,N,\hfill \end{array}\right.\hfill \end{array}$$

(15)

the continuity constraints of the state trajectories:

$$\begin{array}{ccc}\hfill y^i\left(t_f^i\right)-{\mathbf{x}}_{y_0}^{i+1}& =& 0\hfill \\ \hfill z^i\left(t_f^i\right)-{\mathbf{x}}_{z_0}^{i+1}& =& 0\hfill \\ \hfill i& =& 1,\cdots ,N-1,\hfill \end{array}$$

(16)

as well as the initial conditions consistency constraints (6) in the new parametrized form:

$$\begin{array}{ccc}\hfill g({\mathbf{x}}_{y_0}^i,{\mathbf{x}}_{z_0}^i,{\mathbf{x}}_u^i,p,t_0^i)& =& 0\hfill \\ \hfill i& =& 1,\cdots ,N.\hfill \end{array}$$

(17)

The presented reformulation is a consequence of the multiple-shooting approach. Moreover, the continuous model equations can be solved, independently in each subinterval, by an outer numerical procedure. It is worth noticing that the application of the outer DAE solver is easy to implement and more flexible, because various numerical procedures can be used, such as ode15s in MATLAB or NDSolve in Mathematica.

On the other hand, the full parametrization of the state trajectories leads us to a problem formulation that is independent of the outer DAE system solution procedure. Unfortunately, it needs, in many cases, much more computational effort and experience. Then, to obtain a feasible solution, an efficient large-scale nonlinear programming procedure is necessary. In the full-parametrization approach, computational experience seems to be crucial for an efficient solution algorithm design and implementation. Therefore, in the present work, the following assumptions were made.

Assumption 1.

The system of differential-algebraic equations (15) should be solved by an outer numerical procedure.

Assumption 2.

The$DA{E}^i\left(\mathbf{X}\right)$systems (15) cannot be solved for such initial conditions:

$${\left\{\left[{\mathbf{x}}_{y_0}^i{\mathbf{x}}_{z_0}^i{\mathbf{x}}_u^i\right]\right\}}_{i=1}^N$$

(18)

which are far from the feasible solution.

Assumption 3.

The co-operated nonlinear programming procedure enables us to solve small- and medium-sized tasks.

Assumptions 1 and 2 indicate that an appropriate numerical procedure for solving the differential-algebraic system (15) is available. Moreover, the consistent initial conditions should be given as well, because a complete (or dense) survey of a multi-dimensional solution space is not an appropriate way for solving real-life technological design and optimization tasks (Assumption 2). Finally, Assumption 3 blocks the possibility of a full parametrization of the optimization problem.

According to the presented multiple-shooting parametrization procedure, as well as Assumptions 1–3, which reflect the available computing resources, the $\alpha$-model parametrization algorithm is proposed. The $\alpha$-model parametrization is based on influencing the variability and nonlinearity of the differential-algebraic system:

$$\begin{array}{ccc}\hfill \left(\alpha DAE\right)& & \left\{\begin{array}{ccc}\hfill {\dot{y}}_{\alpha }\left(t\right)& =& \alpha f(y_{\alpha }\left(t\right),z\left(t\right),u\left(t\right),p,t)\hfill \\ \hfill 0& =& g(y_{\alpha }\left(t\right),z\left(t\right),u\left(t\right),p,t)\hfill \\ \hfill t& \in & \left[t_0{t}_f\right]\hfill \\ \hfill y_{\alpha }\left(t_0\right)& =& y_{\alpha 0}\hfill \\ \hfill z\left(t_0\right)& =& z_0\hfill \\ \hfill u\left(t_0\right)& =& u_0\hfill \end{array}\right.\hfill \end{array}$$

(19)

with a constant $\alpha \in \left[01\right]$.

3. The α-Model Parametrization Algorithm

The differential equation in ordinary form:

$${\dot{y}}_{\alpha }\left(t\right)=\alpha f(y_{\alpha }\left(t\right),z\left(t\right),u\left(t\right),p,t)$$

(20)

with $\alpha \in \left[01\right]$, can be treated according to an explicit definition, as the variability of the state trajectory $y_{\alpha }\left(t\right)$. In other words, the right-hand side of Equation (20) indicates how rapidly the state $y_{\alpha }\left(t\right)$ varies or changes its value. This interpretation seems to be valid for these particular equations, especially if the state variability is constant or strictly constrained. Moreover, a comparison between the original dynamical equation with its approximation can result in some insight into the range of their similarity.

In a general nonlinear case, the variability rate, as well as its direction cannot be assumed to be constant even on a relatively small subinterval. This situation is often met in the design of heat and mass transfer processes. The proposition discussed in this work is to influence the variability by the $\alpha$-parameter multiplication. Then, the obtained solution, which is feasible for a given value of $\alpha$, can be used as an initial point in further computations with higher $\alpha$ values.

The $\alpha$-model parametrization approach enables us to consider the $\alpha$DAE model in three special cases:

• If $\alpha =0$, then the DAE model (2) takes the form:

  $$\begin{array}{ccc}\hfill \left(\alpha DAE\right)& & \left\{\begin{array}{ccc}\hfill {\dot{y}}_{\alpha }\left(t\right)& =& 0\hfill \\ \hfill 0& =& g(y_{\alpha }\left(t\right),z\left(t\right),u\left(t\right),p,t)\hfill \\ \hfill t& \in & \left[t_0{t}_f\right]\hfill \\ \hfill y_{\alpha }\left(t_0\right)& =& y_{\alpha 0}\hfill \\ \hfill z\left(t_0\right)& =& z_0\hfill \\ \hfill u\left(t_0\right)& =& u_0\hfill \end{array}\right.\hfill \end{array}$$

  (21)
  In this case, the dynamics of the system is completely reduced. For the assumed values $y_{\alpha 0}$, the problem is transformed into an optimization task with a pure algebraic system of constraints;
• If $\alpha =1$, then the $\alpha$DAE model (19) is equivalent to the original one (2);
• If $\alpha \in (0,1)$, then the $\alpha$DAE system (19) can be characterized by a dynamics similar to one of the extreme cases (1 or 2). The similarity can be influenced directly by the $\alpha$-parameter. Moreover, the value of $\alpha$ can reflect the robustness and efficiency of the applied numerical method for solving the differential-algebraic equations, as well as the progress of the optimization procedure.

The similarity between the original DAEs (2) and $\alpha$DAE models (19) can be approximated with the trapezoidal rule:

$$\begin{array}{ccc}\hfill {\int }_{t_0}^{t_f}\left(\dot{y}\left(t\right)-{\dot{y}}_{\alpha }\left(t\right)\right)dt& \le & {\int }_{t_0}^{t_f}|\dot{y}\left(t\right)-{\dot{y}}_{\alpha }\left(t\right)|dt\hfill \\ \\ \hfill & \approx & \sum _{i=0}^{m-1}\frac{1}{2}\left(|\dot{y}\left(t_i\right)-{\dot{y}}_{\alpha }\left(t_i\right)|+|\dot{y}\left(t_{i+1}\right)-{\dot{y}}_{\alpha }\left(t_{i+1}\right)|\right)\frac{|t_f-t_0|}{m}\hfill \\ \\ \hfill & =& \frac{|t_f-t_0|}{m}\sum _{i=0}^{m-1}\frac{1}{2}\left(|f(·;t_i)-\alpha f(·;t_i)|+|f(·;t_{i+1})-\alpha f(·;t_{i+1})|\right)\hfill \\ \\ \hfill & =& \frac{|t_f-t_0|}{m}\sum _{i=0}^{m-1}\frac{1}{2}\left(|f(·;t_i)|(1-\alpha )+\left|f(·;t_{i+1})\right|(1-\alpha )\right)\hfill \\ \\ \hfill & =& \frac{|t_f-t_0|}{m}\sum _{i=0}^{m-1}\frac{1}{2}\left(|f(·;t_i)|+|f(·;t_{i+1})|\right)(1-\alpha )\hfill \\ \\ \hfill & =& \frac{|t_f-t_0|}{m}(1-\alpha )\sum _{i=0}^{m-1}\frac{1}{2}\left(|f(·;t_i)|+|f(·;t_{i+1})|\right)\hfill \end{array}$$

(22)

with an assumed value of $m\in {\mathcal{N}}_{+}$, and ${\mathcal{N}}_{+}$ denotes a set of natural numbers greater than zero. The terms $(1-\alpha )$ and $\left(|f(·;t_i)|+|f(·;t_{i+1})|\right)$ clearly indicate that the similarity is dependent on the following features:

• The variability of the original system of DAEs (2);
• The actual considered value of the $\alpha$-parameter.

It can be clearly observed that the value of $\alpha$ can be used to influence the similarity between the original and $\alpha$DAE, as well as to change the variability of the considered $\alpha$DAE model (19). The $\alpha$-parameter can be modified iteratively, depending on the available numerical procedures.

3.1. An Extension for a Multiple-Shooting Method

For an assumed value of the parameter $\alpha$, the $\alpha$DAE (19) model can be solved with a multiple-shooting approach. The reformulation takes the following form:

$$\begin{array}{ccc}\hfill \left(\alpha DA{E}^i\left({\mathbf{X}}_{\alpha }\right)\right)& & \left\{\begin{array}{ccc}\hfill {\dot{y}}_{\alpha }^i\left(t^i\right)& =& \alpha f(y_{\alpha }^i\left(t^i\right),z^i\left(t^i\right),u^i\left(t^i\right),p,t^i)\hfill \\ \hfill 0& =& g(y_{\alpha }^i\left(t^i\right),z^i\left(t^i\right),u^i\left(t^i\right),p,t^i)\hfill \\ \hfill y_{\alpha }^i\left(t_0^i\right)& =& {\mathbf{x}}_{y_{\alpha 0}}^i\hfill \\ \hfill z^i\left(t_0^i\right)& =& {\mathbf{x}}_{z_0}^i\hfill \\ \hfill u^i\left(t^i\right)& =& {\mathbf{x}}_u^i\hfill \\ \hfill t^i& \in & \left[t_0^i{t}_f^i\right]\hfill \\ \hfill i& =& 1,\cdots ,N.\hfill \end{array}\right.\hfill \end{array}$$

(23)

The matrix of decision variables is similar to the matrix $\mathbf{X}$ (13), designed for the original DAE in the multiple-shooting formulation:

$$\begin{array}{ccc}\hfill {\mathbf{X}}_{\alpha }& =& \left[\begin{array}{ccc}\hfill {\left({\mathbf{x}}_{y_{\alpha 0}}^1\right)}^T& {\left({\mathbf{x}}_{z_0}^1\right)}^T& {\left({\mathbf{x}}_u^1\right)}^T\hfill \\ \\ \hfill {\left({\mathbf{x}}_{y_{\alpha 0}}^2\right)}^T& {\left({\mathbf{x}}_{z_0}^2\right)}^T& {\left({\mathbf{x}}_u^2\right)}^T\hfill \\ \\ ⋮& ⋮& ⋮\\ \\ \hfill {\left({\mathbf{x}}_{y_{\alpha 0}}^N\right)}^T& {\left({\mathbf{x}}_{z_0}^N\right)}^T& {\left({\mathbf{x}}_u^N\right)}^T\hfill \end{array}\right]\hfill \end{array}\in {\mathcal{R}}^{N\times (n_y+n_z+n_u)}$$

(24)

3.2. The Context of the Homotopy Method

The $\alpha$-model parametrization approach can be considered as a special case of the homotopy method. In its classical approach, homotopy is treated as a way to solve a difficult task with an appropriate initial solution. The initial solution should be a result obtained for an easier task, which in some sense is similar to the original one. Then, the solution of the difficult task can be obtained iteratively by solving combined difficult and easy problems. The homotopy for a system of dynamical equations in the formula (19) is defined as follows.

Let f, $\tilde{f}$, and H be continuous transformations such that:

$$f,\tilde{f}:{\mathcal{R}}^{n_y}\times {\mathcal{R}}^{n_z}\times {\mathcal{R}}^{n_u}\times {\mathcal{R}}^{n_p}\times \mathcal{R}\to {\mathcal{R}}^{n_y}$$

(25)

$$\alpha \in \left[01\right]\subset \mathcal{R}$$

(26)

$$H(f,\alpha )=\alpha f(·)$$

(27)

$$H(f,\alpha ):{\mathcal{R}}^{n_y}\times \mathcal{R}\to {\mathcal{R}}^{n_y}$$

(28)

with:

$$\dot{y}\left(t\right)=f(·),{\dot{y}}_{\alpha }\left(t\right)=\alpha f(·),{\dot{y}}_{{\mathbf{0}}_{n_y\times 1}}\left(t\right)\equiv {\mathbf{0}}_{n_y\times 1}=\tilde{f}(·)$$

(29)

and:

$$H(f,\alpha =0)=\alpha f(·)\equiv {\mathbf{0}}_{n_y\times 1}=\tilde{f}(·)$$

(30)

$$H(f,\alpha =1)=\alpha f(·)=f(·)$$

(31)

then:

$$f\sim \tilde{f}$$

(32)

Example 1.

The presented approach can be treated as a special case of the general homotopy relation:

$$H(f,\alpha )=\tilde{f}(·)+\alpha \left(f(·)-\tilde{f}(·)\right)={\mathbf{0}}_{n_y\times 1}+\alpha \left(f(·)+{\mathbf{0}}_{n_y\times 1}\right)=\alpha f(·)$$

(33)

where:

• for$\alpha =0$:    $H(f,\alpha =0)\equiv {\mathbf{0}}_{n_y\times 1}=\tilde{f}(·)$;
• for$\alpha =1$:    $H(f,\alpha =1)=f(·)$.

The homotopy relation of the dynamical parts of $\alpha$DAE (19) and the original DAE (2) will result in a valuable guess of the initial solutions $y_{\alpha }^i\left(t_0^i\right)$ and $z^i\left(t_0^i\right)$ for $i=1,\dots ,N$. The appropriate guess of the decision variables ${\mathbf{X}}_{\alpha }$ is crucial for the effective performance of the optimization procedure. Then, the obtained solution can be iteratively improved together with the changed value of the $\alpha$-parameter.

4. The New Solution Procedure

The stated optimization problem subject to the nonlinear differential-algebraic constraints takes a new form according to the multiple-shooting approach. Then, the optimization task is reformulated by the $\alpha$-model parametrization procedure. Finally, the ordered processing and computational steps of the designed procedure are presented as the $\alpha$DAE model optimization algorithm.

The $\alpha$DAE model optimization algorithm:

Step 1. Define the optimized objective function according to Equation (1) with the system

            of differential-algebraic constraints (2)

Step 2. Define $N\in \mathcal{R}$ and apply the multiple-shooting approach to obtain:

            - a parametrized form of the objective function (12),

            - the matrix of the initial solution (13).

            - the systems of the $DA{E}^i$, for $i=1,\cdots ,N$, (15)

            - the continuity constraints (16),

            - the initial conditions’ consistency constraints (17)

Step 3. Apply the $\alpha$-model parametrization algorithm to obtain:

            - the systems $\alpha DA{E}^i$, for $i=1,\cdots ,N$, (23)

            - a matrix ${\mathbf{X}}_{\alpha }$ (24)

Step 4. Define $n\in {\mathcal{N}}_{+}$ and a sequence ${\left\{{\alpha }_k\right\}}_{k=1}^n$ with ${\alpha }_1=0$ and ${\alpha }_n=1$

Step 5. For $k=1$ to n, solve

                  ${min}_{{\mathbf{X}}_{{\alpha }_k}}F\left({\mathbf{X}}_{{\alpha }_k}\right)$

            subject to

                  ${\alpha }_{k}DA{E}^i\left({\mathbf{X}}_{{\alpha }_k}\right)$, for $i=1,\cdots ,N$, (23)

                  the continuity constraints (16),

                  the initial conditions’ consistency constraints (17),

            to obtain ${\mathbf{X}}_{{\alpha }_k}^{\star }$, and substitute ${\mathbf{X}}_{{\alpha }_{k+1}0}={\mathbf{X}}_{{\alpha }_k}^{\star }$

Step 6. The matrix ${\mathbf{X}}_{{\alpha }_n}^{\star }$ is a final solution.

The finite number of n iterations needs to be assumed. Then, the unknown values in the decision variables’ matrix ${\mathbf{X}}_{{\alpha }_n}^{\star }$ can be obtained.

The computational complexity of the proposed algorithm is related to two main aspects:

• The complexity of the algorithm applied to solve finite-dimensional optimization tasks with pointwise-continuous constraints;
• The number n of $\left\{{\alpha }_k\right\}$ with $k=1,\cdots ,n$.

The computational effort related to the application of the $\alpha$DAE model optimization algorithm is dependent on the sequence of the ${\alpha }_k$ parameters, the number of multiple-shooting subintervals, as well as the size of the DAE system:

$$\underset{\mathrm{sequence}\mathrm{of}\mathrm{the}\left\{{\alpha }_k\right\}\mathrm{parameters}}{\underbrace{n}}\times \underset{\mathrm{shooting}\mathrm{subintervals}}{\underbrace{N}}\times \underset{\mathrm{size}\mathrm{of}\mathrm{the}\mathrm{DAE}\mathrm{system}}{\underbrace{(n_y+n_z+n_u)}}$$

(34)

The computational effort related to the problem solving with the proposed algorithm can improve the efficiency of the applied numerical optimization procedure. The appropriate values of the initial solution can prevent long-term calculations and the possibility of the premature termination of the algorithm as a result of finding a local minimum. The efficiency of the proposed procedure was tested in the task of a heat and mass transfer system’s design.

5. An Application in the Design of a Heat and Mass Transfer Process

The $\alpha$DAE model optimization algorithm was implemented in the MATLAB R2021b environment. Although the considered counter-flow exchanger design task has been discussed previously in other articles [26,35,36], the problem formulation, as well as the solution procedure considered in this work were substantially different.

The problem consisted of the objective function and the system of differential-algebraic equations in a semi-explicit form:

$$\underset{y_1\left(0\right)}{min}{(y_1\left(t_f\right)-30)}^2$$

(35)

subject to

$$\begin{array}{ccc}\hfill {\dot{y}}_1\left(t\right)& =& -B·(z_1\left(t\right)-y_1\left(t\right))/t_f\hfill \\ \\ \hfill {\dot{y}}_2\left(t\right)& =& C·(z_2\left(t\right)-y_2\left(t\right))/t_f\hfill \\ \\ \hfill {\dot{y}}_3\left(t\right)& =& C·(z_3\left(t\right)-y_3\left(t\right))/t_f\hfill \\ \\ \hfill 0& =& -E_{1}B·(z_1\left(t\right)-y_1\left(t\right))/t_f-C·(z_2\left(t\right)-y_2\left(t\right))/t_f-E_{2}C·(z_3\left(t\right)-y_3\left(t\right))/t_f\hfill \\ \\ \hfill 0& =& z_2\left(t\right)-(z_1\left(t\right)-D·(y_1\left(t\right)-z_1\left(t\right)))\hfill \\ \\ \hfill 0& =& z_3\left(t\right)-0.622·\frac{z_4\left(t\right)}{P_b-z_4\left(t\right)}\hfill \\ \\ \hfill 0& =& z_4\left(t\right)-6.107·e^{0.0726·z_2\left(t\right)-2.912·{10}^{-4}·{\left(z_2\left(t\right)\right)}^2+8.33·{10}^{-7}·{\left(z_2\left(t\right)\right)}^3}\hfill \\ \end{array}$$

(36)

The physical interpretation of the variables in the DAE model (36) is presented in

Table 1. The physical interpretation of the state variables in the model of a counter-flow exchanger.

Variable	Description	Type
$y_1\left(t\right)$	temperature in the 1st channel (°C)	differential state
$y_2\left(t\right)$	temperature in the 2nd channel (°C)	differential state
$y_3\left(t\right)$	humidity ratio in the 2nd channel (kg/kg)	differential state
$z_1\left(t\right)$	temperature of the plate surface in the 1st channel (°C)	algebraic state
$z_2\left(t\right)$	temperature of the plate surface in the 2nd channel (°C)	algebraic state
$z_3\left(t\right)$	humidity ratio on the plate surface in the 2nd channel (kg/kg)	algebraic state
$z_4\left(t\right)$	the static pressure (h Pa)	algebraic state

.

Additionally, a vector of the initial conditions was considered:

$$y\left(0\right)=\left[\begin{array}{c}c{y}_1\left(0\right)\\ 24.0\\ 10.4·{10}^{-3}\end{array}\right]$$

(37)

with global parameters:

$$p=\left[\begin{array}{c}B\\ C\\ D\\ E_1\\ E_2\\ P_b\end{array}\right]=\left[\begin{array}{c}40.0\\ 40.0\\ 0.058\\ 1.0\\ 2.5\times {10}^3\\ 1000.0\end{array}\right]$$

(38)

Moreover, a range of the independent variable domain was assumed:

$$t\in \left[0.0t_f\right]=\left[0.01.0\right].$$

(39)

The considered objective function with the system of differential-algebraic equations was solved according to the rules of the $\alpha$DAE model optimization algorithm:

• The number of shooting subintervals $N=20$;
• It was assumed that the $\alpha$ parameter takes the following values:

  $$\alpha =\{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\};$$

  (40)
• The solution trajectories of the parametrized $\alpha DA{E}^i$ models were computed by an outer function ode15s, which belongs to the MATLAB computational environment. The applied solver delivers support to the calculations of the consistent initial conditions;
• The MATLAB function fmincon was applied as a numerical optimization procedure to solve the new optimization task with the parametrized model constraints. The numerical values of the objective, as well as constraint functions were calculated with the outer ode15s solver.

The chosen value of $N=20$ subintervals resulted in:

$$N\times n_{y_1}+(N-1)\times n_{y_2}+(N-1)\times n_{y_3}=58$$

(41)

decision variables related to the initial values of the differential variables. In this special case, the number of decision variables was not affected by the size of $z\left(t\right)$ because the initial values of the algebraic state trajectories were computed by the ode15s procedure. However, to improve the computations, it is worth delivering the approximated values of $z^i\left(t_0^i\right)$ for $i=1,\cdots ,N$.

The parameters B and C are related to the rate of change of the state variables. Therefore, they can represent the physical features of the exchanger, such as, e.g., the number of transfer units. Unfortunately, the approaches such as single- or multiple-shooting were unable to solve the considered task for the higher values of the parameters B and C. The applicability range of the classical procedures has been clearly indicated in the previous research. In particular, in [26], it was numerically tested that a single-shooting procedure is efficient for B and C less than 16. Recently, a multiple-shooting approach was used to solve this task for values of B and C equal to 30 [35]. Now, this task is for the first time solved for higher values of the parameters B and C. Therefore, an extension of the computational capabilities in the multiple-shooting approach is a milestone of this research.

The obtained solution trajectories, for various values of $\alpha$, are graphically presented in Figure 1, Figure 2 and Figure 3, where the shooting subintervals are depicted by the dotted lines. Then, the $\alpha$DAE${}^i$ equations were solved by the applied DAE solver. Finally, the continuity of the trajectories was forced by the imposed continuity constraints. The presented results cannot be compared with other solutions, because the computations were not performed for these special values in the vector p (38). The presented results have a very original character and can be treated as a reference for further research in this field in the future.

6. Conclusions

In the present study, two main computational difficulties were considered: the problem dimensionality and the lack of an appropriate initial solution. The problem dimensionality was related to the selected way of the task parametrization. The mere fact of parametrization has the effect that an infinitely dimensional problem has a finite size. Then, the size of the problem can be adjusted according to the solution procedure used, as well as the available computing resources. The combination of the multiple-shooting method with the sequential approach for the optimization results with medium-sized nonlinear optimization problems can be solved on a personal computer. It is important to notice that the application of an outer DAE solution approach is necessary. The numerical optimization algorithm should co-operate with the outer DAE solver to obtain information about an objective function and its gradient. The appropriate initial conditions can be provided by the iterative steps of the homotopy-based approach.

Finally, the presented components were used to create a general method for solving difficult dynamical optimization tasks. The $\alpha$DAE model optimization algorithm can be implemented according to the main rules given in this work, but a specialized numerical procedures can be freely chosen by the user. The steps of the solution method were strictly defined, but this does not eliminate the possibility of adapting it to a specific real-life task.

The effectiveness of the new algorithm was tested on a heat and mass transfer design problem. This is one of a large number of engineering tasks where physical laws can be used to build a computer simulation model. This classical problem is difficult from a computational point of view, because the wanted solution features are reflected in the nonlinear problem functions. The observed nonlinear relations, as well as the lack of a trustworthy initial solution justify the need for the research presented in this work, as well as and the necessity to develop advanced numerical optimization algorithms.