#### 1.2 Optimization of Complex Technical Devices

##### 1.2.1 Design Hierarchy

Block-hierarchical representation of the design process, implemented with the creation of complex technical devices, leads to a problem of such complexity that can be effectively resolved by means of modern computing, and the results of the decision – understood and analyzed by experts. Typically, the design hierarchy of tasks is formed along functional lines for turbine can have the form shown in Fig. 1.1.

The uniformity of mathematical models of the subsystems of the same level and local optimality criteria make it possible to organize the process of multi-level design, providing maximum global quality criterion of the whole system, in our case – the turbine. This process is based on the idea of so-called multilevel optimization approximation scheme that involves aggregation of mathematical models of the subsystems in the hierarchy when moving upward and disaggregation based on optimization results when moving downwards.

The problem of optimization the subsystem parameters described by OMM has the form (1.5). It can be solved by the methods of nonlinear programming and optimal control, depending on the form of the equations and the optimality criterion of the OMM.

Consider the solution order for the problems hierarchy of the system parameters optimization. Input parameters of k-level subsystem form of the set of internal and external parameters of the higher (k–1)-level subsystem. Feedback is carried out at the expense of the influence of the output parameters B ⃗ “k–1 of the subsystem of k-level which with respect to the (k–1)-th subsystem is external. Complete vector of (k–1)-level external parameters, thus consists of a vector of external parameters B ⃗ ‘k–1 coming from the higher-level and  lower-level subsystems of vectors B ⃗ “k–1 ( Fig. 1.2).

Moving from the bottom up, we solve the problem of the form (1.6) at each k-level for all possible values of the vector of external parameters coming from a higher level. In this phase k-level variables are excluded from the internal parameters of the (k–1)-level model by effect of equations describing the k-level subsystem, and control – as a result of optimization. Thus, at each level above information is transmitted not about all, but only about the best projects of lower-level subsystems:

At the top, the 1-st level, from the problem (1.5) output parameters found, and predetermine external parameters of the level 2 subsystems, which makes it possible to restore the optimum parameters of the 2-nd level, solving the same problem (1.5). This disaggregation process extends to the lowest level, with the result that the optimal parameters are determined by all the subsystems that make up the complex technical systems.

To implement practically the described scheme is possible using FMM subsystems. In terms of the FMM problem (1.5) is written in a form similar to (1.6):

which immediately follows

which is quite similar to (1.8), but has the advantage that it is a known polynomial of the form (1.2).

Methods based on the use of FMM is characterized in that before starting to solve the optimization problem on (k–1)-th level, it is replaced from the OMM to FMM according to the condition (1.9). Driving multilevel optimization using FMM, is very flexible, allowing you to change the setting if necessary optimization tasks at any level due to changes in the components of vectors Q ⃗ (u ⃗, B ⃗’k1).

##### 1.2.2 A Numerical Method for the Implementation of the Multilevel Optimization Approximation Schemes

The current level of possibilities of computer technology and mathematics allow for a new approach to the organization of the block-hierarchical representation of the process of optimal design of axial turbine flow path (Fig. 1.1) and the information exchange between adjacent levels (Fig. 1.2). The essence of this approach lies in the application of the principle of recursion, provides automatic bypass facilities at all levels and solution for each object its local optimization problem in accordance with a predetermined scenario.

On the basis of this method created invariant subsystem of recursive object-oriented multi-criteria, multi-mode and multi-parameter optimization, providing solution of optimization problems, taking into account various types of parametric, structural, technological and functional limitations. Designed for its optimization techniques are universal, and the search for the optimal solution for each object is carried out in accordance with the scenarios of computing processes optimization.

Optimization scripts for all objects of all levels are formed and defined by set of components of the following vectors and lists:

1. optX – address list parameters to be optimized
2. lXmin, lXmax – vectors defining the allowable range of variation of parameters to be optimized
3. lYcq – address list of the object settings and quality criteria
4. lYw – object quality criteria weight vector
5. lYfl – address list of parameters and functional limitations
6. flMin, flMin – functional limitations permissible change vectors
7. lYd – address list of settings – parametric constraints
8. dlMin, dlMin – parametric constraints permissible change vectors
9. lReg – list of regime (changing during the operation of the facility) parameters
10.  sRegim – list of lines with the data on the values of operating parameters and the appropriate time of the object for these values;
11. lLine – address list of parameters whose values are changed in the process of optimization by linear interpolation between the same type of parameters to be optimized nearby objects
12. optM – method for solving the optimization problem of the local object

Forming all the lists, enumerated above, for all level objects and calling a recursive function, which includes a set of corresponding optimization algorithms, an automatic objects bypass and solving optimization problems for each of them is carried out.