# Collaborative Optimization of Aerodynamics and Wind Turbine Blades
## 2 Collaborative Optimization  
The optimization procedure is the core component of MDO. Representative procedures are the multidisciplinary feasible method (MDF), individual discipline feasible (IDF), simultaneous analysis and design (SAND), collaborative optimization (CO), concurrent subspace optimization (CSSO), analytical target cascading (ATC), and bi-level integrated system synthesis (BLISS). In the study, CO is employed since it has the advantages of easy integration of software and parallel analyzing of different disciplines [48]. Unlike direct optimization, CO is characterized by a bi-level structure. A system-level problem coordinates the interdisciplinary coupling and consistency between the discipline-level subsystems is emphasized to improve the system-level performance objective.  
A typical CO framework is shown in Figure 1. It divides a coupled system into a two-level optimization structure consisting of a system level and a discipline level. At the system level, the coupling relationship between disciplines is processed and the globally optimal solution is solved. At the discipline level, each discipline independently undergoes optimization without considering the coupling relationship between disciplines. The information of design variables circulates between the system level and the discipline level until the optimization goal is achieved. At the discipline level, optimization is performed according to the desired design variables given by the system level. The obtained results then return to the system level to allow analysis of the constraint and optimization goal at the system level. After the two-level optimization is completed, the optimal design variables can be obtained and resultant disciplinary performance can be evaluated.  
The mathematical model at the system level optimization can be described as:  
\[\left\{\begin{array}{l}\underset{\text{d.v.}}{\text{Min}}\ f(z)\\ \text{s.t.}\ J_{i}^{*}(z)=\sum\limits_{j=1}^{N}\left(x_{ij}^{*}-z_{j}\right)^{2 }=0\\ \text{d.v.}\ z=\left[z_{j}\right]\end{array}\right., \tag{1}\]  
where \(f(z)\) is the optimization objective; \(J^{*}(z)\) is the constraint; \(x_{ij}^{*}\) is the optimal design variable; and \(z_{j}\) is the design variable.  
At the discipline level, the optimization formulation for the \(i\)-discipline can be described as  
\[\left\{\begin{array}{l}\underset{\text{d.v.}}{\text{Min}}\ J_{i}(x_{i})= \sum\limits_{j=1}^{N}\left(x_{ij}-z_{j}^{*}\right)^{2}\\ \text{s.t.}\ g_{i}(x_{i})\leq 0\\ \text{d.v.}\ x_{i}=\left[x_{ij}\right]\end{array}\right., \tag{2}\]  
where \(J_{i}(x_{i})\) is the optimization objective at the discipline level; \(x_{ij}\) is the design variable at the discipline level; \(z_{j}^{*}\) is the desired design variable obtained from the system level; and \(g_{i}(x_{i})\) is the constraint.  
As can be seen from (1), the consistency constraint at the system level is an equality constraint, which leads to poor robustness and slow convergence in optimization. Moreover, because the Karush-Kuhn-Tucker necessary condition is difficult to satisfy, a locally optimal solution might be obtained. To this end, relaxation of the constraint is required. In the study, a slack factor is introduced to relax the constraint. Combined with a slack factor, the constraint in (1) is rewritten as  
\[J_{i}^{*}(z)=\sum\limits_{j=1}^{N}\left(x_{ij}^{*}-z_{j}\right)^{2}\leq\xi \tag{3}\]  
where \(\xi\) is the slack factor, selected as a small positive constant.