## 4 Pareto Based Controller Parameter Optimization  
### Multi-Objective Optimization Based on Pareto Algorithm  
Multi-objective optimization problems refer to problems that require simultaneous optimization when multiple objectives cannot be met simultaneously. It is difficult to achieve optimal results for multiple objectives under constraint conditions, but rather to obtain a set of Pareto optimal solutions that converge infinitely to the true Pareto frontier and are uniformly distributed. The elements in the set are called Pareto optimal solutions or non-inferior solutions. When studying the suppression of low-frequency oscillations in the power grid, the key is to combine decision information with the optimization process to obtain a satisfactory compromise solution.  
The traditional method adopts a decision-before optimization approach, where decision-makers first assign weight values to each objective, transforming multi-objective problems into single-objective problems. In recent years, with the development of various algorithms, there has been a method of optimization before decision-making, which involves first finding the multi-objective Pareto optimal solution set during the optimization process, and then selecting the optimal solution of one or more multi-objective problems from the Pareto optimal solution set by decision-makers according to specific decision-making methods.  
Pareto domination: For any two decision variables a and b:  
(1) Pareto domination: a domination b: a < b, if and only if any i, there is \(f_{i}(a)\leq f_{i}(b)\) and i exists, such that \(f_{i}(a)<f_{i}(b)\).  
(2) There is no difference between a and b: \(a\sim b\), if and only if \(a\
ot\subset b\) and \(b\
ot\subset a\) If the decision vectors a and b are Pareto-dominated, then all objective function values of decision vector a are less than or equal to all objective function values of decision vector b.  
Pareto optimal solution: The decision variable \(a\in X_{\rm f}\) is called Pareto optimal solution, any \(b\
otin X_{\rm f}\), \(f(b)\geq f(a)\).  
Pareto optimal set: The set of all Pareto optimal solutions.  
Optimization objective:  
Total output power of each unit  
\[F_{1}=\min\sum P_{\rm i}(1\leq i\leq n) \tag{45}\]Frequency offset:  
\[F_{2}=\min\Delta f \tag{46}\]  
Frequency change rate:  
\[F_{3}=\min\frac{\Delta f}{\Delta t} \tag{47}\]  
F1, F2, F3 are objective functions. \(P_{i}\) is the active power, \(\Delta f\) is the variation of frequency, and \(\Delta t\) is the variation of time.  
The optimal solution sorting method based on the Pareto dominance relationship classifies population solutions and allocates individual fitness values. For individuals \(p\) in a population \(P\), it is necessary to calculate the number of dominant \(p\) solutions \(p\) and the set of \(p\) dominated solutions \(S_{\rm p}\). The optimization process for the optimal solution set of the fitness function is as follows:  
Step 1: \(S_{\rm p}=\varnothing\), \(N_{\rm p}=0\);  
\(\forall q\in P\), \(if(p<q)\), \(S_{\rm p}=S_{\rm p}\cup\{q\}\);  
\(elseif(p>q)\), \(N_{\rm p}=N_{\rm p}+1\);  
\(if\big{(}N_{\rm p}=0\big{)}\), \(p_{\rm rank}=1\), \(F_{1}=F_{1}\cup\{p\}\).  
Step 2: \(i=1\), \(F_{i}\
eq\varnothing\), \(\forall p\in F_{i}\), if \(\exists q\in S_{\rm p}\), \(N_{\rm q}=N_{\rm q}-1\);  
\(if\big{(}N_{\rm q}=0\big{)}\), \(q_{\rm rank}=i+1\), \(Q=Q\cup\{q\}\);  
\(i=i+1\), \(F_{i}=Q\).  
Step 3: If \(F_{\rm i}=\varnothing\), the loop ends. Otherwise, move on to step two.  
Q is a non-empty set.