## 4 Optimal Tuning of Damping Controllers
### Formulation of Optimization Problem  
For power system stability issues, the transition process time should be shorter, the oscillation amplitude should be small and disappear faster. In order to enhance system damping, the optimal constraints should be constructed for the purpose of rapid attenuation of the power-angle oscillation of the system generator after the fault and return to the steady state as soon as possible [19]. In the parameter optimization process, the main goal is to suppress power oscillation and maintain the  
Figure 4: Block diagram of the dual-channel additional damping controller. overall stability of the system. Second, torsional oscillation should be taken into account to stabilize the dynamic characteristics of the DFIG in the objective function. The objective function can be expressed as:  
\[\mathrm{J}=\int_{0}^{t}\sum\
olimits_{k=1}^{n}t\Big{[}\big{|}\delta_{k}(t)- \delta_{k0}\big{|}\Big{]}dt+\alpha\int_{0}^{t}t\big{|}T_{e}(t)-T_{0}(t)\big{|}dt, \tag{14}\]  
where \(t\) is the total simulation time, \(\delta_{k}(t)\) is the rotor angle of the \(k_{th}\) generator in the inter-area oscillation mode operation, \(\delta_{k0}\) is the rotor angle of the \(k_{th}\) generator in the rated condition, \(n\) is the number of generators, \(T_{e}\) is the electromagnetic torque of the DFIG, \(T_{0}\) is the electromagnetic torque steady state value, and \(\alpha\) is the corresponding weight coefficient. In this paper, the value of \(\alpha\) is 0.35. The objective is to minimize \(\mathrm{J}\) such that it satisfies the following inequality constraints:  
\[\left\{ \begin{array}{l}K_{s}^{min}\leq K_{s}\leq K_{s}^{max}\\ T_{1}^{min}\leq T_{1}\leq T_{1}^{max}\\ T_{2}^{min}\leq T_{2}\leq T_{2}^{max}\\ T_{3}^{min}\leq T_{3}\leq T_{3}^{max}\\ T_{4}^{min}\leq T_{4}\leq T_{4}^{max}\end{array} \right. \tag{15}\]  
Equation \(K_{s}\) represents the gain of the damping signal (usually 1-50). Equations \(T_{1}\), \(T_{2}\), \(T_{3}\), and \(T_{4}\) are the time constant of the lead-lag block in the damping controller (usually 1-10 s). The designed controller needs to have the optimization parameter as follows:  
\[\mathrm{H}=\Big{[}K_{Q},K_{P},T_{Q1},T_{Q2},T_{Q3},T_{Q4},T_{p1},T_{P2},T_{P3}, T_{P4}\Big{]}. \tag{16}\]  
The dynamic performance of the system is ensured by finely tuning the DFIG controller parameters. The development of the particle swarm optimization algorithm is rapid owing to its comprehensibility, ease of implementation, and applicability in many fields. The time-varying inertia weighting factor is introduced into the particle swarm optimization algorithm (PSO), which enables the algorithm to effectively control the local search and provide sufficient convergence for the global optimal solution [20]. According to the principle of particle swarm optimization, the basic idea of additional damping parameter optimization is that a population of size n is formed first and then the particle is optimized according to the trend of the objective function. The particle variable can be continuously updated so that the objective function is continuously reduced until the convergence condition is satisfied. The basic steps of the optimization process are as follows:  
(1) Modal analysis. The negative damping and weakly damped oscillation modes that threaten the system's safety and stability are determined through small interference analysis and the identification on the interconnected system. (2) Feedback signal preselection. In the analysis system, the state variable that can be measured by the wide-area measurement system is observable for the main mode, and the state variable with a large residue is selected as the candidate signal. For interval oscillation, the best wide-area signal is the rotor angle difference [21,22]. In order to determine the controllability and observability of the interval oscillation mode, the optimal installation position of the damping controller using the residue method are the generator 4 and the wind farm [23].