It is shown that in uncontrollable linear system ẋ = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A − BK has the desired eigenvalues. The procedure is illustrated by simple example.
In the paper, the results of investigations on the location of generating units most affecting the angular stability of a large power system (PS) are presented. For their location, the eigenvalues of the PS model state matrix associated with electromechanical phenomena (electromechanical eigenvalues) were used. The eigenvalues were calculated on the basis of the analysis of the disturbance waveforms of instantaneous power of the generating units operating in the PS. The used method of calculating eigenvalues consists in approximation of the disturbance waveforms of generating units by the waveforms being the superposition of modal components. The parameters of these components depend on the sought eigenvalues and their participation factors. The objective function was defined as the mean square error between the approximated and approximating waveforms. To minimize it, a hybrid algorithm, being a combination of genetic and gradient algorithms, was used. In the instantaneous power waveforms of generating units most affecting the PS angular stability, the least damped or undamped modal components dominate. They are related to eigenvalues with the largest values of real parts. The impact of individual modal components on the disturbance waveforms of subsequent generating units was determined with the use of participation factors and correlation coefficients of electromechanical eigenvalues.
The grid integration of large-scale wind and solar energy affects the power flow of wind-PV-thermal-bundled power transmission systems and may introduce an unpredicted threat to the power system’s small signal stability. Meanwhile, a power system stabilizer (PSS) and static synchronous series compensator (SSSC) play an important role in improving the static and dynamic stability of the system. Based on this scenario and in view of the actual engineering requirements, the framework of wind-PV-thermal-bundled power transmitted by an AC/DC system with the PSS and SSSC is established considering the fluctuation of wind and photovoltaic power output and the characteristics of the PSS and SSSC. Afterwards, the situation model is constructed in the IEEE 2-area 4-unit system, and the influence of the PSS and SSSC on the system stability under different operating conditions is analyzed in detail through eigenvalue analysis and time-domain simulation. Finally, an index named the gain rate is defined to describe the improvement of the stability limitations of various wind-PV-thermal operating conditions with the PSS and SSSC. The results indicate (K) that the damping characteristics, dynamic stability and stability limitations for various wind-PV-thermal operating conditions of the wind-PV-thermal-bundled power transmission system can be significantly improved by the interaction of the PSS and SSSC.