231 主=f0x)+28,(x4 [y=,,v (23) -1 It is interesting to note that the system input-output y1=h,(x) (17) relation becomes decoupled and linear under the above control law (21). IV.Simulations y=h (x) In the previous sections,the nonlinear controller is For convenience,Let's define developed for power systems.It has been shown u=tu....u theoretically that the design of a nonlinear controller amounts to the design of a linear controller based on a y=[y... (18) linear system which is obtained from feedback g(x)=81(x),,8n(x)] linearization.In this section,the performance of the h(x)=[h,(x),,hn(x】 nonlinear control strategy is illustrated.It should be mentioned that the sixth order power system model is which is in a more compact form used in the simulation.The nonlinear controller is 元=f(x)+g(x)H employed to stabilize the system and track the desired (19) y=h(x) operating point and load angles. Similar to the SISO system,the relative degree in MIMO systems with m outputs can be defined by m In Fig.1,the system becomes stable within seconds integers. under the disturbance of an increase of infinite bus Definition 6 (Relative degree):The system is said to voltage from 1.0 to 1.2.Furthermore,the operating point after 2.5 seconds is very close to the values before have a relative degree,..)at a point xo if the the occurrence of the disturbance.In order to see the following two conditions are satisfied for all x in a maximum ability of the nonlinear controller to stabilize neighborhood of x: the sixth order system,we increase the disturbance from ·L,L,*h(x)=01≤i,j≤mk<n:-1 1.0 to 2.4.From Fig.2,the nonlinear control is still able to stabilize the full order system.But,the disturbance 「LL,4-h(x)…L.Lh( larger than 2.4,the system becomes unstable quickly. Similarly for the tracking cases,the results are presented A(x)= … in Figs 3 and 4.The tracked load angle are 0.7 radian L.L."h (x) and the 1.2 radian,respectively.All the graphs clearly illustrate the ability of handling the control problem is nonsingular at o over a large range operation. We can obtain the following multi-machine system extended from the third order SMIB system with the Stabilization of full ordar model with 7415] form: (20) 05 15 25 The formula for nonlinear control law for MIMO system has the form of u=a(x)+B(x)v: [4,,.了=A(x,-L,f,，。-,f (21) 25 Applying the nonlinear transformation and the control law (21)results in a linear and controllable system represented by the m sets of third order equations: 讲= 15 25 站=z对 (22) =v; Fig.1 Stabilization of the full order nonlinear model On the other hand,a simple relation between the input v and the output y becomes:23 1 [y;3', ..., y?']T =[VI, .--, V,lT ... Y, =h,(x) u = [u,, ..., U,IT g(x) = [g,(x), -*-1 g,(x)l h(x) = [h,(x), ---, h,(x)l For convenience, Let's define : Y =[YI'.*-I Y,IT (1 8) which is in a more compact form i = f(x) + g(x)u Y = h(x) (19) Similar to the SISO system, the relative degree in MIMO systems with m outputs can be defined by m integers. Definition 6 (Relative degree): The system is said to have a relative degree { r, , . . . , r,,, } at a point x if the following two conditions are satisfied for all x in a neighborhood of x : L,,L,'~,(~)=o I<i,j<m k<ri-l 1 Lgt L, 't-lhl (x) . . . Lgm L, 'I-%, (x ) ... ... is nonsingular at x We can obtain the following multi-machine system extended from the third order SMIB system with the form: i'= fl'(X), fl(x), fgi(x)+,. ui ]- [ (20) The formula for nonlinear control law for MIMO system has the form of u = a(x) + p(x)v : (21) Applying the nonlinear transformation and the control law (21) results in a linear and controllable system represented by the m sets of third order equations: .. i; = 2; i; = 2; i; =vi (22) .. On the other hand, a simple relation between the input v and the output y becomes: It is interesting to note that the system input-output relation becomes decoupled and linear under the above control law (21). IV. Simulations In the previous sections, the nonlinear controller is developed for power systems. It has been shown theoretically that the design of a nonlinear controller amounts to the design of a linear controller based on a linear system which is obtained from feedback linearization. In this section, the performance of the nonlinear control strategy is illustrated. It should be mentioned that the sixth order power system model is used in the simulation. The nonlinear controller is employed to stabilize the system and track the desired operating point and load angles. In Fig. 1, the system becomes stable within seconds under the disturbance of an increase of infinite bus voltage from 1.0 to 1.2. Furthermore, the operating point after 2.5 seconds is very close to the values before the occurrence of the disturbance. In order to see the maximum ability of the nonlinear controller to stabilize the sixth order system, we increase the disturbance from 1.0 to 2.4. From Fig. 2, the nonlinear control is still able to stabilize the full order system. But, the disturbance larger than 2.4, the system becomes unstable quickly. Similarly for the tracking cases, the results are presented in Figs 3 and 4. The tracked load angle are 0.7 radian and the 1.2 radian, respectively. All the graphs clearly illustrate the ability of handling the control problem over a large range operation. Stabilization of full order model with k+20 74 151 il :OS5 0 u, 0.5 1 1-1 1.5 2 2.5 3 3.5 4 Time(secs) 1r l a x - 8377 - I 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(=-) I 0.9 Et -,, , ,I 0 0.5 1 1.5 2 2.5 3 3.5 4 Time(-) $0.85 Fig. 1 Stabilization of the full order nonlinear model