, TEQ(1,1,1)=k1*(-sin(delta10)/2)+k12*(-sin(delta120)/2)

, TEQ(1,1,2)=k12*sin(delta120)

, TEQ(1,2,2)=k12*(-sin(delta120)/2)

, TEQ(2,1,1)=k12*(-sin(delta210)/2)

, TEQ(2,1,2)=k12*sin(delta210)

, 2,2)=k2*(-sin(delta20)/2)+k12*(-sin(delta210)/2), TEQ, issue.2

, TEC(1,1,1,1)=k1*(-cos(delta10)/6)+k12*(-cos(delta120)/6)

, TEC(2,1,1,1)=k12*(cos(delta210)/6)

, TEC(2,2,2,2)=k2*(-cos(delta20)/6)+k12*(-cos(delta210)/6)

, %%Auxilliary tensors of quadratic and cubic coefficients Gg=zeros

, TEC

(. A=zeros, N. , and N. ). ,

(. B=zeros, N. , and N. ). ,

(. C=zeros, N. , and N. ). ,

(. Alpha=zeros, N. , and N. ). ,

(. Beta=zeros, N. , and N. ). ,

(. Gamma=zeros, N. , and N. ). ,

, OMEGA(i)^3*xi(i)^3*xi(p)*OMEGA(p)-8*OMEGA(i)^3*xi(p)*OMEGA(p)*xi(i)

*. Omega(i and ). , *xi(p)^2*OMEGA(p)^2*OMEGA(i)^2-12*xi(p)*OMEGA(p), p.3

^. *. , OMEGA(i)+OMEGA(p)^4)*G(p,i,i)/(64*OMEGA(i)^6*xi(i)^2-32*OMEGA(i)^4*xi(i)

, OMEGA, p.4

^. *. , OMEGA(i)^5*xi(i)^3*xi(p)*OMEGA(p)+192*OMEGA(i)^4*xi(i), +OMEGA, pp.6-128

^. , *xi(i)^3*OMEGA(i)^3*OMEGA(p)^3*xi(p)-12*xi(i)*OMEGA(i), pp.2-96

^. *. , OMEGA(p)^5*xi(p)+32*xi(i)^2*OMEGA(i)^2*OMEGA(p)^4*xi(p)^2-64*OMEGA(i)

, OMEGA(p)*xi(i)-64*OMEGA(i)^3*xi(p)^3*OMEGA(p)^3*xi(i))

*. , , pp.2-12

^. , OMEGA(p)*xi(i)*OMEGA(i)+4*xi(p)^2*OMEGA(p)^2)/(64*OMEGA(i)^6*xi(i)

*. , , pp.2-20

*. , , pp.2-16

^. *. , OMEGA, pp.6-128

^. *. , OMEGA(p)+192*OMEGA(i)^4*xi(i)^2*xi(p)^2*OMEGA(p)^2-96*xi(i)

^. *. , OMEGA(i)^3*OMEGA(p)^3*xi(p)-12*xi(i)*OMEGA(i)*OMEGA(p)^5*xi(p)+32*xi(i)

, OMEGA(i)^5*xi(p)*OMEGA(p)*xi(i)-64*, pp.2-64

, A.2. Parameters and Programs 219

, OMEGA(i)^3*xi(p)^3*OMEGA(p)^3*xi(i))

^. *. , i)^3*xi(i)+16*xi(i)^3*OMEGA(i)^3-24*xi(i

, OMEGA(i)^2*xi(p)*OMEGA(p)+xi(i)*OMEGA(i)*OMEGA(p), OMEGA, p.3

. ^. , OMEGA(p)^2*xi(i)*OMEGA(i))/(64*OMEGA(i)^6*xi(i)^2-32*OMEGA(i), p.2

^. *. , xi(i)^2*OMEGA(p)^2+64*xi(i)^4*OMEGA(i)^4*OMEGA(p)^2+20*xi(i)^2*OMEGA(i)

, OMEGA, p.2

*. Omega(i and ). , OMEGA(i)^5*xi(i)^3*xi(p)*OMEGA(p), pp.6-128

*. , OMEGA(i)^4*xi(i)^2*xi(p)^2*OMEGA(p)^2-96*xi(i)^3*OMEGA(i)^3*OMEGA(p), p.3

*. , , p.4

, OMEGA(i)^5*xi(p)*OMEGA(p)*xi(i)-64*OMEGA(i)^3*xi(p)^3*OMEGA(p), *... xi(i), p.3

). , ALPHA

*. , BETA

, A(p,i,j)=0

, B(p,i,j)=0

, C(p,i,j)=0

, C(p,j,i)=0

, ALPHA(p,i,j)=0

, BETA(p,i,j)=0

, GAMMA(p,i,j)=0

^. *. , GAMMA(p,j,i)=0; else A2=[OMEGA(j)^2+OMEGA(i)^2-OMEGA(p)^2,-2*OMEGA(j)^2*OMEGA(i)^2,-2*OMEGA(j)

;. *. +2*xi, OMEGA(p)*OMEGA(j)^2,-2*OMEGA(i)^3*xi(i)+2*xi(p)*OMEGA(p)

, OMEGA(i)^2 ;-2,-4*xi(i)^2*OMEGA(i)^2-8*xi(i)*OMEGA(i)*xi(j)*OMEGA(j)-4*

*. , xi(j)^2*OMEGA(j)^2+OMEGA(i)^2+OMEGA(j)^2-OMEGA(p)^2+4*xi(p)*OMEGA(p)

+. , OMEGA(i)+4*xi(p)*OMEGA(p)*xi(j)*OMEGA(j), 2*xi(i)*OMEGA(i)

, OMEGA(j)-2*xi(p)*OMEGA(p), 4*xi(i)*OMEGA(i)+2*xi(j)*OMEGA(j)-2*

, OMEGA(p) ; 2*xi(j)*OMEGA(j)-2*xi(p)*OMEGA(p),-2*OMEGA(i)^3*xi(i)

(. Aa=zeros, N. , N. , and N. ). ,

(. Bb=zeros, N. , N. , and N. ). ,

(. Cc=zeros, N. , N. , and N. ). ,

(. Aa2=zeros, N. , N. , and N. ). ,

(. Bb2=zeros, N. , N. , and N. ). ,

(. Cc2=zeros, N. , N. , and N. ). ,

, N for i=1:N for j=1:N for k=1:N V1=squeeze

. V2=squeeze, A(i:N,j,k))

. V4=squeeze, A(1:i,j,k))

, AA(p,i,j,k)=V1'*V2 + V3*V4

. V6=squeeze,

, A.2. Parameters and Programs 221

, V8=squeeze(B(1:i,j,k))

). Bb(p,i,j,k and . V8,

, V12=squeeze(C(1:i,j,k))

, CC(p,i,j,k)=V9'*V10 + V11*V12

(. R=zeros, N. , N. , and N. ). ,

(. S=zeros, N. , N. , and N. ). ,

(. T=zeros, N. , N. , and N. ). ,

(. U=zeros, N. , N. , and N. ). ,

(. Lambda=zeros, N. , N. , and N. ). ,

(. Mu=zeros, N. , N. , and N. ). ,

(. Nu=zeros, N. , N. , and N. ). ,

(. Zeta=zeros, N. , N. , and N. ). ,

, different loops to express all the particular cases for p=1:N for i=1:N if i~=p D1=(OMEGA(p)^2-OMEGA(i)^2)*(OMEGA(p), pp.2-9

, LAMBDA(p,i,i,i)=0

, ZETA

*. Omega(i)^3*xi, OMEGA(i)^4 ; 0,-OMEGA(p)^2+12*xi(p)*OMEGA(p)*, 226 Appendix A. Interconnected VSCs

R. (p,i,j,k and ). =0,

*. , k)^2+OMEGA(j)^2+OMEGA(i)^2, 0, 2*xi(p, p.2

^. , OMEGA(i)^2-2*OMEGA(i)^3*xi(i), 2*xi(p)*OMEGA(p)*OMEGA(j), OMEGA

*. Omega(j)^3*xi, ;. , and .. , 2*xi(p)*OMEGA(p)*OMEGA(k)^2-2*OMEGA(k)^3*xi(k)

, OMEGA(i), p.2

*. , OMEGA(i)^2+ OMEGA(j)^2-OMEGA(p)^2+OMEGA(k)^2+4*xi(p)*OMEGA(p), vol.0

*. , OMEGA(i)+4*xi(p)*OMEGA(p)*xi(j)*OMEGA(j)+4*xi(p)*OMEGA(p)*xi(k)

^. , OMEGA(k)-4*xi(i)^2*OMEGA(i)^2-4*xi(j)^2*OMEGA(j)^2-4*xi(k)^2*OMEGA(k)

*. , OMEGA(i)*xi(j)*OMEGA(j)-8*xi(i)*OMEGA(i)*xi(k)*OMEGA(k)-8*xi(j)

, OMEGA(j)*xi(k)*OMEGA(k),-2,-2,-2, 4*xi(j)*OMEGA(j)+4*xi(k)*OMEGA(k)

;. , OMEGA(p)+2*xi(i)*OMEGA(i), 4*xi(i)*OMEGA(i)+4*xi(k)*OMEGA(k)

;. , OMEGA(p)+2*xi(j)*OMEGA(j), 4*xi(i)*OMEGA(i)+4*xi(j)*OMEGA(j)

;. and .. , OMEGA(p)+2*xi(k)*OMEGA(k) ;-2*xi(p)*OMEGA(p)+2*xi(i)*OMEGA(i)

^. and .. , OMEGA(i)^2-4*xi(i)^2*OMEGA(i)^2+OMEGA(j)^2+OMEGA(k)^2, 2*OMEGA(j)

^. , OMEGA(k)^2, 0, 2*xi(p)*OMEGA(p)*OMEGA(k)^2-4*xi(i)*OMEGA(i)*OMEGA(k)

*. , OMEGA(k)^3*xi(k), 2*xi(p)*OMEGA(p)*OMEGA(j)^2-4*xi(i)*OMEGA(i)

*. , OMEGA(j)^2-2*OMEGA(j)^3*xi(j) ;-2*xi(p)*OMEGA(p)+2*xi(j)*OMEGA(j),-2*OMEGA(k)^2*OMEGA(i)^2, 2*OMEGA(i)^2,-OMEGA(p)^2+4*xi, OMEGA

, OMEGA(j)+OMEGA(j)^2-4*xi(j)^2*OMEGA(j)^2+OMEGA(i)^2+OMEGA(k), p.2

^. , OMEGA(k)^2, 2*xi(p)*OMEGA(p)*OMEGA(k)^2-4*xi(j)*OMEGA(j)*OMEGA(k)

*. , OMEGA(k)^3*xi(k), 0, 2*xi(p)*OMEGA(p)*OMEGA(i)^2-4*xi(j)*OMEGA(j)

*. , ) ;-2*xi(p)*OMEGA(p)+2*xi(k)*OMEGA(k),-2*OMEGA(j)^2*OMEGA(i)^2, 2*OMEGA(i)^2, 2*OMEGA(j)^2, OMEGA, pp.2-4

^. , OMEGA(p)*xi(k)*OMEGA(k)+OMEGA(k)^2-4*xi(k)^2*OMEGA(k)

*. , OMEGA(i)^2+OMEGA(j)^2, 2*xi(p)*OMEGA(p)*OMEGA(j)^2-4*xi(k)*OMEGA(k)

*. , OMEGA(j)^2-2*OMEGA(j)^3*xi(j), 2*xi(p)*OMEGA(p)*OMEGA(i)^2-4*xi(k)

*. , OMEGA(i)^2-2*OMEGA(i)^3*xi(i), 0 ;-2, 2*xi(p)*OMEGA(p), OMEGA

*. , OMEGA(i)^2-2*OMEGA(i)^3*xi(i)-4*xi(j)*OMEGA(j)*OMEGA(i)^2-4*xi(k)

*. , OMEGA(i)^2, 0,-2*xi(p)*OMEGA(p)+4*xi(j)*OMEGA(j)+2*xi(k), OMEGA

, OMEGA(k),-2*xi(p)*OMEGA(p)+4*xi(k)*OMEGA(k)+2*xi(j)*OMEGA(j),-OMEGA(p), p.2

. ^2+, OMEGA(p)*xi(j)*OMEGA(j)+4*xi(p)*OMEGA(p)*xi(k)*OMEGA(k)+OMEGA(i)

, ^2+OMEGA(k)^2-4*xi(j)^2*OMEGA(j)^2-8*xi(j)*OMEGA(j)*xi(k)*OMEGA(k), OMEGA(j)

*. , k)^2, 2*OMEGA(i)^2, 2*OMEGA(i)^2 ;-2, 2*xi(p)*OMEGA(p

*. , OMEGA(j)^2-4*xi(i)*OMEGA(i)*OMEGA(j)^2-2*OMEGA(j)^3*xi(j)-4*xi(k)

^. and .. , OMEGA(p)+4*xi(i)*OMEGA(i)+2*xi(k)*OMEGA(k), 0,-2*xi(p)*OMEGA(p)+4*xi(k)*OMEGA(k)+2*xi(i)*OMEGA(i), 2*OMEGA(j)

A. , Parameters and Programs, vol.227

;. *. Omega(p)^2+4*xi, OMEGA(p)*xi(i)*OMEGA(i)+4*xi(p)*OMEGA(p)*xi(k)

*. , OMEGA(k)+OMEGA(j)^2+OMEGA(i)^2+OMEGA(k)^2-4*xi(i)^2*OMEGA(i)^2-8*xi(i)

, OMEGA(i)*xi(k)*OMEGA(k)-4*xi(k)^2*OMEGA(k)^2, 2*OMEGA(j)^2 ;-2

;. *. , OMEGA(p)*OMEGA(k)^2-4*xi(i)*OMEGA(i)*OMEGA(k)^2-4*xi(j)

+. , OMEGA(k)^2-2*OMEGA(k)^3*xi(k),-2*xi(p)*OMEGA(p)+4*xi(i)*OMEGA(i), OMEGA

, OMEGA(j),-2*xi(p)*OMEGA(p)+4*xi(j)*OMEGA(j)+2*xi(i)*OMEGA(i, p.0

+. , OMEGA(k)^2, 2*OMEGA(k)^2,-OMEGA(p)^2+4*xi(p)*OMEGA(p)*xi(i)*OMEGA(i)

. ^2-, OMEGA(p)*xi(j)*OMEGA(j)+OMEGA(k)^2+OMEGA(i)^2+OMEGA(j)

, i)^2-8*xi(i)*OMEGA(i)*xi(j)*OMEGA(j)-4*xi(j)^2*OMEGA(j, vol.2

. Blastijkp=, H(p,i,j,k)+AA(p,i,j,k)+AA(p,k,i,j)+AA(p, p.0

, CC(p,k,j,i)+CC(p,j,k,i)

, CC(p,k,i,j)+CC(p,i,k,j)

, CC(p,j,i,k)+CC(p,i,j,k)

, BB

, BB

, BB

, Alastijkp); solrstu=Alastijkp\Blastijkp, p.1

, OMEGA(j), vol.2

, OMEGA(i)+xi(j)*OMEGA(j))*

, OMEGA(i)^2*T(p,i,j,k)

. Mu(p,i,j,k)-;-*, =U(p,k,i,j)+U(p,j,i,k)+U(p,i,j,k)-2*(xi(i)*OMEGA(i)+xi(j)

, OMEGA(j)+xi(k)*OMEGA(k))*S(p,i,j,k)

, =R(p,i,j,k)-2*xi(j)*OMEGA(j)*T(p,j,i,k)-OMEGA(i)^2*

, OMEGA(k)^2*

, OMEGA(k)^2*

). Zeta(p,j,i,k, j)^2*S(p,i,j,k)+T(p,k,i,j)+T(p,i,j,k)-2*(xi(i)*OMEGA(i)

, OMEGA(k))*

. Sym,

, Vf= sym('Vf

, Vm= sym('Vm

, Vr= sym('Vr

. Yr=zeros-;-*gn,

, Gn for m=1:Gn Yr(2*l-1,2*m-1)=Gt(l,m), vol.1

. Yr, *l-1,2*m)=-Bt(l,m)

. Yr, *l,2*m-1)=Bt(l,m)

, Yr(2*l,2*m)=Gt(l,m)

, %% Transformation matrices %% T1: from the $dq_{i}$ reference frame to the common reference frame %% T2: from the bus terminal to the generator transient axes %% T1 and T2 are defined in Chapter

, T2(2*l-1,2*l-1)=Ra(l)

, T2(2*l-1,2*l)=-Xlq(l)

, T2(2*l,2*l-1)=Xld(l)

, T2(2*l,2*l)=Ra(l)

, B1(l,m)=Y(2*l,2*m-1)

, for j=1:Gn eldq(2*j-1)=eld(j)

, *j)=elq(j)

, Idq=Ydelta*eldq

, %% Auxiliary variable-id,iq for j=1:Gn id(j)=Idq(2*j-1)

, iq(j)=Idq(2*j)

, end for j=1:Gn Vt(j)=sqrt((elq(j)-Xld(j)*id(j)/9)^2+(eld(j)+Xlq(j)*iq(j)/9)^2)

, Pe(j)=eld(j)*id(j)+elq(j)*iq(j)

, elx(j)=sin(delta(j))*eld(j)+cos(delta(j))*elq(j)

, ely(j)=-cos(delta(j))*eld(j)+sin(delta(j))*elq(j)

. Qe0=,

. Vref=zeros(gn, , vol.1

. Vref=,

. Vm0=,

. Vr0=,

. Vf0=,

, Gn-1 ddelta_Gn(j)=omega_s*(omega(j)-omega(Gn)), vol.1

, j))*(Pm(j)/9-Pe(j)/9-D(j)*(omega(j)-1)

, delq(j)=1/Tld0(j)*(Vf(j)-(Xd(j)-Xld(j))*id(j)/9-elq(j))

, deld(j)=1/Tlq0(j)*((Xq(j)-Xlq(j))/9*iq(j)-eld(j))

, dVm(j)=(Vt(j)-Vm(j))/T_r(j)

, dVr(j)=(K_A(j)*(1-T_1(j)/T_2(j))*(Vref(j)-Vm(j))-Vr(j))/T_2(j)

, T_1(j)/T_2(j)*(Vref(j)-Vm(j))+Vf0(j)-Vf(j))/T_e(j); end for j=Gn domega(j)=1/(2*H(j))*(Pm(j)/9-Pe(j)/9-D(j)*(omega

, delq(j)=1/Tld0(j)*(Vf(j)-(Xd(j)-Xld(j))*id(j)/9-elq(j))

, deld(j)=1/Tlq0(j)*((Xq(j)-Xlq(j))/9*iq(j)-eld(j))

C. Appendix, Kundur's 4 Machine 2 Area System F2c(p,l)=F2c(p,l)+F2(p,l,k)+F2(p,k,l)

+. , C3(p,l,m,n)=F2H2(p,l,m,n)+F3(p,l,m,n); i=1:(M/2) for j=1:(M/2) if j~=i c3, vol.3

, C3

+. , C3

+. , C3

+. , MN(2*i),MN(2*j-1), C3

, C3

+. , MN(2*j))=C3(MN(2*i-1), c3(MN(2*i-1)

+. , MN(2*j),MN(2*i-1),MN(2*j-1)), C3(MN(2*i-1)

+. , MN(2*j),MN(2*j-1),MN(2*i-1)), C3(MN(2*i-1)

+. , MN(2*i-1), C3(MN(2*i-1)

+. , MN(2*j-1), C3(MN(2*i-1)

, MN(2*j-1), C3(MN(2*i-1)

+. , MN(2*i))=C3(MN(2*i)

+. , C3

, MN(2*i-1), C3(MN(2*i)

+. , MN(2*i))=C3(MN(2*i-1), c3(MN(2*i-1)

+. , MN(2*i-1), C3(MN(2*i-1)

, C3(MN(2*i-1)

, i-1)=conj(x(2*i)); p=1:M for l=1:M for m=1:M for n=1:M

C. , Calculation of Normal Form Coecients %%%%%%%%%%%%%%%%AVR Initialization%%%%%%%%%%%%%%%%%%%%%%% Tr=ones

. Ka=zeros,

. Kp=zeros,

. Ki=zeros,

. Kd=zeros,

. Td=ones,

. Ka=ones,

. Kad=zeros,

, %for DC4B Efd0 initialization Ta=ones

. Ke=ones,

. Aex=zeros,

. Bex=zeros,

. Te=ones,

. Kf=zeros,

. Tf=ones,

. Efdmin=zeros,

. Efdmax=zeros,

. Efdmin_dc=zeros,

. Efdmax_dc=zeros,

, len_exc=size(exc_con

. Vref_manual=ones,

, for i=1:1:len_exc(1) Exc_m_indx=exc_con(i,2);%present machine index Vref_Manual(Exc_m_indx)=0

, if(exc_con(i,3)~=0)

, Tr(Exc_m_indx)=exc_con(i,3)

, if(exc_con(i,5)~=0)

. Ta, Exc_m_indx)=exc_con(i,5)

, if(exc_con(i,1)==1)

, Kp(Exc_m_indx)=exc_con(i,16)

, Kd(Exc_m_indx)=exc_con(i,17)

K. , Exc_m_indx)=exc_con(i, vol.18

. Td, Exc_m_indx)=exc_con(i,19)

, Ke(Exc_m_indx)=exc_con(i,8)

, Te(Exc_m_indx)=exc_con

D. Appendix, New England New York 16 Machine 5 Area System Kf(Exc_m_indx)=exc_con(i,14)

, Tf(Exc_m_indx)=exc_con(i,15)

. Bex, Exc_m_indx)=log(exc_con(i,11)/exc_con(i,13))/(exc_con(i,10)-exc_con(i,12))

. Aex, Exc_m_indx)=exc_con(i,11)*exp(-Bex(Exc_m_indx)*exc_con(i,10))

K. , Exc_m_indx)=exc_con(i,4)

, Kad(Exc_m_indx), p.1

, Efdmin_dc(Exc_m_indx)=exc_con

, Efdmax_dc(Exc_m_indx)=exc_con(i,6)

, KA(Exc_m_indx)=exc_con(i,4)

, Efdmin(Exc_m_indx)=exc_con

, Efdmax(Exc_m_indx)=exc_con(i,6)

. Ks=zeros,

. Tw=ones,

. T11=ones,

. T12=ones,

. T21=ones,

. T22=ones,

. T31=ones,

. T32=ones,

. Vs_max=zeros,

. Vs_min=zeros,

, );%present machine index Ks(Pss_m_indx)=pss_con(i,3);%pssgain Tw(Pss_m_indx)=pss_con(i,4);%washout time constant T11(Pss_m_indx)=pss_con(i,5);%first lead time constant T12(Pss_m_indx)=pss_con(i,6);%first lag time constant T21(Pss_m_indx)=pss_con(i,7);%second lead time constant T22(Pss_m_indx)=pss_con(i,8);%second lag time constant T31(Pss_m_indx)=pss_con(i,9);%third lead time constant T32(Pss_m_indx)=pss_con(i,10);%third lag time constant Vs_max(Pss_m_indx)=pss_con(i,11)

, Sm= sym('Sm

. Eq_dash=sym, Eq_dash

. Ed_dash=sym,

. Eq_dash=sym, Eq_dash

. Edc_dash=sym,

. Psi1d=sym,

. Psi2q=sym,

, D.1. Programs, vol.263

. Efd=sym, Efd

. Pss3=sym,

. V_r=sym, V_f

. V_ki=sym, V_Ki

. V_kd=sym, V_Kd

. V_a=sym,

. V_f=sym, V_f

. Efdd2=sym,

+. , ig=(Eq_dash.*(xddd-xls)./(xdd-xls)+Psi1d.*(xdd-xddd)./(xdd-xls)

, 1i*(Ed_dash.*(xqdd-xls)./(xqd-xls)

. Psi2q, *(xqd-xqdd)./(xqd-xls)+Edc_dash)).*Yg

, delta))).*exp(-j*delta)

. Vq=real,

. Vd=imag,

+. , idq=(Eq_dash.*(xddd-xls)./(xdd-xls)+Psi1d.*(xdd-xddd)./(xdd-xls)

, xqdd-xls)./(xqd-xls)-Psi2q.*(xqd-xqdd)./(xqd-xls)+Edc_dash)-(Vq+1i*Vd)).* iq=real(idq)

, id=imag(idq

;. +. Teg=eq_dash, *(xddd-xls)./(xdd-xls)+Ed_dash.*id.*(xqdd-xls)./(xqd-xls)

, *id.*iq-Psi2q.*id.*(xqd-xqdd)./(xqd-xls)+Psi1d.*iq.*(xdd-xddd)./(xdd-xls)

, %%% Differential Equations of the Electromechanical Part ddelta=wB*Sm

-. *id, xq-xqd).*(-iq+(xqd-xqdd)./(xqd-xls).^2.*((xqd-xls)

, Ed_dash-Psi2q)))

. ^. , xd-xdd).*(id+(xdd-xddd)./(xdd-xls)

D. Appendix, New England New York 16 Machine 5 Area System *(Psi1d-(xdd-xls), *id-Eq_dash)))

, dPsi1d=1./Td0dd.*(Eq_dash+(xdd-xls)

, *iq-Psi2q

, %% Differential Equations of the PSS KPSS1=

, PSS_in1=Ks.*Sm, p.1

, PSS_in2=PSS_in1+(PSS_in1-PSS2), *KPSS1

, *KPSS3, pp.2-3

, PSS_in3=PSS_in2+(PSS_in2-PSS3), *KPSS3

, Vs=PSS_in3+(PSS_in3-PSS4), *KPSS6

, %% This vector records the numer of machines on which the PSSs are equipped. for i=1:1:len_pss(1) Pss_m_indx=pss_con(i,2);%present machine index dPSS1(Pss_m_indx)=1/Tw(Pss_m_indx)*(Sm(Pss_m_indx)*Ks(Pss_m_indx)-PSS1

, dPSS2(Pss_m_indx)=1/T12(Pss_m_indx)*(PSS_in1(Pss_m_indx)-PSS2

, dPSS3(Pss_m_indx)=1/T22(Pss_m_indx)*(PSS_in2(Pss_m_indx)-PSS3

, %dPSS3(i)=1/T22(i)*(((Ks*Sm-PSS1)*T11./T12-PSS2(i))*(T11(i)-T12(i))/T12(i)-PSS3(i)

, dPSS4(Pss_m_indx)=1/T32(Pss_m_indx)*(PSS_in3(Pss_m_indx)-PSS4

, N_PSS=[N_PSS i

, %% Differential Equations of the Exciter

, %% This vector record the number of machines equipped with exciter type ST1A

, %% Tihs vector record the number of machines equipped with exciter type DC4B for i=1:1:len_exc(1)

, if(exc_con(i,1)==1)%% The differential equations for exciter type DC4B

, DC4B_in1(Exc_m_indx)=Vs(Exc_m_indx)+Vref(Exc_m_indx)-V_r(Exc_m_indx)

. Kf, Exc_m_indx)

+. , DC4B_in2(Exc_m_indx)=DC4B_in1(Exc_m_indx)*Kp(Exc_m_indx)+V_Ki(Exc_m_indx)

, DC4B_in1(Exc_m_indx)*Kd(Exc_m_indx)/Td(Exc_m_indx)-V_Kd(Exc_m_indx)

+. , Exc_m_indx)=1/Te(Exc_m_indx)*(V_a(Exc_m_indx)-(Ke(Exc_m_indx)*Efd(Exc_m_indx)

. Efd, Exc_m_indx)*Aex(Exc_m_indx)*exp(Bex(Exc_m_indx)*Efd(Exc_m_indx))))

D. , Programs 265 dV_r(Exc_m_indx)=1/Tr(Exc_m_indx)*(Vt(Exc_m_indx)-V_r(Exc_m_indx))

, dV_Ki(Exc_m_indx)=DC4B_in1(Exc_m_indx)*Ki(Exc_m_indx )

. V_kd, Exc_m_indx))*1/Td(Exc_m_indx)

, dV_a(Exc_m_indx)=(DC4B_in2(Exc_m_indx)*Ka(Exc_m_indx)

. V_a, Exc_m_indx))*1/Ta(Exc_m_indx)

, dV_f(Exc_m_indx)=(Efd(Exc_m_indx)-V_f(Exc_m_indx))*1/Tf(Exc_m_indx)

, else dEfdd2(Exc_m_indx)=(Vt(Exc_m_indx)-Efdd2(Exc_m_indx))*1/Tr(Exc_m_indx)

. Efd, Exc_m_indx)=KA(Exc_m_indx)*(Vs(Exc_m_indx)+Vref(Exc_m_indx)-Efdd2(Exc_m_indx))

, end end V_r0=Vg; V_f0=Efd0.*Kad, vol.0

. Pss20=zeros,

. Pss30=zeros,

. Pss40=zeros,

E. , C. G. , and C. ). ,

, F=[ddelta;dSm;dEd_dash;dEdc_dash;dEq_dash;dPsi1d;dPsi2q;dEfd(N_DC4B)

, dV_r(N_DC4B).';dV_Ki(N_DC4B).';dV_Kd(N_DC4B).';dV_a(N_DC4B).';dV_f(N_DC4B)

, N_ST1A).';dPSS1(N_PSS).';dPSS2(N_PSS).';dPSS3(N_PSS).';dPSS4(N_PSS), p.2

, X=[delta;Sm;Ed_dash;Edc_dash;Eq_dash

. V_r, , vol.4

. V_ki, , vol.4

. V_kd, , vol.4

. V_a, , vol.4

. V_f, , vol.4

. Efdd2, , vol.1

, N_PSS, p.1

, N_PSS, p.2

, N_PSS, p.3

. V_r0, , vol.4

. V_ki0, , vol.4

. V_kd0, , vol.4

. V_ka0, , vol.4

. V_f0, , vol.4

. Efdd20, , vol.1

, N_PSS, p.10

, N_PSS, p.20

, N_PSS, p.30

. Eiv=eig,

L. Tu, . F2n, and . F3n]=nlcoefjordan,

, damping_ratioP=-real(EIV)./abs(EIV)

, N_State=size(EIV, vol.1