PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS
A.3 Linear Ordinary Differential Equations
A.3.4 Complex and Distinct Roots
It is known from the theory of polynomials that if polynomial (A.57) with real coefficients a1, . . . ,an−2,an−1,an has complex roots then the roots form complex conjugate pairsλi, λ∗i and so on.
Assume the following notation:
λi =αi+ji and λ∗i =αi−ji. (A.72) Obviously the condition of distinct roots (A.64) is satisfied for this pair asλi =λ∗i. Vandermonde’s determinant can be expressed using (A.62) as
1≤i≤j≤n
(λj−λi)=(λn−λn−1)(λn−λn−2). . .(λi−λ∗i). . .(λ3−λ2)(λ3−λ1)(λ2−λ1)=0, (A.73) and it is different from zero because (λi−λ∗i)=j2i =0. This makes it possible to assume the following fundamental system of solutions:
eλ1t, . . . ,eλit, eλ∗it, . . . ,eλnt, (A.74) which contains exponential functions ofλiandλ∗i.
Using Equation (A.67) for given integration constantsA1, . . . ,Ai, . . . ,An makes it possible to find the particular solution. As Vandermonde’s matrix in Equation (A.67) and its determinant are complex, it may be expected that the integration constants in the fundamental set of solutions will also be complex, that is
x(t)=. . .+Aieλit+Bieλ∗it+. . . , (A.75) where variables x,t∈Real and the integration constants Ai,Bi ∈Complex. Differentiation of (A.75) gives
˙
x(t)=. . .+λiAieλit+λ∗iBieλ∗it+. . . . (A.76) Integration constantsAi,Bican be calculated from the initial conditions assuming
x(t=0)=. . .+ xi+. . .= x
˙
x(t=0)=. . .+0+. . .=0. (A.77) Substituting these initial conditions into Equations (A.75) and (A.76) gives the following two simple equations:Ai+Bi= xiandλiAi+λ∗iBi =0. Solving these equations requires care because both
Ai,Biandλi,λ∗i are complex numbers. Expressing the equation in matrix form gives 1 1
λi λ∗i
Ai
Bi
= xi
0
or Ai
Bi
= 1
−j2i
λ∗i −1
−λi 1 xi
0
, (A.78)
where, according to (A.72),iis the imaginary part ofλi. Now one gets Ai= x 1
−j2i
λ∗i = xi+jαi
2i
Bi= x 1
−j2i
(−λi)= xi−jαi
2i
= A∗i.
(A.79)
This shows thatBi = A∗i. The general important conclusion is that for each pair of solutions eλit and eλ∗itthe integration constants resulting from the initial conditions form a complex conjugate
pairAi,A∗i. Hence the solutions of (A.75) is
x(t)= . . .+Aieλit+A∗ieλ∗it+. . . (A.80) where
Aieλit+A∗ieλ∗it=Aieαit(cosit+j sinit)+A∗ieαit(cosit−j sinit)
=eαit
(Ai+A∗i) cosit+j(Ai−A∗i) sinit
. (A.81)
Obviously (Ai+A∗i)=2 ReAiand j(Ai−A∗i)= −2 ImAiare real numbers equal to the real and imaginary parts of the integration constantAi, respectively. Hence Equation (A.81) is now
Aieλit+A∗ieλ∗it=eαit[2 ReAiãcosit−2 ImAiãsinit]. (A.82) Note that the left hand side of the equation contains operations on real numbers and the right hand side contains operations on imaginary numbers. This means that appropriate operations on complex numbersAi,Ai∗, eλit, eλi∗tmust result in the imaginary part of the termAieλit+A∗ieλ∗itbeing equal to zero so that the overall result is a real number. This is an important observation leading to the conclusion that for the discussed case of complex conjugate pairs of roots, the particular solution is of the form
x(t)=. . .+2 ReAiãeαitcosit−2 ImAiãeαitsinit+. . . . (A.83) Hence it can be concluded that operations on complex numbers connected with looking for the particular solution are unnecessary because, instead of the fundamental system of solutions given by (A.74), one can consider a fundamental system of solutions of the form
eλ1t, . . . ,eαitcosit, eαitsinit, . . . ,eλnt, (A.84) consisting of real functions. As sine and cosine functions are orthogonal, the solutions eαitcosit and eαitsinitare linearly independent. This can be checked by calculating the Wronskian of the fundamental system of solutions (A.84) and the corresponding Vandermonde’s determinant. The latter will contain terms proportional to (cosit−sinit)=0.
These considerations lead to an important conclusion:
Each complex conjugate pair of the rootsλi, λ∗i in the solutionx(t) of the differential equation (A.52) corresponds to real exponential functions eαitcositand eαitsinitbecause the imaginary parts of the solutions corresponding to the pairsλi, λ∗i cancel each other out.
There is another proof of the above statement using the theorem that if a complex function is a fundamental solution of a linear ordinary differential equation, then both the real and imaginary parts of that function also form the general solution. Proof of this can be found in a number of textbooks including Arnold (1992).
An examination of Equation (A.84) shows that the real rootsλiof the characteristic equation will produce exponential terms of the form eλitso that the roots are the reciprocals of time constants of the exponential terms. The complex conjugate root pairsλi=λ∗i =αi+ji of the characteristic equation will produce oscillatory terms eαitcositand eαitsinit. The imaginary parts of the roots are therefore equal to the frequencies of oscillation of each term and the real parts of the roots are the reciprocals of time constants of the exponential envelope of the oscillatory terms. The overall solution is stable if the real parts of all the roots are negative.
For the dynamics considered in this book, of particular interest is a second-order scalar equation corresponding to the equation of motion for the synchronous generator (Section 5.4.6), but now a solution to the second-order equation will be discussed when the roots of the characteristic equation are complex.
Example A3.4
Solve a second-order equation ¨x−2αx˙+(α2+2)x=0 with the initial conditions given by (A.66).
The characteristic equation isλ2−2αλ+(α2+2)=0. The roots areλ1=α+jandλ2= λ∗1=α−j. The fundamental system of solutions eλ1t, eλ∗1t results in the following Vander- monde’s determinant:
det 1 1
λ1 λ∗1
=λ∗1−λ1= −j2=0, (A.85) which shows that the fundamental system of solutions was well chosen and the general solutions is of the form
x(t)= A1eλ1t+B1eλ∗1t. (A.86) Equation (A.78) takes the form
1 1 λ1 λ∗1
A1
B1
= x
0
or A1
B1
= 1
−j2
λ∗1 −1
−λ1 1 x
0
. (A.87)
Hence
A1= xã+jα
2 and B1= xã−jα
2 =A∗1. (A.88)
After substituting (A.88) into (A.86) simple algebra gives the following particular solution:
x(t)= x
eαt[ωcost−αsint]. (A.89)
Obviously the solution can be obtained in a simpler way by assuming at the outset the funda- mental system of solutions given by (A.84), eαtcost, eαtsint, and the general solution
x(t)=C1eαtcosωt+C2eαtαsint. (A.90) Substituting the initial conditionx(t0)= xleads toC1= x/. Differentiating (A.90) and substituting ˙x(t0)=0 givesC2= −C1. Substituting the calculated constantsC1= −C2= x/
into Equation (A.90) gives the solution given by (A.89).
The solution (A.89) contains an expression [cost−αsint]. It corresponds to the cosine of angle differences: cos(t+φ)=[costcosφ−sintsinφ]. In order to obtain that form exactly, it is necessary to transform Equation (A.89) in the following way:
x(t)= x eαt
2+α2 √
2+α2cost− √ α
2+α2 sint
, (A.91)
where the expression in front of the square brackets was multiplied by√
2+α2 while the components in the square brackets were divided by the same term. Assuming the notation
sinφ= √ α
2+α2 and cosφ= √
2+α2, (A.92)
it is easy to check that sin2φ+cos2φ=1. With this definition of angle φ, Equation (A.91) becomes
x(t)= x
cosφeαtcos(t+φ). (A.93)
This form of the second-order equation is convenient because Equation (A.93) clearly shows that the solution is in the form of a cosine function with exponentially decaying amplitude forα <1 and exponentially increasing amplitude forα >1 or a constant amplitude forα=0. Inspection of (A.93) shows that the solution satisfies the initial conditionx(t=0)= x.
Second-order equations represent many physical problems. It is convenient to express a second- order equation in thestandard forminvestigated in the next example.
Example A3.5
Consider the standard form of a second-order equation ¨x+2ζ natx˙+2natx=0 wherenatis thenatural frequency of oscillationsandζis thedampingratio. The initial conditions are given by (A.66). The characteristic equation isλ2+2ζ natλ+2nat=0. When = −42nat(1−ζ2)≥0, that is the damping ratioζ≥1, the roots are real and the solution will contain the exponential terms discussed in Example A3.2 and Example A3.3. In this example the case of theunderdamped second-order systemwill be discussed when 0≤ζ <1. The characteristic equation will then have two roots forming a complex conjugate pair:
λ1,2= −ζ nat±jnat
1−ζ2 or λ1,2= −ζ nat±jd, (A.94) whered=nat
1−ζ2is thedamped frequency of oscillation(in rad/s) asnatis the natural frequency of oscillations (in rad/s) when damping is neglected, that is whenζ =0 andλ1,2=
±jnat. The solutionx(t) can be obtained in the same way as in the previous example or by using the solution (A.93) and substituting=dandα= −ζ nat. Hence
x(t)= x
cosφe−ζ nattcos(dt+φ), (A.95) whereφ= −arcsinζ.