The linearized model of the non-linear dynamic system

State equations, eigen-analysis and applications

3.3 The linearized model of the non-linear dynamic system

The equations describing the power system and its dynamics are non-linear. For analyzing the dynamic performance of the non-linear plant and the system, typically following a large disturbance such as a fault, a step-by-step integration of the non-linear equations is carried out to calculate the time-domain responses of the system variables. Such variables are gen- erator speeds and rotor angles, bus voltages, controller outputs, etc. Because the dynamic behaviour of the system depends very much on the location and the severity of the distur- bance, as well as the operating conditions, it is necessary to conduct a large number of so- called transient stability studies to characterise the dynamics of the system [4].

Linearizing the set of non-linear equations for a selected operating condition results in a new set of equations in a new set of variables. These variables are the perturbations about the steady-state quantities . The variable in the non-linear equations is related to the former pair by . The advantages of forming the linearized equations of a system are:

• All the powerful analytical methods developed in linear control theory are available for the analysis of the linearized dynamic system.

A

a₁₁ a₁₂  a_1n a₂₁ a₂₂  a_2n

. . . .

a_n1 a_n2  a_nn

= B

b₁₁ b₁₂  b_1m b₂₁ b₂₂  b_2m

. . . .

b_n1 b_n2  b_nm

=

n n n m

C

c₁₁ c₁₂  c_1n c₂₁ c₂₂  c_2n

. . . .

c_p1 c_p2  c_pn

= D

d₁₁ d₁₂  d_1m d₂₁ d₂₂  d_2m

. . . .

d_p1 d_p2  d_nm

=

p n p m

n1 x t = x₁ t x₂  t x_n t ^T m1 u t = u₁ t u₂  t u_m t ^T p1 y t = y₁ t y₂  t y_p t ^T

x

x₀ x

x = x x+ ₀

Sec. 3.3 Linearized models 69

• If the linearized system is stable at the selected steady-state operating point then, according to a theorem by Poincaré [5], the non-linear system is also stable at that operating point.

• The dynamic performance of the linearized system can be characterized by the loca- tion of its poles in the complex s-plane. Based on results in the theory covered in Sec- tions 2.8 and 2.9, the real parts of these poles (or damping constants) provide the information on stability, how well damped the modes¹ are, the nature of the transient response, etc. Such information cannot be gleaned directly from the results of time- domain analysis of the non-linear system.

If the transient responses of the linearized system to a disturbance are calculated, a question is ‘how accurate are the responses’. Poincaré proved that the response of the linearized sys- tem to a disturbance is exactly the same as that of the original non-linear system if the dis- turbance is vanishingly small. As explained in Section 1.1, the responses predicted by the linearized model are often sufficiently accurate for practical analysis and design purposes.

However, care must be exercised to take into account the nature of the non-linearities, the operating point and the size of the perturbation when deciding if the linearized model is practically applicable to the analysis being performed.

3.3.1 Linearization procedure

In [6] the set of nonlinear differential-algebraic equations (DAEs) describing the dynamic behaviour of the integrated power system are derived and are shown to be of the form:

, , ,

where the vector represents the n states of the system, the r algebraic variables, the m system input variables, and the ^p output variables.

At the steady-state operating condition, which is the equilibrium point about which the sys- tem is to be linearized, implies by definition that all rates of change are zero, , thus

, , . (3.12)

Assume the system is subjected to a small perturbation from the steady state such that

, , , . (3.13)

The perturbed the variables must satisfy (3.12). For example, in the case of the output y in (3.13)

.

Because the perturbations are small, the nonlinear function can be expressed as a first-order Taylor’s series expansion. Consider the i^th output, , :

1. See a note on eigenvalues, modes and stability in Section 3.5.2.

x· = f x   u 0 = g x   u y = h x 

x  u

y

x· = 0 f x ₀,₀,u₀ = 0 0 = g x ₀,₀,u₀ y₀ = h x ₀,₀

x = x₀+x  = ₀+ u = u₀+u y = y₀+y

y = y₀+y = h x ₀+x, ₀+ y = h x ,

y_i i = 1,,p

70 State equations; eigen-analysis Ch. 3

where the partial derivatives , ; and , are evaluated at the in- itial steady-state operating point .

Because in the above equation, it reduces to an equation in the perturbed variables:

. (3.14) Similar expressions can be derived for the two remaining functions in (3.12), for j=1, ... , n:

, (3.15)

. (3.16)

The sets of linearized equations, (3.14) to (3.16) are more conveniently expressed in matrix form,

(3.17) or more compactly:

(3.18)

where the element of the sub-matrix of the system Jacobian matrix is eval- uated at the initial steady-state operating point. The elements of the other sub-matrices are similarly defined.

The formulation of the linearized equations of the system as a set of DAEs possesses a num- ber of significant advantages, including:.

• The ‘natural’ formulation of the equations for devices and their controllers is exploited when building the set of system equations;

y_i y_i0+y_i h_ix₀, ₀ h_i x₁

---x₁  h_i x_n

---x_n h_i

₁

---₁  h_i

_r

---_r

+ +

+ + + +

= =

h_i

 x_a

--- a = 1,,n h_i

_b

--- b = 1,,r x₀,₀,u₀

 

y_i0 = h_ix₀, ₀

y_i h_i x₁

---x₁  h_i x_n

---x_n h_i

₁

---₁  h_i

_r

---_r

+ +

+ + +

=

x·

j

f_j

 x_a

---x_a f_j

_b

---_b f_j u_c

---u_c

c=1 m



+

b=1 r



+

a=1 n



=

0 g_k

x_a

---x_a g_k

_b

---_b g_k u_c

---u_c

c=1 m



+

b=1 r



+

a=1 n



=

x· = J_fxx J+ _f+J_fuu 0 = J_gxx J+ _g+J_guu

y = J_hxx J+ _h+J_huu

x· 0

y

J_fx J_f J_fu J_gx J_g J_gu J_hx J_h J_hu

x



u

=

i a

 ^th J_fx f_j

x_a

---

Sec. 3.4 Solution of the State Equations 71

• The equations for large systems are inherently highly modular and extremely sparse (i.e. the Jacobian matrix contains very large numbers of zeros). Highly efficient com- putational algorithms for processing modular and sparse matrices are exploited when computing frequency responses, transfer-function residues - as well as computing a subset of the system eigenvalues within a selected region of the complex s-plane.

• The modularity and sparsity of the system equations can be exploited when comput- ing eigenvalue sensitivities.

Elimination of the algebraic variables from the DAEs yields the conventional ‘ABCD’ form of the state equations, i.e.

, , (3.19)

,where . (3.20)

Equations (3.19) are the state and output equations and are a form commonly used in the literature on linear control theory to describe a system.