Research Journal of Applied Sciences, Engineering and Technology 2(8): 784-788, 2010
ISSN: 2040-7467
© M axwell Scientific Organization, 2010
Submitted date: September 10, 2010
Accepted date: October 27, 2010
Published date: December 10, 2010
The Realization of a Discrete-Time Deterministic Regulator for
Energy Renewable Multivariable Systems
M.S. Okundamiya and E.C. Obinabo
Departm ent of Electrical and Electronic Engin eering, Ambrose Alli U niversity,
P. M . B., 14 , Ekpom a, Edo State, Nig eria
Abstract: This study proposes an optimal control law for a linear constant energy renewal multivariable system
supplying power to protection and signaling circuits for switchgears and substation operations. The energy
storage mechanism creates a control source that dictates the frequency of the pow er supply voltage. The e ntire
analy sis was based essentially on the use of a vector of inputs and outputs, and a matrix of operations
characterizing the storage mechanism, assuming the system is both controllable and observable under arbitrary
state feedback. The property of the output regulator which was desirable from the view point of feedback
control desig n was clarified.
Key w ords: Decouplab le linear sy stems, energy ren ewal multivariable system , optimal discrete regulators,
switchgears and substation operations
INTRODUCTION
Electric pow er gen eration has rapidly increased in
recent years due essentially to the pressure of limited
energy resources and environmental policies. When the
load demand of the system is less than generation, there
is the need to store the excess en ergy in an efficient
system. However, due to the intermittent nature of energy
renew al, use of energy storage such as batteries becomes
inevitable to compensate for the fluctuations of power
when there is shortage of generation. Electric power
generation is categorized into generation, transmission
and distribution. The system is a complex multi-machine
interconnection with voltage and frequency dependent
loads; the main componen ts being generating stations,
transmission lines and distribution systems. Substations
are necessary for change of voltage (up or down) between
the generating, transmission, primary distribution and
second ary distribution systems, and for providing
switching for control of the network under both normal
and fault conditions. In cases where no voltage change is
involved the substation is called a switching station.
Batteries are used as auxiliaries for switchgears and
substations (Fig. 1) for supplying power to control
protection and signaling circuits. They require a charging
system, usually a rectifier which converts a mains (A.C .)
supp ly to D.C. supply. The usual mode is the standby
parallel operation, i.e., with mains supply available, the
charger unit sup plies the D.C . control voltage requ irement
while at the same time charging the batteries. In the event
of mains failure the batteries take over to supply the
control voltag e requ irement.
One of the major features of linear constant
multivariable system s is that each inp ut variable affects
the output variables. This has led to representations for
multivariable systems based on state or output feedback
and the use of a vector of inputs, a vector of outputs, and
a matrix of operations to characterize the input-o utputs
relation. It has been shown (Gao and Wang, 2004;
W ang, 1970) that for a subclass of decouplable linear
systems satisfying the criterion,
where m
is the number of inputs and outputs and n the dimension
of the state space, a minimum order realization can be
found such that the set {di }; i = 1,… , m defines a
necessary and sufficient condition for decoupling the
system by output feedback. This condition implies that the
system is both controllable and observable under arbitrary
state feedback. For this class of systems, it can be easily
seen that the set {d i }; i = 1,…, m is a complete set of
independent invariants, which thus defines a unique
canonical form (Silverman and Payne, 1971; Sato and
Lop resti, 1971) for finding a control law such that the
subsets of the output vector of the closed-loop system are
controllable by subsets of the input vector in an
independent man ner. The po ssible variations of this
approach pose the pro blem of finding existence of a
deco upling control law and characterization of the class of
decoupling control law (Majumdar and Choudhury, 1973;
Crem er, 1971; M orse and W onham, 1970), thereby
satisfying the necessa ry con ditions for acceptab le
dynamic response namely, steady state accuracy and
asymptotic decoupling (Hirsch a nd Sm ith, 2005).
Corresponding Author: E.C. Obinabo, Department of Electrical and Electronic Engineering, Ambrose Alli University, P. M. B.
14, Ekpoma, Edo State, Nigeria
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Res. J. Appl. Sci. Eng. Technol., 2(8): 784-788, 2010
Fig. 1: linear constant multivariable sub-transmission station
The discrete-time deterministic optimal control
problem is well established in the existing literature for
linear constant dynamic systems (Hirsch and Smith, 2005;
Anderson a nd M oore, 1968; Shaked , 1976 ). For this
system, the performance criterion is related to the
smallness of a norm represented by the relation:
was defined, where pi represent the adjoint or costate
variab les.
In this study, we propose an optimal control law for
a linear co nstant energy rene wal system utilizing the
degrees of freedom available in the closed-loop input
vectors, as well as the associated pant description using
state feedback where the process is assumed detectable
and contro llable.
MATERIALS AND METHODS
The energy renewal multivariable system considered
was a linear, time-invariant process assum ed co ntrollable
and represented by the state equations:
(1)
(3)
The requiremen t here was to determine an optimal
control function u(t), t , [to , t1 ] to minimize J, where the
matrices Q(t), and R(t) are assumed systematic and
nonnegative, and p ositive d efinite respectively, and G is
a non negative definite m atrix.
For optimal solution, the Hamiltonian:
(4)
The process S(A,B,C) was also assumed to be squ are
since the actual plant outputs and those defined by Eq. (4)
need not necessarily coincide (Chow et al., 1989). The
matrices B and C have full rank since the system has
independent inputs and o utputs. In addition CB has full
rank, that is, rank CB = m so that the system is therefore
clearly outpu t controllable.
The initial condition was given by:
(5)
(2)
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Res. J. Appl. Sci. Eng. Technol., 2(8): 784-788, 2010
For controllability, it was desired to take the system to a
terminal state
, where tf is the terminal time.
This
was
(13)
(14)
done by applying a suitable control
which is optimal in some sense.
Combining (13) and (14), the following equation was
obtained:
Conseq uently a quadratic pe rformance criterion of the
from:
(15)
(6)
The result (15) holds for all nonzero
and h ence S(t)
is an n×n symmetric matrix satisfying the m atrix Riccati
equation.
was obtained and minimized. Without loss of generality,
Q(t), R(t) and S f were assumed to be symmetric matrices.
Further S f, Q(t) and R(t) are respectively, positive semidefinite, positive definite matrices w ith Q(t) and R(t)
being continuous in t. an appropriate choice of S f, Q(t) and
R(t)
must
be
made
for obtaining acceptable
performance of the system (Krichman et al., 2001;
Lehtomaki et al., 1981 ; Grim ble, 1978). to obtain a linear
feedback law a cost functional of the form given in (6)
was desired to keep the problem mathematically tractable.
For optimal solution, the Hamiltonian:
(16)
W ith the terminal condition:
S(t f) = Sf
(17)
the optimal control law becomes:
(18)
where K(t) is the n×n feedback gain matrix given by:
(7)
K(t) = R-1 (t) B T (t) S(t) (19)
was defined, and by the maximum principle of Pontryagin
(Krichman et al., 2001) the following was obtained:
The corresponding optimal cost function was obtained as:
(20)
(8)
An optimal regulator (Xu and Yu, 2006; Tempo et al.,
1997) was then required to bring a subset of the system
states to given non-zero constant set po ints. Consequently,
we described the linear constant energy renewal
multivariable system by the general representation (4).
The and matrices were defined from Obinabo (2008) as:
(9)
Terminal condition:
(10)
and
Hence fro m (8),
and , respectively using the performance index of the
form:
(11)
Assuming that the solution to
as (10),
(t) has the sam e structure
(t) was represented by:
The model was shown to be reducible to an equivalent
interconnection of subsystems, where each representation
for a given system has the same input-output relations. the
steady state Riccati equation was obtained as follows:
(12)
Hence:
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Res. J. Appl. Sci. Eng. Technol., 2(8): 784-788, 2010
The state equation s we re obtained as:
Using these data, the steady state error between the states
and their set points w as found to b e zero.
RESULTS AND DISCUSSION
The com mon trend in the design of op timal regulators
places special premium on some aspects of the
performance in order to minimize energ y losse s. In this
study, this was done by quantifying the cost savings using
a defined performance index. The controller that
minimizes this function gives the optimal solution to the
problem. The adva ntage of the line ar qua dratic optimal
control appro ach is th at if the state feedb ack system is
realizable, the resulting closed-loop configuration known
as the optimal regulator is expected to have some
desirable sensitivity and robustness properties as in the
case
of
finite-dimensiona l systems. The work
reported in the existing literature (Xu and Yu, 2006;
Gorecki et al., 1989) presents fundamental results on the
existence and charac terization of optimal control by
Ricc ati equations with delay in state. For finite values of
(tf – t0 ), the direct solution method was used to compu te
the matrix Riccati Eq. (12) from the terminal time t = tf
with the condition S (tf) = Sf. from the solution S(tf), K(t)
was com puted and stored for t0 # t # tf and used when
required.
The regulator problem is a feedback control system
in which the reference input was constant for long
duration (Philips and Harbor, 2000), and often for the
entire duration of system operation. Succinctly, the
regulator corresponds to a terminal controller with
(tf – t0 ) ÷ 4. For ge neral time varying system s it is
impractical to solve for, and store K(t) for 0 # t-t # 4 by
the direct method considered in this study. For stationary
systems and period ic systems the regulator problem was
solved by co nsidering the A an d B matrice s as co nstant.
W hen R and Q were chosen as constant matrices
satisfying all the conditions stated earlier, then the matrix
Riccati Equation was obtained as (Payne and
Silverman , 1973):
The steady state Riccati equations were computed as
follows:
whe re S 0 is a constant matrix. Hence S 0 satisfies the
equation:
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Res. J. Appl. Sci. Eng. Technol., 2(8): 784-788, 2010
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(22)
which, is the steady state algebraic matrix Riccati
equation.
CONCLUSION
A generalized optimal regulator problem for
application to an energy renew al multivariable system
supplying power for substation operations was
formulated. This enabled optim al regulators for the
transient response to be computed for both open- and
closed-loop configurations. Conditions were developed
for the resulting state feedback to be realizable. The
quadratic performance criteria determined for the process
is the optimal regulator with the desirable sensitivity and
robustness of the form associated with the finitedimensional linear constant multivariable system. the
property of the output regulato r which was required for
optimal perform ance was obtained by defining the
weighting matrice s. This enab led the pole location
problem to be brought into the optimal regulator design,
and it was shown that if the generalized weighting
matrices were chosen appropriately, the generalized
optimal regulator prob lem could be reduced to one which
could be solved only by compu ting solu tions to
finite-dimensional Riccati equations (Burns, 1990;
Gibson, 1983).
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