Tuning Structurally Constrained Stabilizers for Large Power Systems

TUNING STRUCTURALLY CONSTRAINED STABILIZERS FOR LARGE POWER
SYSTEMS VIA NONSMOOTH MULTI-DIRECTIONAL SEARCH
Paulo C. Pellanda∗, Pierre Apkarian†, Nelson Martins‡
∗
†
IME - Instituto Militar de Engenharia - Electrical Engineering Dept.
Praca General Tibúrcio, 80, Praia Vermelha
22290-270 - Rio de Janeiro, RJ, Brazil
ONERA - Control Dept. and Paul Sabatier University - Maths. Dept.
2, Avenue Edouard Belin
31055 - Toulouse, France
‡
CEPEL - Centro de Pesquisas de Energia Elétrica
Avenida Hum s/n - Cidade Universitária, PO 68007
20001-970 - Rio de Janeiro, RJ, Brazil
Emails: [email protected], [email protected], [email protected]
Abstract— This paper proposes the use of a recently available algorithm combining multi-directional search
techniques with nonsmooth optimization methods to design structurally constrained controllers for large power
systems. The main aim is to provide stability with guaranty of a feasible user-defined minimum damping ratio for
the closed-loop dynamics. The multivariable Brazilian North-South interconnected power system model, which
has more than 16 hundred open-loop states, is used to test the proposed methodology. Three decentralized
lead-lag power system stabilizers are simultaneously tuned considering time performance specifications, which is
difficult to reach through available conventional control techniques.
Resumo— Este artigo propõe o uso de um algoritmo recentemente disponı́vel que combina técnicas de busca
multi-direcional com métodos de otimizacão não-suave no projeto de controladores com restricões estruturais
para sistemas de potência de grande porte. O objetivo principal é estabilizar o sistema com a garantia de um
fator de amortecimento mı́nimo viável, predefinido pelo projetista, para a dinâmica de malha fechada. O modelo
do sistema de potência interconectado norte-sul brasileiro, que possui mais de 1600 estados em malha aberta,
é utilizado para testar a metodologia proposta. Três controladores descentralizados do tipo atraso-avanco são
simultaneamente ajustados considerando especificacões de desempenho no domı́nio do tempo, o que é uma tarefa
difı́cil utilizando-se as técnicas de controle convencionais existentes.
Key Words— Nonsmooth Optimization, N P -Hard Design Problems, Pattern Search Algorithm, Moving Polytope, Power System Stabilizers, Multivariable Systems, Small-Signal Stability, Large Scale Systems, Structurally
Constrained Controllers, Fixed-Order Synthesis, Simultaneous Stabilization.
1
Introduction
In (Apkarian and Noll, 2006), the authors describe
an algorithm combining Multi-Directional Search
(MDS) (Torczon, 1991; Torczon, 1997) with nonsmooth techniques to solve several difficult synthesis problems in automatic control. More specifically, they show how to combine Direct Search
(DS) techniques with nonsmooth descent steps in
order to ensure convergence in the presence of nonsmoothness. Typical nonsmooth criteria appearing in control problems include the spectral abscissa, the maximum eigenvalue function and the
H∞ -norm. The proposed algorithm, here named
MDSN, is intended to solve several nonconvex
and even N P -hard problems, for which LMI techniques or algebraic Ricatti equations are impractical. Hence, the algorithm can be applied to constrained and unconstrained optimal control problems, including static and fixed-order output feedback controller design, simultaneous stabilization
and mixed H2 /H∞ synthesis. As the approach
avoids using Lyapunov variables, it is suitably applied in the synthesis of small and medium size
controllers for plants with large state dimension,
constituting an alternative to LMI- or BMI-based
nonlinear programming algorithms.
This paper presents the first large-scale application results of the MDSN algorithm, exploring the design of multiple fixed-parameter stabilizers in interconnected power system models.
The problem of obtaining decentralized lead-lag
controllers that provide good damping enhancement to interarea oscillation modes of large power
systems is a typical nonconvex control problem
solved in practice through conventional (smallsignal) analysis/design techniques (Yang and Feliach, 1994; Martins et al., 1999). However, the
efficiency and effectiveness of this procedure is
strongly dependent on the designer experience and
can lead to a great amount of trial-and-error before obtaining satisfactory specifications in terms
of time-domain properties. The MDSN algorithm
provides a systematic framework to solve this
problem and is used here to compute structurally
constrained Power Systems Stabilizers (PSS) by
maximizing the minimum closed-loop damping ratio.
The MDSN algorithm is briefly described in
Section 2. Section 3 presents the test system used
to verify the effectiveness of this method in solving large-scale problems. In section 4, numerical
results are discussed. Section 5 concludes.
2
The MDSN algorithm
This section contains a brief description of the
MDSN algorithm. For an in-depth discussion of
MDS in the smooth case the interested reader is
referred to (Torczon, 1991).
The MDS algorithm requires a ‘seed’ or base
point v0 and an initial simplex S in Rn with vertices v0 , v1 , . . . , vn . The vertices are then relabeled so that v0 becomes the best vertex, that is,
f (v0 ) ≤ f (vi ) for i = 1, . . . , n, where f (·) : Rn →
R is a C 1 function to be minimized. The initial
S is chosen from one of the three different shapes
shown in Figure 1. The scaled simplex is used
when prior knowledge on the problem scaling is
available, but right-angled and regular simplices
are generally preferred in the absence of information.
The algorithm updates the current simplex S
into a new simplex S + by performing two types of
linear transformations and driving the search towards a point having a lower function value (better point): reflection and expansion/contraction
(Figure 2).
First vertices v1 , . . . , vn are reflected through the current best vertex v0 to give
r1 , . . . , rn . If a reflected vertex ri gives a better function value than v0 , the algorithm tries an
expansion step. This is done by increasing the
distance between v0 and ri for i = 1, . . . , n and
yields new expansion vertices ei for i = 1, . . . , n.
The current simplex S is then replaced by either
S + = {v0 , r1 , . . . , rn } or S + = {v0 , e1 , . . . , en },
depending on whether the best point was among
the reflection or expansion vertices. If neither reflection nor expansion provide a point better than
v0 , a contraction step is performed. This is done
by decreasing the distances from v0 to v1 , . . . , vn .
If a point better than v0 is found among the contraction vertices c1 , . . . , cn , the simplex S is replaced by S + = {v0 , c1 , . . . , cn }. To complete one
iteration (or sweep) of the algorithm, v0+ is taken
to be the best vertex of S + .
tion but may fail at a point of nonsmoothness for
general nondifferentiable functions. In our tests
we have observed that it is beneficial in such a
situation to switch between the geometries (regular, right-angled) in order to give MDS some additional help to move on. But all these considerations are clearly heuristic, depend on the context
and will need further testing.
We sum up the above discussion in the following pseudo-code.
MDS with nonsmooth steps (MDSN)
1.
2.
3.
4.
5.
Select initial simplex S = {v0 , . . . , vn },
where v0 is the best vertex. Fix an
expansion factor µ ∈ (1, ∞) and a
contraction factor θ ∈ (0, 1), and an
intervention tolerance ω > 0.
Stop if the relative size of S is below
threshold ε
Perform a reflection step
ri = v0 − (vi − v0 ). Compute f (ri ).
If improvement f (ri ) < f (v0 )
perform expansion step
ei = (1 − µ)v0 + ri . Compute f (ei ).
If improvement f (ei ) < f (v0 )
put S + = {v0 , e1 . . . , en }.
Goto step 5.
else
put S + = {v0 , r1 . . . , rn }.
Goto step 5.
else
perform contraction step
ci = (1 + θ)v0 − θri . Compute f (ci ).
Put S + = {c0 , . . . , cn }.
Compare best vertex in S + to f (w). If w
is better, replace S + by new simplex
containing w as a vertex. Otherwise
accept S + . Go back to step 2 to loop on.
The adopted stopping criterion is based on the
relative size of the current simplex:
1
max kvi − v0 k < ε ,
max(1, kv0 k) 1≤i≤n
(1)
where v0 is the current best vertex of S =
{v0 , . . . , vn } and ε > 0 is a specified tolerance.
Figure 1: Selection of initial simplex
Note that the MDS algorithm is guaranteed
to converge to a local minimum for smooth func-
The choice of the initial simplex S is a relatively unexplored topic. The convergence proof
in (Torczon, 1991) only requires that S be nondegenerate, which means that the n + 1 points
{v0 , v1 , . . . , vn } defining the simplex must span
Rn . Otherwise MDS would only search over the
subspace spanned by the degenerate simplex.
0
2000
4000
6000
8000
10000
Figure 2: Reflection, expansion and contraction of
current simplex
12000
0
3
Test system
The large system model utilized to test the MDSN
algorithm is the Brazilian Interconnected Power
System (BIPS) (Martins et al., 1999; Gomes Jr.
et al., 2003). The North-Northeast and SouthSoutheast subsystems were interconnected in 1999
through a 1,000 km long, series-compensated 500
kV transmission line (Figure 5). Thyristor Controlled Series Compensators (TCSC) were placed
at the two ends of this line and equipped with
Power Oscillation Damping (POD) controllers in
order to damp the North-South (NS) mode, a
low-frequency, poorly damped interarea oscillation mode associated with this interconnection.
The BIPS model for the year 1999 has about
60 GW of generating capacity, 2,370 buses, 3,401
lines/transformers, 2,519 voltage dependent loads,
123 synchronous machines, 122 excitation systems, 46 PSS, 99 speed-governors, 4 static Var
compensators, 2 TCSCs equipped with POD controllers, 1 HVDC link with two bipoles. Each synchronous machine and associated controls is the
aggregate model of a whole power plant. All system equipment relevant to the study were modeled in detail, yielding a system Jacobian matrix
with 13,165 rows/columns and 48,532 nonzero elements (Gomes Jr. et al., 2003). The system has
1,676 state variables and more than 11,500 algebraic variables. The pattern of nonzero elements
of this large descriptor system matrix is pictured
in Figure 3.
QR routines from standard mathematical libraries (Patel et al., 1994; Mathworks, 1984-2005),
have performed quite reliably for power system
state matrices of about 2,000 states, considering
practical controller parameters and operating conditions. The CPU time for the QR eigensolution
of the 1,676-state Brazilian system matrix is about
50 seconds on a Pentium personal computer (CPU
1.86 GHz - Intel Centrino).
2000
4000
6000
8000
10000
12000
Figure 3: Sparse structure of BIPS matrix
Figure 4 pictures the multivariable feedback
control system focusing on the decentralized PSS
loops of three major plants located at the Northeast Brazilian region. The system outputs are the
machine rotor speeds (ωi (t), i = 1, 2, 3) at the
same power plants. The output of the PSS controllers are supplementary control signals to be
applied to the machine voltage regulator terminals (viref (t)). For the application of MDSN algorithm, these three PSSs are considered disconnected together with the POD controllers of the
BIPS model, yielding an open-loop system model
with 1,637 states.
Figure 4: Multivariable feedback control system
including PSSs at three Northeast power plants
Figure 6 shows the full eigensolution of the
open-loop BIPS model. It is seen that the NS
mode becomes unstable (0.1089 ± 1.2052j) in the
absence of the three PSSs depicted in Figure 4 and
the two POD controllers. The objective is to stabilize the NS mode only through improved designs
of these three PSSs, so that the POD controllers
would not be needed in the small signal stability
context.
Figure 5: Brazilian North-South interconnected power system - geographical location
15
ζ = 0.0538
10
NS mode
Imaginary Axis
5
0
−5
Adequate interarea oscillation damping is
achieved in practice by combining classical frequency response techniques with efficient methods for small signal stability analysis. Most of the
available methods are based on the use of a descriptor state form, exploiting the Jacobian matrix sparsity (See Figure 3), and rely on iterative
steps to obtain the dominant eigenstructure (one
or a set of eigenvalues and associated eigenvectors
at a time). In spite of the fact that some of these
methods allow dealing with multivariable systems
in the analysis step, the design is normally performed by tuning one stabilizer loop at a time,
which may require a number of redesigns.
In the next section, we show that the design of
multivariable structurally constrained controllers
for large systems can be performed in a more direct way while considering time domain specifications.
−10
4
Results
ζ = 0.0538
−15
−5
−4
−3
−2
Real Axis
−1
0
Figure 6: QR eigensolution for the open-loop
BIPS model (1,637 states)
This sections presents numerical results from the
application of MDSN algorithm (Section 2) to design the three decentralized PSSs in Figure 4.
The control synthesis objective is to maximize
the closed-loop damping ratio (ζ) by tuning the
controller parameters, i.e., the gains Ki and the
time constants T1i , T2i , T3i , T4i , i = 1, 2, 3, what
is equivalent to minimize the function
¶¾
½
µ
Imλ
f (A(K)) = max cos(α) : α , tan−1
Reλ
where λ ∈ spec(A(K)), A is the closed-loop dynamic matrix and K is the vector of controller
8
ζ = 0.0538
7
Open loop
Iterations
Closed loop
uncontrollable
mode
6
5
Imaginary Axis
parameters.
initial simplex: right-angled simplex
initial edge length: 5
expansion factor µ = 2
contraction factor θ = 0.5
stop criterion ε = 1e − 5
Initial controller parameters (Seed 1): random ∈ [0,30]
For stabilization: 1 iteration (cpu time about
30 min)
max ζ: cpu time 23h42min
4
3
2
NS mode
1
0
Table 1: MDSN algorithm iterations (seed 1)
Iteration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Simplex transformations
−
1 expansion
2 contractions
1 expansion
1 reflection
3 contractions
1 expansion
1 expansion
2 contractions
1 expansion
1 expansion
1 reflection
1 expansion
2 contractions
1 contraction
1 expansion
1 expansion
1 reflection
1 expansion
1 reflection
1 reflection
1 contraction
1 expansion
ζ
-0.105
-0.079
-0.076
-0.073
-0.052
-0.052
-0.044
-0.027
-0.022
-0.018
-0.010
-0.003
0.005
0.010
0.010
0.012
0.016
0.019
0.035
0.042
0.049
0.049
0.054
−1
−0.6
−0.5
−0.4
0.1
Initial controller parameters (Seed 2): random ∈ [0,1]
cpu time 3h13min
Table 2: MDSN algorithm iterations (seed 2)
Iteration
1
2
3
Simplex transformations
−
1 contraction, 1 reflection
1 expansion
ζ
-0.032
-0.029
0.054
Final controller parameters:

K1
 T11

 T21

 T31
T41
K2
T12
T22
T32
T42
 
K3
7.9186
 0.8462
T13 
 

T23 
= 5.5252
T33   0.2026
T43
0.6721
0.8381
0.0196
0.6813
0.3795
0.8318

0.5028
0.7095 

0.4289 

0.3046 
0.1897
Conclusions
References
NS mode
5
Imaginary Axis
0
difficult step: full QR eigensolution
ζ = 0.054
10
0
−5
ζ = 0.054
−4
−3
−2
Real Axis
−1
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−10
−15
−5
−0.1
Figure 8: Effectiveness of the MDSN algorithm in
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15
−0.3
−0.2
Real Axis
0
Figure 7: QR eigensolution for the closed-loop
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