Boston university; operations research presentation; 2013

OPERATIONS RESEARCH PRESENTATION
OPERATIONS RESEARCH
PRESENTATIONPRESENTATION
Al i Y ZhAlvin Yuan Zhang
Center of Information and System Engineering
Boston University
b t @b dyzboston@bu.edu
Yuan Zhang Boston University yzboston@bu.edu OPERATIONS RESEARCH PRESENTATION

OUTLINE
1 Project I:  Sustainable Ecosystem (SE) Planning Based on Discrete
Stochastic Dynamic Programming (DSDP) and Evolutionary Game
Theory (EGT)
Project II: Research on the Locational‐Marginal‐Price (LMP) Based
Distribution Power Network
2
Project III:  Optimization Approach to Parametric Tuning of Power
System Stabilizer (PSS) Based on Trajectory Sensitivity (TS) Analysis
3

Project I: SE Planning Based on DSDP and EGT
Introduction
Why investigating SE Planning?
Ecosystems have been faced with server threats under the impacts of climate and humankind together
Different patterns of resource utilization could directly influenced ecosystem heath
Sustainability is an important target of developing nature ecosystem, i.e., SE
Difficulties
Ecosystems are usually influenced by many factors which are difficult to define and quantify
Research related to ecosystem is rather difficult due to its complex structures and metabolic processes
Direction: To represent multi‐subsystems and their dynamic interactions in an analytical form using a
reasonable number of equations and parameters!reasonable number of equations and parameters!
Drawbacks of Previous Work
Fundamental weakness is that they use strictly deterministic and quantitative approaches to describe systems
that are full of uncertainty and only qualitatively understoodthat are full of uncertainty and only qualitatively understood
Mainly focus on economically developed and densely populated areas, but neglected regions with adverse
weather conditions, such as Loess Plateau
Merely focused on analysis of overall resource planning among multi‐subsystems, but ignore impacts of dynamic
relationship among them, namely evolutionary game relations
Motivation: To explore some feasible applications of decision theory/method into SE planning, with a
specific area of ecological resource planning, such as water resource planning problem!
p g , y y g

Brief Overview of Loess Plateau
Extensive region (530,000 km2 ‐ larger than Spain and almost as large as France)
Extreme loss of soil fertility and reduction in arability
Natural and human factors threat the sustainability of Loess Plateau, especially the
shortage of water resource

Simplified DSDP Model for SE Planning
Definition 1: Resource and User
Define the total kinds of concerned resource as m,
Definition 2: Time Horizon
Define time horizon as k=1 2 N which represent,
which is utilized by n user subsystems (or known as
users). These users can be regarded as residents,
companies, governments, agriculture firms, etc.
Define time horizon as k=1, 2, …, N, which represent
the period when each user begin to utilize the
resource.
Definition 3: State Variable
Define state variable at time k as follows:
11 12 1( ) ( ) ( )x k x k x k 
Definition 4: Decision Variable
Define decision variable at time k as follows:
( ) ( ) ( )u k u k u k 

   

11 12 1
21 22 2
1 2
( ) ( ) ( )
( ) ( ) ( )
, 1,...,
( ) ( ) ( )
n
n
k
m m mn
x k x k x k
x k x k x k
k N
x k x k x k
 
 
   
 
 
 
X



   

11 12 1
21 22 2
1 2
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
n
n
k k k
m m mn
u k u k u k
u k u k u k
u k u k u k
 
 
 
 
 
 
U U X
where xij(k) denotes as the case of whether the i‐th
resource is used by the j‐th user. If xij(k)=1, the i‐th
resource is assigned to the j‐th user; otherwise not.
1 2( ) ( ) ( )m m mn 
where uij(k) denotes as the amount of resource that
the j‐th user decide to use from the i‐th one.

111u
Define reward function at time interval [k, k+1] as
Definition 6: Reward Function
1
2
2
21u
12u
22u
u
ijC
ijS
[ , ]
follows


V
11 12 1
21 22 2
( ) ( ) ( )
( ) ( ) ( )
n
n
k
r k r k r k
r k r k r k
 
 
 
 
2
3
13u
23u
where rij (k) can be express as
   

V
1 2( ) ( ) ( )
k
m m mnr k r k r k
 
 
 
Definition 5: Transition Probability Matrix
Define state variable at time as follows:
h S d t th d f th j th th t
( ), ( ) 0
( )
0, ( ) 0
ij ij ij ij
ij
ij
S C u k x k
r k
x k
  
 




   
1| 1
11 12 1
21 22 2
( | , )
( ) ( ) ( )
( ) ( ) ( )
k k k k k
l
l
p k p k p k
p k p k p k
 
 
 
 
 
X XP P X X U
where Sij denote the reward of the j‐th user that
utilized the i‐th resource=; denote the cost Cij of the
j‐th user that utilized per‐unit amount of the i‐th
resource. Assume Sij =S~|j and Cij =Ci|~ .
   
1 2( ) ( ) ( )l l llu k u k p k
 
 
 

Mathematically speaking, there are 2mxn kinds of
Remark 1
Based on Remark 2, P(Xk+1|Xk,Uk)=P(Xk+1|Xk, Uk),
Remark 3
y p g,
possible selection of Xk. However, it is obviously
that we can’t select every element of Xk as zero,
which means there is no resource is assigned to any
user. Assume that each resource will be assigned to
arbitrary user; and each use can get at least one
, ( k+1| k, k) ( k+1| k, k),
which is a stochastic matrix that can’t be easily
derived from analytical modeling. Referring C. C. Lin
et al*, we will use the statistic data of water
resource bulletin** to determine PXk+1|Xk. Using the
maximum likehood estimator, PXk+1|Xk could be
kind of resources. Thus, each row and each column
of Xk will have at least an integer 1 for any k=1,2, …,
N .
Moreover, a stationary Markov chain is used to
h bl h h d
Xk 1|Xk
estimated as the observation data as follows:
h h b f f h


ˆ ( ) , 1,...,ij
ij
i
N
p k k N
N
  
generate the state variable Xk, which is assumed to
take on a finite number of values
(1) (2) ( ) ( )
{ , ,..., ,..., }i l
k k k k k kX X X X X X
where Nij is the number of occurrences of the
transition from Xk
(i) to Xk
(j) at time k, and Ni is the
total number of times that has occurred at time k.
Remark 2
For any xij(k)=0, uij=0; xij(k)=1, 0<uij≤max(uij). Then,
Uk is dependent of Xk with a similar matrix structure.
W ill hi f i h f ll i di i
* C. C. Lin, et al, “A stochastic control strategy for hybrid electric vehicles,”
Proceedings of the American Control Conference, vol. 5, pp. 4710–4715, 2004.
** http://www.sxmwr.gov.cn/gb-zxfw-news-3-dfnj-28873
We will use this fact in the following discussions.

DSDP model (Using DP Algorithm)
X( )k kJ
X X X(1) (2) (3)
0 0 1 0 0 1 0 1 0
, ,
1 1 0 1 1 1 1 0 1k k k
     
       
     
     



X X
U U
X X
U U
P X
X
1| 1 1
1 1
1 1
max ( ) ( , ) ( )
max ( ) ( ) ( )
k k
k k
k k
m n
ij k k
i j
m n
ij ij k k
r k i j g
r k p k J
  

   
 
 
  
 
 
  
 
 
 
X X X(4) (5) (6)
0 1 1 0 1 1 0 1 0
, ,
1 0 0 1 0 1 1 1 1k k k
     
       
     
X X X(7) (8) (9)
0 1 1 0 1 1 1 0 0
, ,
1 1 0 1 1 1 0 1 1k k k
     
       
     
U U
X X1 1k k
k k
j j
i j

    
 
Water Resource Planning based on the
Proposed DSDP Model
f d d i 2
X X X(10) (11) (12)
1 0 1 1 0 1 1 1 0
, ,
0 1 0 0 1 1 0 0 1k k k
     
       
     
X X X(13) (14) (15)
1 1 1 1 1 0 1 1 1
, ,
0 0 1 0 1 1 0 1 0k k k
     
       
     surface water and ground water, i.e., m=2
Users subsystems can be classified as three
parts: agricultural firms, industrial usage and
daily usage, i.e., n=3
0 0 1 0 1 1 0 1 0     
X X X(16) (17) (18)
1 1 1 1 0 0 1 0 1
, ,
0 1 1 1 1 1 1 1 0k k k
     
       
     
X X X(19) (20) (21)
1 0 1 1 1 0 1 1 1
, ,
1 1 1 1 0 1 1 0 0k k k
     
       
     As indicated in 2011 Water Data Bulletin,

 

| |
3
3 4 1 ,
5j iS C 
 
      
   
1 1 1 1 0 1 1 0 0k k k     
     
X X X
X
(22) (23) (24)
(25)
1 1 1 1 1 0 1 1 1
, , ,
1 0 1 1 1 1 1 1 0
1 1 1
k k k
     
       
     
 
 
X(25)
1 1 1k   
 We list all the 25 possible cases of Xk as follows:

Transition probability matrix PXk+1|Xk as
k=9
*Optimal results of water planning of L.P.
0.8
0.2
0.4
0.6
1|kk+XXP
10
15
20
25
0
5
10
15
20
0
( )iI
( )j
kXI
5
25
( )i
kXI
* Yuan Zhang. “Sustainable Ecosystem Planning Based on Discrete Stochastic Dynamic Programming and Evolutionary Game Theory”,
arXiv:1305.1990v2 [math.OC], May 2013.

Evolutionary Game Analysis of Water Resource Planning of L. P.
*Optimal results of water planning of L.P.
(cont…)
Evolutionary game theory as a supplemen‐
tation of the proposed SDP model
Two participants to do game playing in group A and B
Each payoff equals to 1 or 0, and u, v, (u>1, v>1 )
denote as the payoff of A and B, under cooperation
case, respectively
Two strategies in the decision games namely CTwo strategies in the decision games, namely, C
(sustainable usage), D (unsustainable usage)
p as the ratio of participant who choose strategy C
among group A; q as the ratio of choosing strategy D
among group B.a o g g oup
(p,q) can represent the evolutionary dynamics of the
system, which can satisfies** :
/ (1 )( 1)
/ (1 )( 1)
dp dt p p uq
dq dt q q vp
  

*
/ (1 )( 1)dq dt q q vp  
** D. Friedman, “Evolutionary games in economics,” Econometrica, vol. 6, no. 3, pp.637–660, 1991.

Evolutionary Game Analysis of Water Resource Planning of L. P.
*Evolutionary game theory as a supplementation of the proposed SDP model (Cont…)
Two ESS points: Q1=(0,0) & Q4=(1,1)
Three unstable points: Q2=(0 1) Q3=(1 0) Q5=(1/v 1/u)Three unstable points: Q2 (0,1), Q3 (1,0), Q5 (1/v, 1/u)
4(1,1)Q2(0,1)Q
5Q
Increasing v & u
1(0, 0)Q 3(1, 0)Q

Conclusion
Conclusion
SE planning of the Loess Plateau area has been analyzed based on DSDP model and EGT
The concept of SE planning is introduced with specifications in ecological resource planning
Transition probability matrix is calculated in a statistic sense so as to derive the DSDP model
Although the approach is applied to the water resource planning of Loess Plateau as an example, the
methodology of using DSDP and EGT is applicable to other complex systems
Further reading: Yuan Zhang ‐‐ http://arxiv.org/abs/1305.1990

Project II: Research on LMP-Based Distribution Power Network
Background Introduction
Necessity of investigating LMP in distribution network
Integration of smart grid in the electricity networks allows for the expansion of the real time marginal cost g f g y p g
based pricing to the distribution network
Due to increasing demands of energy generation and consumption, standard network structures will not be
sufficient to provide state‐of‐the‐art security of supply under increasing cost pressure
Power losses in the middle and cascading failures on customer side usually take place in distribution network g y p
with most of loads or electronics connected
Transaction of utilization and provision of real and reactive power by participants requires the improvement
of pricing in distribution network
Overall goal of LMP‐based distribution network
Propose a redesigned market that could embrace the distribution level and extend the clearing prices accounting
for the marginal costs that occur in this level, i.e., LMP
Consider effects of power consumers/ producers on LMP, when connected at the low voltage level ff f p / p , g
Direction: Investigate distribution level LMP that are incorporating marginal costs of real and reactive power,
transformer loss of life, and voltage control limits
Possibility: Propose certain novel optimization approach for distribution market clearing problem.

Distribution network market clearing problem
2 2
2 2 2 2
0 ( ) ( )
sin(arccos( ))
( ) ( ) ( ) ( )
i i i
i i i
i i i i i
g g g
b b b
d d d
b b b
e e e e e
P Q C
Q P A
C P Q C P
  

  
Objective function & constraints* Power
Constraints of
generator,
d b d
, ,
, ,
,min i i i
i
b m b m
b m b m
g d dP
b g b b b
b i i
f f
f f
P c u P
tM

 
 
 


  

2 2 2 2
( ) ( ) ( ) ( ) ,
0,
0,
0,
i i i i i
i
e e e e e
b b b b b i
i
e
b i i
i i
C P Q C P e
if e is standalone
P if e is associated withd
if e is associated withg
     


 

distributed
loads and
electronic
devices
Cost of real power production of
the slack bus minus real power
consumption
Cost of transformer loss of life
 
 
,
, (1)
2 2
, (1)
2
1
b m
M
M
P
b
P
b
V
P
C C Q
c V


 
   
 

  
 
,
,
i i i
i i i
g e d
b b b b
i i i
g e d
b b b b
P P P P b
Q Q Q Q b
     
     
  
  
Overall
real and
reactive
balance at
Cost of real power procured at substation
Opportunity cost compensation generator
of reactive power at the substation
Cost of required voltage increase at the  
2
, , , ,
. .
cos( ) sin( ), ( , )b m b b m b m b m b m b m b m b m
st
P V G VV G A A VV B A A b m     
,
,
,
2
1500 1500
exp ,
383 273b m
b m
f b mH
f
H A
f

 
    
  
i i i
each bus
Transformer
Cost of required voltage increase at the
substation for voltage control
, , , ,
2
, , , ,
,
,
cos( ) sin( ), ( , )
,
,
b m b b m b m b m b m b m b m b m
b m b m n b m b m b m b m b m b m
b b m
m
b b m
Q V B VV B A A VV G A A m n
P P b
Q Q b
      
 
 


, , , ,
2
1, 2, , 3, , ,
2 2
, , ,
,
, ( , )
b m b m b m b m
H A
f f f b m f b m b m
b m b m b m
k k S k S f
S P Q b m
V V V b
     
  
  
loss of life
l l d f l
Real /Reactive power flow
on any line and its injections
at any bus
m ,
0
b b bV V V b
A
  

Voltage limitation & default
angle value for slack bus
at any bus
* E. Ntakou, M. C. Caramanis, “Price Discovery in Dynamic Power Markets with Low-Voltage Distriution-Network Participants,” Manuscript , Mar. 2013.

Objective function & constraints (cont…)
Nonlinear objective function under constraint of a non‐convex setj
Using KKT condition to obtain the dual variable             , which denotes as LMP of real and reactive power at each
bus in the distribution network
,P Q
b b 
M t QP Q  
 ,
, ,
, , (, (1) , (1)1)
2 2
, (1)
2 1m n Mm n M
m n m
M
n M
f b b m
b b
f bP P
b b
P V
b m
f mf b b
M t Q
c V
C
P Q V V
P P PQ P P
   


  


 
 
  
    
  
 
  
 
 , , , (, (1) , (1)1)
2 1m n Mm n M Mf b b mf bQ P P V
M t Q
c V
P Q V V
     
  
    
   
, ,
2 2
, (1)
2 1
m n m n M
b b b b
b m
f mf b b
c V
CQ Q QQ Q Q
   
    
 
  
  
 
  
 
h d t i l l ffi i t f l/ ti
, (1) , (1) , (1) , (1)M M M Mb b b bP Q P Q      
where                                                                          denote as marginal loss coefficients of real/reactive power;
, ( ) , ( ) , ( ) , ( )
, , ,M M M Mb b b b
b b b bP P Q Q   
denote as marginal cost of transformer loss of life;                        denote as marginal, ,
,m n m nf f
b bP Q
 
 
,m m
b b
V V
Q P
 
 
cost of voltage control that increases voltage at each bus as well as meets constraints in the problem.

Objective function & constraints (cont…)
Using Matlab to do power flow calculation and then solve the aformentioned LMP in distribution networkUsing Matlab to do power flow calculation and then solve the aformentioned LMP in distribution network
Analyzing LMP based on some numerical results obtained from a give distribution level network
Related considerations of LMP in distribution network
Uniqueness of the solution: Radial power network (YES, unique); Meshed power network (NO, may be multiple…)Uniqueness of the solution: Radial power network (YES, unique); Meshed power network (NO, may be multiple…)
Multi‐period consideration: Evolution of LMP varied with Time & Space
Simplification approach: Linearization…

Convex Relaxation: An Interesting Idea for Solving Market Clearing Problem
Conexify*
L. W. Gan, et al., proposed a convex relaxation method for optimal power flow in tree networks**

opt
x

( )f x
This form can then be transformed
into Second‐Order‐Cone constraint
* Oral communication with Prof. M. C. Caramanis.
** L. W. Gan, N. Li, U. Topcu, S. Low, “On the exactness of convex relaxation for optimal power flow in tree networks,” IEEE 51st Conference on Decision and Control, Dec.
2012 Caramanis.

Reference
M. C. Caramanis, et al., “Provision of Regulation Service Reserves by Flexible Distributed Loads,” IEEE 51st Annual
Conference on Decision and Control, Dec. 2012.
M T Wishart et al “Smart demand‐sided management of LV distribution networks using multi‐objectiveM. T. Wishart, et al, Smart demand sided management of LV distribution networks using multi objective
decision making,” Manuscript for IEEE PES Transactions on Smart Grid.
M. C. Caramanis, “It is time for power market reform to allow for retail customer participation and distribution
network marginal pricing ” IEEE Smart Grid Mar 2012network marginal pricing, IEEE Smart Grid, Mar. 2012.
S. M. M. Agah, H. A. Abyaneh, “Distribution transformer loss‐of‐life reduction by increasing penetration of
distributed generation,” IEEE Transaction on Power Delivery, Apr. 2011.
M. C. Caramanis, R. E. Bohn and F. C. Schweppe, “Optimal spot pricing: price and theory,” IEEE Transactions on
PAS, vol. 101, 1982.
C. Y. Lee, H. C. Chang, H. C. Chen, “A method for estimating transformer temperatures and elapsed lives
considering operation loads”, WSEAS Transactions On Systems, Issue 11, vol. 7, pp.1349‐1358, Nov. 2008.considering operation loads , WSEAS Transactions On Systems, Issue 11, vol. 7, pp.1349 1358, Nov. 2008.
M. Thomson, D. G. Infield, “Network power flow analysis for a high penetration of distributed generation,” IEEE
Transactions and Power Systems, vol. 22, no. 3, pp. 1157‐1162, Aug. 2007.
E Nt k M C C i “P i Di i D i P M k t ith L V lt Di t i ti N t k
E. Ntakou, M. C. Caramanis, “Price Discovery in Dynamic Power Markets with Low‐Voltage Distriution‐Network
Participants,” Manuscript for IEEE Conference on decision and Control, Mar. 2013.

Project III: Optimization Approach to Parametric Tuning of PS Based on TS
Research Background of Optimal PSS Parametric Tuning
Why Introducing PSS?
Grid interconnection of P.S. lead to oscillation that inhibits its long‐term stability
PSS is introduced as a feedback controller to decrease oscillations, and increase the reliability
Optimal PSS parametric tuning is crucial to P.S., and become a focal point of much on‐going research
Drawbacks of Previous Work
Merely focused on local equilibrium point/orbit, i.e., small disturbance ‐based
P.S. is essentially a hard (nonlinear and nonsmooth) dynamic system undergoing large disturbance (LD)
Traditional PSS optimization methods fail to obtain globally optimal parameter setTraditional PSS optimization methods fail to obtain globally optimal parameter set
Motivation: A LD‐based Optimal PSS parameter tuning approach should be explored!
DifficultiesDifficulties
Discontinuous change of P.S. structural dynamics under LD
Hybrid Power System (HPS): A mix of continuous‐time, discrete‐time and discrete‐event dynamics
TS analysis can focus around transient flow trajectory
Direction: Exploring from LD based optimization approach to evaluate TS under constraints of HPS model!
Direction: Exploring from LD‐based optimization approach to evaluate TS under constraints of HPS model!

Modeling of PSS and HPS
Major parameter set of PSS
1 2 3 4( , , , , )sK T T T Tl
TS information will be obtained from
Is iV ω
Definition 1: Switching Event
Switching event SE(i)
is defined as any event that
d l h h f l bcan directly trigger the change of algebraic states y
at the i‐th period, which can then form a switching
event set ASE, with its index set denoted as ISE.
Definition 2: Reset Event
Reset event RE(j)
is defined as any event that can
directly trigger the change of discrete states z at the
j‐th period which can then form a reset event set
j th period, which can then form a reset event set
ARE, with its index set denoted as IRE.

Modeling of PSS and HPS (Cont…)
* Compact HPS model of parameter‐dependent differential‐algebraic‐discrete (DAD)
[ , , ] n m p
c
 
 x x z l ( , )x f x y
: n l p m n  
f
: n l p m m  
g
: ( )j n l p m l  
h
(0)
( ) ( )
( ) ( )
( , )
( , ), ;
( , ), ; SE
i i
SE
Ai i
SE
SE A
i I
SE A



 
  

0 g x y
g x y
0
g x y
* Ian A. Hiskens and M. A. Pai. “Trajectory Sensitivity Analysis of Hybrid Systems” IEEE Trans. Power Sys. 47 (2), 2000. NOT GENERAL!
p
l
Incorporating parameters λ into the state x

( ) ( )
( )
( , ), ;
, ;
RE
RE
j j
RE A
j
RE A
RE A j I
RE A j I
  
   
   
z h x y
z 0
Ian A. Hiskens and M. A. Pai. Trajectory Sensitivity Analysis of Hybrid Systems IEEE Trans. Power Sys. 47 (2), 2000. NOT GENERAL!
Mapping SE(i)
and RE(j)
into two triggering hypersurfaces H(i)(x,y) and S(j)(x,y)
 ( )x f x y ( ) ( )t tx xy
(0)
( ) ( )
( ) ( )
( , )
( , )
( , ), ( , ) 0;
{1,2}
( ) ( ) 0;
i i
i i
H
i
H




 
  

x f x y
0 g x y
g x y x y
0
g x y x y
( ) ( , )ot t xx xy
( ) ( , )ot t yy xy
0 0( ) ( , )o ot t xx x xy
Trajectory
Flow

( ) ( )
( )
( , ), ( , ) 0;
, ( , ) 0; RE
j
A
H
S j I

   
g x y x y
z 0 x y
0( , ( , )) ( , )o o o ot y0 g x x g x yy
 0l
Initial
Condition

Optimal PSS Parametric Tuning Based on TS
n l p m+ + +
2
min ( , )
f
K t
f iJ t dt x
Objective Function TS Analysis for HPS
1 1( ( ), ( ))J Jt tx y
 p
x
y 0 0( , )x y
0Dx
1( )Jt+
Dx
t
2 2( ( ), ( ))J Jt tx y
2( )Jt+
Dx

0
1
(0)
( ) ( )
. . ( , )
( , )
( ) ( ) 0;
f it
i
i i
s t
H




 

x f x y
0 g x y
g x y x y
l
0 0t
(1)
( , ) 0H =x y
1Jt
1Jt
1Jt 2Jt
2Jt
(2)
( , ) 0H =x y
2Jt
(1)
SE (2)
SE

( ) ( )
( ) ( )
( )
( , ), ( , ) 0;
{1,2}
( , ), ( , ) 0;
, ( , ) 0;
( )
RE
i i
j
A
H
i
H
S j I
t

 
  

   
g x y x y
0
g x y x y
z 0 x y
TS (red parts)
0
0
( )
( )
, {1,2,..., }
o
o
i i i
t
t
i K


   
x x
y y
l l l
0
0
0
0
( )
)) (
( )
(
tt
t t
  


 
x
x
x
y
x x
y x
TS dynamics equations
1 2 1 2{ , , , }, { , , }k i si i iK T T l l l l l
K is the number of generators
Gradient information can be obtained as
TS dynamics equations
 0 0
0 0
0
(1 ) (1 )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
d d t t t t
t t t t 
     
     
x x y x
x x y x
x / x f x f y
0 g x g y
 ( ) ( ) ( ) ( )d d     / f f
1 2[ , ]J Jt t t 

1
( , ) 2 ( )
f
o
t
i
K
f i
t i
J t t 

  x ll
 0 0
0 0
0
(2 ) (2 )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
d d t t t t
t t t t 
     
     
x x y x
x x y x
x / x f x f y
0 g x g y
2[ , ]J ft t t


Optimal PSS Parametric Tuning Based on TS
TS Analysis for HPS (Cont…)
Refer Ian A. Hiskens et al, jump conditions for
th iti it ft th t t i i t
1 1( ( ), ( ))J Jt tx y
n l p m+ + +
x
y 0 0( , )x y
2 2( ( ), ( ))J Jt tx y
the sensitivity after the event triggering tJ1:
0 0 0
0 0 1
1 1 1
(1 ) 1 (1 )
1
( ) ( ) ( )
( ) [ ( ) ]|
J
J J J
J t
t t t
t 
   
   
     
     
x x x
x y x x
x x f f
y g g x
0 0t
(1)
( , ) 0H =x y
0Dx
1Jt
1Jt
1Jt
1( )Jt+
Dx
t
2Jt
2Jt
(2)
( , ) 0H =x y
2Jt
2( )Jt+
Dx
(1)
SE (2)
SE
Updating the jump condition for the sensitivity
after the event triggering tJ2:
0 0 02 2 2( ) ( ) ( )J J Jt t t   
     x x xx x f f SE SE0 0 0
0 0 2
(2 ) 1 (2 )
2( ) [ ( ) ]|
J
J t
t 
   
     x y x xy g g x
Optimum searching using Conjugate Gradient Method (CGM)p g g j g ( )
1
1 1 1
, 0
( )
k k k k k
k k k kJ
 


  
  
  
d
d dl
l l
l
Powell‐Fletcher‐Reeves Rule
( ) ( ) ( )
[0 1] [0 1]
kk k m k k k kJ J s s J 
 
    
  
d dll l l
Armijo Rule
 1 1
1
( ) ( ) ( )
, 1,..., 1
( ) ( )
k k k
k
k k
J J J
k n
J J
  

  
   
 
l l l
l l
l l l
l l
[0,1], [0,1]   

Application to IEEE Standard Test System
IEEE three‐machine‐nine‐bus standard test system
2G 3G
7 8 9
1
2
5
4
6
3
1G
1
* Yuan Zhang. “Optimization Approach to Parametric Tuning of Power System Stabilizer Based on Trajectory Sensitivity Analysis”, arXiv:1305.0978v2 [cs.SY] , May 2013

Conclusion
Conclusion
Optimal PSS parametric tuning method is studied from the viewpoint of TS, both theoretically and numerically
Discontinuity is a major obstacle to analyze the constraints of this optimization problem
Gradient information of the objective function is obtained from TS of state variables w.r.t. PSS parameters
Objective function considers the transient features under large disturbances, which indicates that the proposed
method can effectively damp the spontaneous oscillation caused by large disturbance
Further reading: Yuan Zhang ‐‐ http://arxiv.org/abs/1305.0978

Thank You!Thank You!
Q & A

Boston university; operations research presentation; 2013

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