1
Mean-Variance Mapping Optimization
Current developmental status and
application to power system problems
Dr. –Ing. José L. Rueda
9 September 2014
Rationale behind MVMO
2
Start
Initialize algorithm and optimization problem parameters
Normalize optimization variables in vector x to range [0, 1]
 Introduced by I. Erlich (University DuisburgEssen, Germany ) in 2010
 Internal search range of all variables
restricted to [0, 1].
 Solution archive: knowledge base for
guiding the searching direction.
 Mapping function: Applied for mutating the
offspring on the basis of the mean and
variance of the n-best population attained
so far.
Fitness evaluation by using de-normalized variables
Termination criteria satisfied?
Yes
No
Fill/update solution archive
(store n-best population
Single parent assignment
The first ranked solution xbest is chosen as parent
Single offspring generation
Selection: Select m (m<D) dimensions of xbest
Mutation: Apply the mapping function for
selected m dimensions
Stop
The hybrid variant: MVMO-SH
Start
Initialize algorithm parameters
Generation and normalization of initial population
i=0
k=1
No
rand<gLS Yes
Fitness evaluation
Local search
i=i+1
i=i+DFE
Fill/Update individual archive
Classification of good and bad solutions
and parent selection
No
Bad particle?
Single parent crossover
based on local best
Yes
Offspring
generation
Multi-parent crossover based
on a subset of good solutions
Mutation through mapping of m selected
dimensions based on local mean and variance
k<NP Yes
No
No
Termination criteria satisfied?
k=k+1
Yes
Stop
3
MVMO-SH: launching local search
Local search performed according to
rand  g LS
(1)
where
g LS : local search probability, e.g. g LS  1.5 /100 / D
D: Problem dimension
Different methods can be used:
 Classical: Interior-Point Method (IPM)
 Heuristic: Hill climbing, evolutionary strategies
4
Ranking
Fitness
x1
1st best
Ranking F1Fitness
Individual 2
x2
x1
...
x2
xD
...
xD
Optimisation
2nd best
F1 F1Fitness
1st best
Ranking
x1 x2 ... xD
Variables
... 2nd best
Optimisation
1st best F1 F1
Variables
Last best ...2nd best
FA
Optimization
F2
Variables
Last best... --- FA x1 x2  xD
Individual 1
Individual NP
MVMO-SH: solution archive
s1
sx21
Last best --- F
Mean
A
--Shape
Mean
d-Factor
Shape
d-factor
s
D
xD
dD
s2
ds21 x
--- d--1
1 x
2
sD
xD
sd12 s
2
dD
sD
--- -----
d1

x2
d1 d 2
dD
5
MVMO-SH: parent selection
6
All individuals ranked according to their particular best
Bad solutions
Good solutions
Global
Best
Pi
Random
Good
xbest
RG
best
xGB
x iparent =x ibest
x i , si , d i
Last First
Good Bad
Global
Worst
Pk
xbest
LG

best
best
xkparent = xbest
RG   xGB  xLG
xk =xkparent , sk , dk
Selection of m dimensions out of D
for mutation via mapping function

MVMO-SH: selection of dimensions for mutation
7
11
Method QP, quadratic progressive
Method L, linear
Method QD, quadratic degressive
10
a) The full range
corresponds with the
number of mutated
variables, e.g. m =7
8
7
6
Range for mutation
No. of variables mutated varied
9
5
4
3
2
Two different strategies are
available:
b) The number of mutated
variables estimated
randomly in the given
range, e.g.
m = irand(7)
1
0
0
200
400
600
800
1000
No. of iteration
8
MVMO-SH: selection of dimensions for mutation
8
Random-sequential selection mode
child*
x
=
Value inherited from x
parent
Generation n
Selected pivot dimension
xchild*=
Generation n+1
Randomly selected dimension
xchild*=
Generation n+2
8
MVMO-SH: mutation based on mapping function
0
1
1
h  xi  (1  e  xs1 )  (1  x i )  e  (1 x )s2
xi  hx  (1  h1  h0 )  xi*  h0
xi
hx  h( x  xi* )
h0  h( x  0)
0
0
9
xi*
1
(2)
(3)
(4)
h1  h( x  1)
xi* and xi in the range  0 1
8
MVMO-SH: mapping function features
1
1
xi
Parameter
shape si1 = si2
xi
0.8
1.0
0.8
0.75
0.6
0.6
0.2
1
xi
xi*
0.5
0.2
0
0
0
1
0.6
0.4
0.4
symmetrical
mapping function
si 1=si 2=10
0.2
1
symmetrical
mapping function
si 1 = si 2 = 10
0.8
0.6
xi*
0.5
1
xi
asymmetrical
mapping function
si 1 = 20
si 2 = 10
0.8
0.5
0.25
0
0
Parameter: xmean
si1 =si2= 10
0.4
0
5
10
15
50
0.4
10
asymmetrical
mapping function
si 1 = 10
si 2 = 20
0.2
0
0
0
0.2
0.4
0.6
0.8
xi* 1
0
0.2
0.4
0.6
0.8 xi* 1
MVMO-SH: assignment of shape and d-factors
si1  si2  si  ln(vi )  fs
if si > 0 then
Δd  1  Δd0   2  Δd0   rand  0.5 
if si > di
di  di  Δd
else
di  di /Δd
end if
if rand < 0.5 then
si1  si ; si2  di
else
si1  di ; si2  si
end if
end if
Dd =
10
.
(5)

randomly
varied
Dd0
Dd0
dr is always oscillating around
the shape sr and is set to 1 in the
initialization stage
Dd0  0.4
The d-factors remain dynamic with the mapping even the
corresponding shape doesn’t change
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MVMO-SH: assignment of shape and d-factors
si1  si2  si  ln(vi )  fs
if si > 0 then
60
Δd  1  Δd0   2  Δd0   rand  0.5 
if si > di
di  di  Δd
else
si1  di ; si2  si
end if
end if
1.E+05
1.E+03
50
1.E+01
40
(5)
1.E-01
30
1.E-03
1.E-05
20
1.E-07
10
1.E-09
0
0
2000
4000
6000
Number of Iteration
8000
1.E-11
10000
CEC2013 function F1, single particle MVMO
without local search, fs=1.0, Dd0=0.15
Fitness
Shape Factors
else
di  di /Δd
end if
if rand < 0.5 then
si1  si ; si2  di
shape factor from variance
shape factor S1
shape factor S2
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Application to power system problems
1. Optimal reactive power dispatch
2. Identification of power system dynamic
equivalent
3. Online optimal control of reactive sources
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Optimal Reactive Power Dispatch
Minimize
Losses
Ploss 
subject to
Operational
constraints
 
kN K
g k Vi 2  V j2  2VV
i j cosij

p  v, θ  pg  pd  0
(17)
q  v, θ  qg  qd  0
(18)
v min  v  v max
(19)
q gmin  q g  q gmax
(20)
qcmin  qc  qcmax
(21)
t min  t  t max
(22)
s  smax
(23)
(16)
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Optimal Reactive Power Dispatch
IEEE 118 bus system
77 dimensions (54 gen, 9 OLTCs, 14 compensators)
15
Optimal Reactive Power Dispatch
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IEEE 118 bus system: Average convergence performance
Optimal Reactive Power Dispatch
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IEEE 118 bus system: Statistics of active power losses
Ploss
(MW)
Algorithms
MVMOS
MVMO
CLPSO
SPSO
UPSO
FDRPSO
DMSPSO-HS
DE
JADE-vPS
Min
117.0802 117.0074 120.2117 121.8049 123.1174 119.1387 123.4717 118.7199
118.1047
Max
118.1662 125.1501 132.0461 125.4654 130.2011 123.5461 128.5504 121.1128
120.2177
Mean
117.4251 119.3353 122.2499 123.6784 125.6709 121.6536 125.0562 119.7737
118.9533
Std.
0.2285
1.9386
2.0533
0.9145
1.7663
1.0501
1.2033
0.6289
0.5321
Identification of dynamic equivalent
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Ecuador-Colombia interconnected system
Opt. problem
22 dimensions
(reactances, gains, time
constants)
Ecuador (study area)
Colombia (external area)
320 buses -- 64 generators
3.23 GW installed capacity
2.66 GW peak load
1729 buses -- 109 generators
11.08 GW installed capacity
8.78 GW peak load
Identification of dynamic equivalent
Dynamic equivalent for Colombia
- Sixth order generator model
- AVR model
- Governor model
Optimization problem
22 dimensions
(Reactances, gains,
time constants)
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Identification of dynamic equivalent
Optimization & Dynamic simulation
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Identification of dynamic equivalent
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Parameter identification problem statement
Minimize
p


OF   np   w1 y1  y1ref

np 1
0

2

System with
component model to
be identified
subject to
x j min  x j  x j max
Parameters of the model


 w n yn  yn ref  dt

2
From PMU
or
simulations
Identification of dynamic equivalent
DE for Colombia: comparison of heuristic methods
22
Identification of dynamic equivalent
DE for Colombia: comparison of dynamic responses
Fault 1
Fault 2
Full system
model
With DE
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Online optimal control of reactive sources
WPP
p+jq
●
●
●
T2
T1
GRID
PCC
●
●
●
Xsh
L1
CONTROLLER
QPCC
Qref
-
DQtotal
PI
Reference value
Distribution factors,
tap positions,
reactor switching status
Optimization
MVMO
every 15min
On/off status and
Loading of WG
( Pi & Qi )
DQ1*
●
●
●
DQi*
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Online optimal control of reactive sources
25
Other applications to power system problems
1. Active-reactive power dispatch
2. Short-term transmission planning
3. Location and tuning of damping controllers
4. Optimal transmission pricing
5. Optimal allocation and sizing of dynamic Var
sources
26
Highlights
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1. Winner of the competition on Expensive optimization at CEC-2014,
Beijing, PR-China, 6-11 July 2014.
2. 4th out of 17 place in the competition on real-parameter single
Objective optimization at CEC-2014, Beijing, PR-China, 6-11 July
2014.
3. 6th out of 21 place in the real-parameter single Objective
optimization at CEC-2013, Cancun, Mexico 21-23 June 2013.
4. Used for benchmarking in 2014 Competition on OPF problems
organized by the Working Group on Modern Heuristic Optimization
(WGMHO) under the IEEE PES Power System Analysis,
Computing, and Economics Committee
(https://www.uni-due.de/ieee-wgmho/)
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Thanks!
Dr. José L. Rueda
J.L.RuedaTorres@tudelft.nl
http://www.uni-due.de/mvmo/
MVMO-SH: parent selection
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All particles ranked according to their local best
Good
Bad
Dynamic Good/Bad Particle Selection:
è The ranking is re-calculated after any function evaluation
è The border between Good/Bad particles is shifted downwards
with the progress of iteration






GP  round Np g p*






(2)
NP: number of particles
i: function evaluation counter
 i / imax (3)



*
*
*
g p*  gp_ini
  gp_final
 gp_ini
(4)





g*
=0.7
g*
=0.2
p_ini
p_final
MVMO-SH: parent selection










xparent =xbest  xbest xbest
RG
GB LG
k










 =2 rand  shift 
(6)
shift =0.5  1   2
(7)


(5)
0
At the beginning
min
max
Range of random
search for 
At the end
min
max

Alternatively:   2.5 rand  0.25 2  0.5

(8)
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