Dynamic e olu ion o a hyd aulic-mechanical-elec ic sys em wi h
andomly luc ua ing speed based on Chebyshe polynomial
app oxima ion me hod
Beibei Xua, Diyi Chen*a,b, Ca aballo, Tomasc
aIns i u e o Wa e Resou ces and Hyd opowe Resea ch, No hwes A&F Uni e si y, Shaanxi
Yangling 712100, P. R. China
bKey Labo a o y o Ag icul u al Soil and Wa e Enginee ing in A id and Semia id A eas, Minis y o
Educa ion, No hwes A & F Uni e si y, Shaanxi Yangling 712100, P. R. China
cDep o. Ecuaciones Di e enciales y Análisis Numé ico, Facul ad de Ma emá icas, Uni e sidad de
Se illa, c/ Ta ia s/n, E-41012 Se illa, Spain
Co esponding au ho : Diyi Chen
Mailing Add ess: Ins i u e o Wa e Resou ces and Hyd opowe Resea ch, No hwes A&F Uni e si y, Shaanxi
Yangling 712100, China
Telephones: 086-181-6198-0277
E-mail: [email p o ec ed]
Abs ac The esea ch p oposed in his pape ocuses on he dynamic e olu ion o a
hyd aulic-mechanical-elec ic sys em unde he e ec o andomly luc ua ing speed. The apid
g ow h o ins alled wind powe capaci y may po en ially a ec he s abili y o powe g ids, causing
la ge luc ua ions o he gene a o speed o hyd opowe s a ions. In his wo k, a p obabilis ic
componen is associa ed o he gene a o speed o a de e minis ic hyd aulic-mechanical-elec ic
sys em p o iding a no el andom model. This la e is analyzed o in es iga e he dynamic e olu ion
o he sys em adop ing he Chebyshe polynomial app oxima ion me hod. A ca e ul compa ison o
he nume ical applica ion esul s ob ained by he de e minis ic and he p obabilis ic app oaches is
ca ied ou . In addi ion, he in luence o he luc ua ion in ensi y (D) on he di e en ial gain (kd) o
he PID is in es iga ed, p oposing a law o kd as unc ion o D. Finally, he ope a ing anges o he
g id wa e hamme and o he elas ic wa e hamme models a e compa ed in o de o alida e he
consis ence o he law. The esul s o he s udy p o ide obus bases o he s able and sa e ope a ion
o hyd opowe s a ions.
Key wo ds: ib a ion cha ac e is ics; hyd aulic-mechanical-elec ical sys em; andomly luc ua ing
gene a o speed; elas ic wa e hamme ; Chebyshe polynomial app oxima ion;
1. In oduc ion
O e he las decades, he hyd opowe sec o has expe ienced a apid de elopmen in China,
wi h he cons uc ions o powe s a ions in e e y co ne o he coun y, eaching an ins alled
capaci y o en hund ed million kilowa s [1-4]. Fu he mo e, acco ding o he 2014 plan ag eed by
mos in e na ional hyd opowe indus ies, he cons uc ion o new hyd opowe ins alla ions will be
p omo ed in o de o double he wo ldwide ins alled capaci y wi hin he nex hi y yea s [5, 6]. Due
o his apid expansion, he eliabili y and sa e y o such ins alla ions ha e nou ished he conce ns o
he public opinion as well as o he egula o s [7-12]. In pa icula , he
hyd aulic-mechanical-elec ical sys em is a c ucial componen which plays an essen ial ole o he
sa e y o indi idual ins alla ions and can hence a ec he s abili y o he whole elec ic powe g id.
The e o e, he analysis o he s abili y o hyd aulic-mechanical-elec ic sys ems is a high-p io i y
opic o esea ch. In spi e o his, he scien i ic li e a u e in his ield is s ill qui e limi ed. S udies
ocusing on hyd aulic-mechanical-elec ic sys ems can be oughly classi ied in wo g oups. The i s
ocuses on he g id wa e hamme heo y, aiming a es ablishing de e minis ic models o he
hyd aulic-mechanical-elec ic sys em [13-16]. This app oach is sui able o desc ibe he dynamic
e olu ion o hyd opowe s a ions wi h sho pens ocks cha ac e ized by uni o m geome y. The
wo ks belonging o he second g oup adop he elas ic wa e hamme heo y o he de ini ion o he
de e minis model [17-25]. This app oach is used o accu a ely analyze he dynamic cha ac e is ics
o hyd opowe s a ions wi h long pens ocks and allows o ake in o accoun e en ual shape changes
in he geome y o he pens ock.
In addi ion o wa e powe , enewable ene gy includes sola , wind, geo he mal o idal powe
amongs o he s [26-31]. Mo eo e , as highligh ed by he las Global Wind Ene gy Ou look, he wind
powe sec o has g own apidly o e he las decades in e ms o bo h echnological and comme cial
compe i i eness: his is expec ed o esul in an ins alled capaci y o nea ly 2 TW by 2030, supplying
be ween 16.7% and 18.8% o global elec ici y [32]. Ob iously, his end may po en ially a ec he
s abili y o powe g ids, leading o la ge andom luc ua ions o hyd o- u bine gene a o s speed
[33-36]. High-in ensi y andom luc ua ions can lead o he b eaking o he o que balance o
hyd o- u bine gene a o uni s and hence h ea en i s ope a ing s abili y. Lo s o s udies ha e ocused
on he inc easing andomness o powe g id. The e o e, he inc easing gene a ion capaci y o wind
powe ins alla ions in oduces new impo an challenges o he s able ope a ion o hyd opowe
s a ions o any size.
A ligh o his, ou wo k p oposes a no el app oach o he s abili y analysis o he
hyd aulic-mechanical-elec ic sys em wi h espec o he exis ing li e a u e. Fi s , a p obabilis ic
componen u is in eg a ed wi h he gene a o speed
w
, ob aining a no el p obabilis ic model o he
hyd aulic-mechanical-elec ic sys em. Second, by compa ing he dynamic e olu ions o he
de e minis ic and p obabilis ic app oaches, he ad an ages and d awbacks o each me hod a e
iden i ied. Thi d, a ma hema ical de ini ion o he di e en ial gain kd o he PID in unc ion o he
luc ua ion in ensi y D is p oposed. Finally, he consis ence o such de ini ion is e i ied h ough a
nume ical applica ion.
The con en o his pape is o ganized as ollows: Sec ion 2 p esen s he nonlinea p obabilis ic
model o he hyd aulic-mechanical-elec ic sys em. Nume ical simula ions along wi h de ailed
analysis o he esul s ob ained a e p esen ed in Sec ion 3. Conclusi e ema ks and discussions a e
included in Sec ion 4.
2. Nonlinea andom models o hyd aulic-mechanical-elec ic sys ems
The hyd aulic-mechanical-elec ic sys em conside ing he elas ic wa e hamme model adop ed
in his s udy is shown in Eq. (1):
12
23
3011223
''
2
''
4
4
1[ sin sin 2 ]
2
1(( ) )
s
qs d q
s
ab ddq
pid
y
xx
xx
xaxaxaxy
EV x x
V
mD
Txxx
yk kxky
T
x
dww
wddw
ww
w
SS
SSS
=
ì
ï=
ï
ï=- - - +
ï
=
ï
ï-
í=- - -
ï
ï
ï
=-+--
ï
ï
ï=-
î
(1)
De ails abou he pa ame e s in Eq. (1) can be ound in Re . [24].
Fig. 1 E olu ion in ime o gene a o speed luc ua ion
Acco ding o he na u e o he luc ua ion a ec ing he gene a o speed, wo di e en ypes can
be iden i ied: he i s e e s o a limi ed ange, o abou 49.5~50.5 /min, as shown in Fig. 1 o
alues o ime lowe han 120 s. This ype o luc ua ion is di icul o de ec om he gene a o
unning sound due o he negligibili y o he associa ed noise compa ed o ha o he ope a ing
en i onmen . The second ype e e s o luc ua ions wi hin he ange 49.5~50.5 /min, as shown in
Fig. 1 o ime alues highe han 120 s. Bo h he ypes o iden i ied luc ua ion ha e he po en ial
o b eak he dynamic balance o o ques be ween he hyd o- u bine and he gene a o . In ligh o his,
a p obabilis ic ep esen a ion o he o que caused by he speed luc ua ion is essen ial o ob ain he
s a e-space ep esen a ion o he gene a o speed. Le
Du
w
be he andom o que, hen he
gene a o speed can be w i en as:
''
2
''
1sin sin 2
2
qs d q
s
ab ddq
EV x x
V
mDDu
Txxx
wddww
SS
SSS
éù
-
=- - --
êú
êú
ëû
(2)
whe e u is he andom a iable in oduced in his s udy. I s p obabili y densi y unc ion, shown in
Fig. 2, can be desc ibed as [37]
2
21; 1.
()
0; 1.
uu
pu
u
p
ì-£
ï
=í
ï>
î
(3)
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
u
p(u)
Fig. 2 The diag am o he p obabili y densi y unc ion
()pu
.
Chebyshe polynomial app oxima ion is used o simpli y he p obabilis ic model o he
hyd aulic-mechanical-elec ic sys em. Such app oxima ion can be p esen ed as:
22
0
(1)( )!
() (2)
!( 2 )!
n
k
nk
n
k
nk
Uu u
kn k
-
=
--
=
-
å
. (4)
I s ecu ence ela ion is
[ ]
11
1
() () ()
2
nnn
nU u U u U u
-+
=+
, (5)
and he app oxima ion p ope y is
12
1
1,
21()()
0,
ij
ij
uU uU udu ij
p
-
=
ì
-=
í¹
î
ò
(6)
Acco ding o he app oxima ion heo y o o hogonal polynomials and he abo e analysis, he
andom a iables o he sys em can be w i en as

( )
( )
( )
( )
( )
( )
11
0
22
0
33
0
0
0
0
44
0
(, ) () ( )
(, ) () ( )
(, ) () ( )
(, ) () ( )
(, ) () ( )
(, ) () ( )
() () ( )
N
i
i
i
N
i
i
i
N
i
i
i
N
i
i
i
N
i
i
i
N
i
i
i
N
iii
i
x u x Uu
x u x Uu
x u x Uu
u U u
u U u
y u y U u
x x Uu
dd
ww
=
=
=
=
=
=
=
ì=
ï
ï
ï=
ï
ï
ï=
ï
ï
ï
ï=
í
ï
ï
=
ï
ï
ï
=
ï
ï
ï
ï=
ï
î
å
å
å
å
å
å
å
, (7)
whe e N is he maximum numbe o Chebyshe polynomials;
( )
1
11
1
() ( ) (, ) ( )
i
i
x pux uUudu
+
-
=ò
;
( )
1
22
1
() ( ) (, ) ( )
i
i
x pux uUudu
+
-
=ò
;
( )
1
33
1
() ( ) (, ) ( )
i
i
x pux uUudu
+
-
=ò
;
1
1
() ( ) (, ) ( )
ii
pu uUudu
dd
+
-
=ò
;
1
1
() ( ) (, ) ( )
ii
pu uUudu
ww
+
-
=ò
;
1
1
() ( ) (, ) ( )
ii
y puy uUudu
+
-
=ò
;
1
44
1
() ( ) (, ) ( )
iii
x pux uUudu
+
-
=ò
.
A ligh o his, he andom s a e-space equa ions o he hyd aulic-mechanical-elec ic sys em can be
ew i en as:
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
12
00
23
00
3011223
00000
0
() ( ) () ( )
() ( ) () ( )
() ( ) () ( ) () ( ) () ( ) () ( )
() ( )
NN
ii
ii
ii
NN
ii
ii
ii
NNNNN
iiiii
iiiii
iiiii
N
i
i
i
dx Uu x Uu
d
dx Uu x Uu
d
dx Uu a x Uua x Uua x Uu y Uu
d
d U u
d
d
==
==
=====
=
éù
=
êú
ëû
éù
=
êú
ëû
éù
=- - - +
êú
ëû
åå
åå
ååååå
( )
( )
( )
( )
( ) ( ) ( ) ( ) ( )
0
'
00
4
00000
() ( )
1
() ( ) () ( )
1
() ( ) ( ( () ( )) () ( ) () ( ) () ( ))
N
si
i
i
NN
i e i
ii
ii
ab
NNNNN
ip iiid i i
iiiii
iiiii
y
U u
d U u m P D Du U u
d T
dd
y Uu k Uu k x Uu k Uu y Uu
d T d
ww
ww
ww
=
==
=====
éù
=
êú
ëû
éùé ù
=--+
êúê ú
ëûë û
éù éù
=-+- -
êú êú
ëû
ëû
åå
åå
ååååå
( ) ( )
4
00
() ( ) () ( )
NN
ii
ii
ii
dx Uu Uu
d
w
==
ì
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
í
ï
ï
ï
ï
ï
ï
ï
ïéù
ï=-
êú
ïëû
îåå
(8)
whe e
( ) ( )
''
2
'
''
00
sin ( ) ( ) sin 2 ( ) ( )
2
NN
qs d q
s
ei i
ii
ii
ddq
EV x x
V
P Uu Uu
xxx
dd
åå
==
ååå
-
éù éù
=-
êú êú
ëû ëû
åå
.
F om Eq. (5), he ollowing ela ionship can be ob ained:
2
00
01 1
12 3 1
()
22
22()
Uu U
UU U
UU U U
=
=
=+
(9)
Combining he exp essions ob ained so a ,
( )
0
( ) ( ) ( )
N
i
i
i
Du U u
w
=
éù
êú
ëû
å
can be simpli ied as:
Du( )
ω
i
( )
( )Ui(u)
i=0
N
∑
"
#
$
$
%
&
'
'
=1
2
D
ω
i
( )
( )Ui−1(u)+Ui+1(u)
"
#%
&
i=0
N
∑=1
2
D
ω
i−1
( )
( )+
ω
i+1
( )
( )
"
#
$%
&
'Ui(u)
i=0
N
∑
, (10)
whe e
1
w
-
=0, and
1N
w
+
=0.
Replacing Eq. (10) in o Eq. (8), he p obabilis ic model o he sys em can be w i en as:
( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )
( )
12
00
23
00
3011223
00000
0
() ( ) () ( )
() ( ) () ( )
() ( ) () ( ) () ( ) () ( ) () ( )
() ( )
NN
ii
ii
ii
NN
ii
ii
ii
NNNNN
iiiii
iiiii
iiiii
N
i
i
i
dx Uu x Uu
d
dx Uu x Uu
d
dx Uu a x Uua x Uua x Uu y Uu
d
d U u
d
d
==
==
=====
=
éù
=
êú
ëû
éù
=
êú
ëû
éù
=- - - +
êú
ëû
åå
åå
ååååå
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0
11
000
4
000
() ( )
11
() ( ) () ( ) () () ( )
2
1
() ( ) ( ( () ( )) () ( ) () (
N
si
i
i
NNN
i e i i
iiii
iii
ab
NNN
ip iiid i
iiii
iii
y
U u
d U u m P D U u D U u
d T
dd
y Uu k Uu k x Uu k Uu
d T d
ww
wwww
ww
=
-+
===
===
éù
=
êú
ëû
éùé ù
éù
=-- - +
êúê ú
ëû
ëûë û
éù
=-+-
êú
ëû
åå
ååå
ååå
( )
( ) ( )
00
4
00
)()())
() ( ) () ( )
NN
i
i
ii
NN
ii
ii
ii
y Uu
dx Uu Uu
d
w
==
==
ì
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
í
ï
ï
ï
ï
ïéù
ï-
êú
ïëû
ïéù
ï=-
êú
ïëû
î
åå
åå
(11)
Le us mul iply he abo e sys em o equa ions by
()
i
Uu
. Conside ing he ma hema ical expec a ion
wi h ega d o he andom a iable u on bo h sides o Eq. (11), and se ing i=0, 1, 2, 3, and 4, he
ini ial p obabilis ic model can be app oxima ed by he sys em in Eq. (12):
10 20
20 30
30 0 10 1 20 2 30 0
00
''
2
00 0 0 1
''
00
0 0 40
11
()sin () () ()sin2 () ()
22
1
s
NN
qs d q
s
ii ii
ii
ab ddq
pi d
y
xx
xx
xaxaxaxy
EV x x
V
mE Uu Uu Uu Uu D D
Txxx
ykkxk
T
dww
wd dww
ww
SS
==
SSS
=
=
=- - - +
=
éù
æö
-
æö æö
=-+--
êú
ç÷
ç÷ ç÷
ç÷
èø èø
êú
èø
ëû
=-+-
åå
( )
00
40 0
11 21
21 31
31 0 11 1 21 2 31 0
11
''
2
11 1 1 0
''
00
1
2
11
()sin () () ()sin2 () ()
22
s
NN
qs d q
s
ii ii
ii
ab ddq
y
x
xx
xx
xaxaxaxy
EV x x
V
mE Uu Uu Uu Uu D D
Txxx
w
dww
wd dww
SS
==
SSS
ì
ï
ï
ï
ï
ï
ï
í
ï
ï
ï
ï-
ï
ï=-
î
=
=
=- - - +
=
æö
-
æö æö
=-+--+
ç÷
ç÷ ç÷
ç÷
èø èø
èø
åå
( )
2
1 1 41 1 1
41 1
14 24
24 34
34 0 14 1 24 2 34 4
44
''
2
44 4 4
''
0
1
1
2
1()sin () () ()s
2
pi d
y
s
N
qs d q
s
ii
i
ab ddq
d
ykkxky
T d
x
xx
xx
xaxaxaxy
EV x x
V
mE Uu Uu Uu
Txxx
w
ww
w
dww
wd
SS
=
SSS
ì
ï
ï
ï
ï
ï
ï
ï
íéù
ïêú
ïêú
ëû
ï
ïæö
=-+- -
ç÷
ïèø
ï
ï=-
î
=
=
=- - - +
=
-
æö
=-+
ç÷
èø
å3
0
4 4 44 4 4
43 4
1
in 2 ( ) ( ) 2
1
N
ii
i
pi d
y
U u D D
d
ykkxky
T d
x
dww
ww
w
=
ì
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
í
ï
ï
ï
ï
ï
ï
ï
ï
ïì
ïï
ïï
ïï
ïï
ïï
ïï
ïï
í
ïéù
æö
æö
ï
ï--
êú
ç÷
ç÷
ç÷
ï
ïèø
êú
èø
ëû
ï
ïïæö
ï=-+- -
ç÷
ï
ïèø
ï
ïï
ï=-
î
î
å
(12)
The o mula o he ma hema ical expec a ion o he elec omagne ic o que
'
e
P
can be hen
w i en as:
( ) ( )
( ) ( )
''
2
'
''
00
''
2
44
1
''
100
sin ( ) ( ) sin 2 ( ) ( )
2
sin ( ) ( ) sin 2 ( ) ( ) ( )
2
NN
qs d q
s
ei i
ii
ii
ddq
qs d q
s
ii
ii
ii
ddq
EV x x
V
EP E Uu Uu
xxx
EV x x
V
U u U u pudu
xxx
dd
dd
åå
==
ååå
åå
-==
ååå
æö
-
éù éù
éù
=-
ç÷
ëû êú êú
ç÷
ëû ëû
èø
æö
-
éù éù
=-
ç÷
êú êú
ç÷
ëû ëû
èø
åå
åå
ò
(13)
Finally, he a e age esponse can be e alua ed acco ding o he Eq. (12):
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
( )
[ ]
4
11 10
0
4
22 20
0
4
33 30
0
4
0
0
4
0
0
4
0
0
44
(, ) () ( ) ()
(, ) () ( ) ()
(, ) () ( ) ()
(, ) () ( ) ()
(, ) () ( ) ()
(, ) () ( ) ()
(, ) ()
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
ii
Ex u x U u x
Ex u x U u x
Ex u x U u x
E u Uu
E u Uu
Ey u y Uu y
Ex u x U
dd d
ww w
=
=
=
=
=
=
==
==
==
==
==
==
=
å
å
å
å
å
å
4
40
0
() ()
i
i
ux
=
ì
ï
ï
ï
ï
ï
ï
ï
ï
ï
ï
í
ï
ï
ï
ï
ï
ï
ï
ï
ï=
ï
îå
(14)
3. Nume ical simula ions
The p oposed model has been applied o a case-s udy in o de o p o e he e iciency and
consis ency o he app oach. The alues adop ed o he pa ame e s in ol ed in he compu a ion
ha e been selec ed wi hin ealis ic anges: he a ed gene a o speed is
s
w
=314; he ine ia ime
cons an o he hyd o- u bine gene a o uni is
ab
T
=8.0; he damping ac o o he gene a o is
D =0.5; he ansien in e nal ol age o he a ma u e is
'
q
E
=1.35; he di ec axis ansien eac ance
is
'
d
xå
=1.15; he quad a u e axis eac ance is
'
q
xå
=1.474; he majo elay connec e esponse ime is
y
T
=0.1 and he bus ol age a in ini y is
s
V
=1.0. The i s -o de pa ial de i a i e alue o low a e
wi h espec o wa e head is
qh
e
=0.5; he i s -o de pa ial de i a i e alue o o que wi h espec
o wicke ga e is ey=1.0; he in e media e a iable is e=0.7; he leng h o he phase o he wa e
hamme wa e is T =1.0; he elas ic ime cons an o he pens ock is hw=2.0; he e e ence inpu is
=0; he p opo ional gain o he PID con olle is kp=2 and he in eg al gain o he PID con olle
ki=1.
The ini ial alues o he de e minis ic model o he hyd aulic-mechanical-elec ic sys em (1) a e
[x1(0), x2(0), x3(0),
(0)
d
,
(0)
w
, y(0), x4(0)]=[0.001, 0.001, 0.001, 0.001, 0.001, 0.001, 0.001].
Simila ly, he ini ial alues o he andom hyd aulic-mechanical-elec ic sys em (11) a e
di e en alues o he in ensi y D and kd a ying om 0 o 6. The end associa ed wi h poin 3,
namely he alue o kd o which he gene a o speed
w
pass om he non- unable o he ib a ion
s a e, is shown in Tab. 3.
-8 -6 -4 -2 0 2 4 6 8
-0.05
0
0.05
0.1
kd
ʍ
-8 -6 -4 -2 0 2 4 6 8
-0.05
0
0.05
0.1
kd
ʍ
D=0.06 D=0.16
-8 -6 -4 -2 0 2 4 6 8
-0.05
0
0.05
0.1
kd
ʍ
-8 -6 -4 -2 0 2 4 6 8
-0.05
0
0.05
0.1
kd
ʍ
D=0.36 D=0.52
Fig. 6 Dynamic e olu ion o he gene a o speed
w
o he p obabilis ic model in Eq. (15) wi h
di e en alues o he andom in ensi y D and kd a ying om -8 o 8. (a) D=0.06; (b) D=0.16; (c)
D=0.36; (d) D=0.52;
Tab. 3 Loca ion o poin 3 acco ding o inc easing alues o he andom in ensi y D
D
0.04
0.06
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.36
0.40
Poin 3
-5.854
-5.646
-5.542
-5.021
-4.604
-3.979
-3.25
-2.521
-1.479
-0.2292
0.083
As shown in Fig. 6 and Tab. 3, inc easing le els o he in ensi y D cause he egion o he
domain associa ed wi h he non- unable s a e shi ing o he igh , while he ope a ing s a e on he
igh o poin 3 emains unchanged. As expec ed, he end egis e ed o he p obabilis ic model in
Eq. (15), conside ing he igid wa e hamme model, ma ches pe ec ly he esul s ob ained o he
p e ious p obabilis ic model o Eq. (11).
In addi ion o his, as shown in Fig. 5(a) and Fig. 6(a), i can be no iced ha when he sys em

ope a es in he egula ib a ion s a e, he ampli ude o he luc ua ions o he p obabilis ic model o
Eq. (11) a ies in he ange o [0.0188, 0.02015], and he adjus able ange o kd is [1.951, 2.563].
Rega dless o he shape changes o he pens ock wall, he ampli ude o he luc ua ions o he
p obabilis ic model in Eq. (15) a ies in he ange [0.01754, 0.02281], which is simila o he
p e ious case. Con e sely, he adjus able ange o kd, which is [0.083, 4.458], esul s much la ge
han ha associa ed wi h he p e ious p obabilis ic model. Mo eo e , when he hyd o- u bine
gene a o uni ope a es in he quasi pe iodic ib a ion s a e, al hough he ampli ude o he
luc ua ions associa ed wi h he model o Eq. (11) is simila o ha o he model o Eq. (15), he
ange o kd is much la ge han ha associa ed wi h he la e p obabilis ic model.
4. Conclusions
In his s udy, a andom a iable u is in eg a ed wi h he gene a o speed o he de e minis ic
model o a hyd aulic-mechanical-elec ic sys em o es ablish he co esponding p obabilis ic model.
Using his la e , he dynamic e olu ion o he sys em is analyzed, and h ee main conclusions can be
achie ed. Fi s , al hough he dynamic e olu ion o he andom sys em is b oadly simila o ha o
he de e minis ic app oach, he wo se s o esul s show signi ican di e ences, ca e ully
in es iga ed in he pape . Second, he poin 3, which highligh s he ansi ion om he no - unable
s a e o he andom ib a ion s a e, shi s o he igh wi h inc easing alues o he andom in ensi y
D, which leads o he dec easing adjus able ange o kd. Thi d, he consis ence o he ends
highligh ed by he p obabilis ic model implemen ed in his pape is compa ed and e i ied h ough
he use o ano he model. Mo eo e , when he shape o he pens ock is assumed no uni o m, he
adjus able ange o kd is na owed om le o igh and he ope a ing s a e o he sys em becomes
less s able. A ligh o he ob ained esul s and o he apid de elopmen o wind powe , i is
ecommendable o selec alues o he di e en ial gain kd o he PID go e no as la ge as possible.
Acknowledgmen -
This wo k was suppo ed by he scien i ic esea ch ounda ion o Na ional Na u al Science
Founda ion--Ou s anding You h Founda ion (51622906), Na ional Na u al Science Founda ion
(51479173), Fundamen al Resea ch Funds o he Cen al Uni e si ies (201304030577), Scien i ic
esea ch unds o No hwes A&F Uni e si y (2013BSJJ095), Science Fund o Excellen Young
Schola s om No hwes A&F Uni e si y and Shaanxi No a p og am (2016KJXX-55).
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