PhD Thesis - Cranfield University

Chapter 5
With the reference power input matrix expanded to 4 x 1, the connection matrix A takes a 4
x 4 dimension to from an agreeable matrix equation as follows,
⎡D
⎢
⎢
D
⎢D
⎢
⎣D
T1
T 2
T 3
T 4
⎤ ⎡ A
⎥ ⎢
⎥ = ⎢
A
⎥ ⎢A
⎥ ⎢
⎦ ⎣A
11
21
31
41
A
A
A
A
12
22
32
42
A
A
A
A
13
23
33
43
A
A
A
A
14
24
34
44
159
⎤ ⎡ Pb
⎥ ⎢
⎥ ⎢ Pb
.
⎥ ⎢Puc
⎥ ⎢
⎦ ⎢⎣
Puc
+
−
+
−
⎤ ⎡ P1⎤
⎥ ⎢ ⎥
⎥ ⎢
P2
+ ⎥
⎥ ⎢P3⎥
⎥ ⎢ ⎥
⎥⎦
⎣P4⎦
(5-35)
By nature that the batteries and ultracapacitors cannot charge and discharge simultaneously,
four diagonal elements in the connection matrix can be set to zero. This results in the
following,
⎡D
⎢
⎢
D
⎢D
⎢
⎣D
T1
T 2
T 3
T 4
⎤ ⎡ 0
⎥ ⎢
⎥ = ⎢
A
⎥ ⎢A
⎥ ⎢
⎦ ⎣A
21
31
41
A
12
0
A
A
32
42
A
A
13
23
0
A
43
+ A14
⎤ ⎡ Pb ⎤ ⎡ P1⎤
⎥ ⎢ − ⎥
A
⎢ ⎥
24 ⎥ ⎢ Pb ⎥ + ⎢
P2
.
⎥
⎥ ⎢ + A ⎥
34 Puc ⎢P3⎥
⎥ ⎢ − ⎥ ⎢ ⎥
0 ⎦ ⎢⎣
Puc ⎥⎦
⎣P4⎦
(5-36)
Further reduction of elements in the connection matrix can be attained by considering the
operating modes of the PES state machine and by eliminating non-applicable conditions.
The four duty cycles are then expressed as,
DT 1
DT 2
DT 3
DT 4
= Pb
+
= Pb
= Pb
= Pb
+
+
+
A
A
A
A
12
12
31
41
+ P1
+ Puc
+ Puc
+ Puc
+
+
+
A
A
A
23
34
43
+ Puc
+ P3
+ P4
−
A
24
+ P2
in state [ 011 ] (5-37)
in states [ 101 , 100 , 110 ] (5-38)
in states [ 110 , 010 ] (5-39)
in states [ 001 , 101 ] (5-40)