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Uncertainty management in decision-making Chapter | 2 47

c
c
P min ðtÞX # P ESS e ðtÞ # P max ðtÞX ; ’tAT; ’eAE ð2:9Þ
char e e;t char e e;t
where P min and P min are the minimum power discharged and charged by the
char
dis
th
e ESS, respectively. P max and P max are the maximum power discharged and
char
dis
charged by the e th ESS, respectively. X d and X c are binary variables for
e;t e;t
th
discharging and charging modes of the e ESS, when charging, the charging
c
d
state X is 1 and the discharging state X is 0, so the minimum and maxi-
e e
mum limits of charging mode are imposed (2.9). When discharging, the dis-
c
d
charging state X is 1 and the charging state X is 0, so the minimum and
e e
maximum limits of discharging are imposed (2.8). To separate the ESS oper-
ation modes (ESS cannot operate in both charging and discharging modes,
simultaneously), another constraint is considered as follows:
d
c
X 1 X # 1; ’eAE ð2:10Þ
e e
Energy storage system state-of-charge (SOC) constraint is calculated as
0 # SOC e;t # SOC max ; ’eAE ð2:11Þ
e
th
where SOC max is the maximum SOC of the e ESS.
e
The energy storage system is subject to minimum charging and dischar-
ging time limits, respectively [33]:

c
T char $ MC e X 2 X c
e e;t e;t21
’eAE; ’tAT ð2:12Þ

d
T dis $ MD e X 2 X d
e e;t e;t21
where MC e and MD e are the minimum charging and discharging time of the
th
e ESS, respectively. T char and T dis are the number of continuous charging
e e
th
and discharging time (per hour) of the e ESS, respectively.
2.3.1.4 Minimum up/down time constraint
As we know, a dispatchable unit is limited by minimum up and down times.
The time during which a unit must be on/off after being startup/shutdown
is called minimum up/down time constraints, respectively, which are as
follows [34]:
on
MUT i X i;t 2 X i;t21 # T i
ð2:13Þ
off
MDT i X i;t21 2 X i;t # T i
th
where MUT i and MDT i are the minimum up and down time of i generator,
th
respectively, X i;t and X i;t21 are the commitment status of i generator at
th th on off
t 2 1 and t time, respectively. T i and T i are the numbers of on and off
th
hours of i generator, respectively.