IEEE Computational Intelligence Magazine - November 2023 - 26

TABLE IV The best-found solution by all runs of the five algorithms for each decision variable and the decision values of two additional
knee points.
ID BESTOBJECTIVE
Cinvest
1
2
3
4
5
6
7
8
9
10
11
12
Cannual
Gtotal
R
bSOC
Ebatt;discharge
Ppeak;supply
tm
Efeed
Ppeak;feed
(knee point 1)
(knee point 2)
aPV
33.63
45.00
45.00
34.65
18.98
29.32
31.57
22.88
28.69
22.99
6.33
27.06
bPV
57.25
185.49
185.49
216.24
199.38
19.57
161.47
16.10
41.50
351.10
254.21
209.21
PPV
10.00
450.00
450.00
415.43
270.19
10.00
18.45
218.87
383.68
226.85
145.55
243.38
thesamesolutionisalsoabletoachieve thelowestlevel of
CO2 emissions (Gtotal,see solution 3).
Solution 4 utilizes a large PV system in combination with a
large battery. Together with the high mean state of charge
(bSOC) of the battery and its overall high charging limit
(bSOC;max), this results in the longest resilience time in cases
when no grid power is available.
Solution 5 requires only a minimum battery SOC
(bSOC;max) and achieves the lowest mean battery state ofcharge.
Solution 6 utilizes the smallest possible PV system, as well
as the smallest possible battery capacity and lowest maximum
charging limit (bSOC;max), thereby eliminating the utilization of
the battery (shown by the lowest value ofEbatt;discharge).
Solution 7 surprisingly achieves the lowest power peak
demand from the grid (Ppeak;supply), despite a very small PV
CB
5.00
465.32
465.32
1000.00
178.78
5.00
348.58
138.39
728.03
469.36
596.61
918.93
bSOC;max
0.5000
0.5200
0.5200
0.8883
0.5039
0.5000
0.8697
0.5634
0.9282
0.9472
0.9090
0.5361
bSOC;min
0.0716
0.3681
0.3681
0.3606
0.0500
0.1242
0.0908
0.1672
0.1711
0.1074
0.3607
0.2591
Pcharge
149.87
-431.47
-431.47
45.10
-358.75
3.61
149.90
48.97
32.86
-475.62
-313.61
60.78
Pdischarge
278.83
698.76
698.76
629.54
195.20
590.26
268.40
595.48
392.67
206.53
539.80
396.57
VCHP
4.8401
4.9332
4.9332
4.8264
4.9628
4.9609
4.6988
4.2494
4.6852
1.0618
3.1487
4.3863
system. The rather small battery (CB)isefficiently discharged
(shown by an above average value for Ebatt;discharge)
to compensate the demand peak. This behaviour is achieved
with the highest possible battery charging threshold (Pcharge)
and a low discharging threshold (Pdischarge). Overall, the solution
seems to utilize the small battery very efficiently for
power peak shaving. Interestingly, the solution does not
rely on a large PV system to produce additional power
when the building demand is high. Hence, an efficiently
used battery seems to be more important for power peak
shaving than a large PV system, in this specificscenario.
Solution 8 achieves the largest time share (tm) in which the
battery SOC is between 30% and 70% due to a combination
of a low maximum battery SOC, high charging and high discharging
thresholds. The battery has a mean state of charge of
roughly 60%. Small charging or discharging processes are likely
to keep the SOC between the desired values.
For solution 9, the lowest possible energy that is fed into
the grid (Efeed) corresponds to a feed-in power peak value of
zero and a high value for tm. In spite of the large PV system,
no excess energy is being produced and fed into the grid. This
can be seen as a result of the low orientation angle of the PV
system (bPV). The orientation leads to an overall low energy
production, in particular during times of higher energy
demand around midday.
Solution 10 can be interpreted in a similar way. The high
value for tm and low value for Efeed are in line with the lowest
possible feed-in power peak. The PV system is directed North,
thus producing only a small amount ofenergy. Note that this is
a very inefficient solution, indicating that multiple objectives
are always needed to be considered.
Several insights into the optimization of the BEM probFIGURE
14 Distribution of the objective values for all Pareto nondominated
solutions: The boxes indicate the 25%-75% percentile
range, and the whiskers mark the minimum and maximum values. The
markers indicate the two knee point solutions.
26 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | NOVEMBER 2023
lem can be further gained from the distribution of the decision
variables and objectives of the Pareto non-dominated
solutions (Figs. 13 and 14). While some decision variables,
such as aPV, PPV, CB, bSOC;max, and bSOC;min
take a wide
range of values, bPV, Pcharge, Pdischarge, and VCHP can only

IEEE Computational Intelligence Magazine - November 2023

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