Modelling the Occupant Behaviour Impact on Buildings Energy Prediction - Nearly Zero Energy Building Refurbishment - page 130

seed k ¼ X

X

n m

; 8 MC k 2 ECM

10 3 p i ; j ð 7 Þ

i ¼ 1 j ¼ 1

This process is illustrated in Fig. 9 a, and it is referred to as open loop seed
discovery. A property of ( 7 ) is that the seed becomes a property of the model, since
it is derived from its parameters. The upside of this method is the simple com-
putational calculation; however, it does not provide any feedback about the gen-
erated sequence. The sequence can be the worst-case prediction. In order to
account for the generated sequence, another algorithm was implemented - the
close-loop seed discover (Fig. 9 b). In this method, the performance of the model is
evaluated by a residual analysis and the model prediction error. The following
residual is applied

s

1
n

X

n

Þ 2 RMSE ¼ ð y i x 1 ð 8 Þ

i ¼ 1

s

1

n P

n

Þ 2 ð

y i x 1

i ¼ 1

RMSRE ¼ ð 9 Þ

y

In the above definitions, RMSE is the Root Mean Square Error and RMSPE is
the Root Mean Square Relative Error. There, x i is measured data, y i is ECM
prediction, y is the measured data mean value and n is the number of data points.
The objective is to select the best seed within an interval that minimises the
RMSE or RMSRE in order to select an optimal state sequence. To accomplish this,
it is necessary to introduce two new metrics. The state distribution error (SDE)
and the prediction error (PE). Considering that the two data sets (estimation and
validation) are two time series, the SDE is the quadratic error between both the
steady-state distributions

SDE ¼ X

#states

Þ 2

ð P s ðÞ P ð v k Þ ð 10 Þ

k ¼ 1

The PE represents the quadratic error between the model and the building

Þ 2 PE ¼ y ECM y B

ð ð 11 Þ

The optimal state metric is calculated by minimising the previous metrics

seed k ¼ argmin seed k f RMSE ; SDE ; PE g ð 12 Þ