Page 149 - IJOCTA-15-1

Modeling the renewable energy development in T¨urkiye with optimization
modeling parameters have been consid-
M
1 X ered. This results in 45 combinations
min MAE = |ˆy i (b) − y i | (1)
b M (N C = 45).
i=1 • Case 2: Including 4 optimization vari-
s.t. ˆy i (b) − y i = 0, i = 1, . . . , M. (2)
ables (3 modeling parameters and an in-
tercept). All possible combinations in-
cluding 3 modeling parameters out of 10
where M is the number of years considered for modeling parameters have been consid-
developing the models. As the data used to de- ered. This results in 120 combinations
velop the models belong to 2005-2019 period, (N C = 120).
M = 15. This is due to the fact that some of the • Case 3: Including 5 optimization vari-
most recent data were missing for some modeling ables (4 modeling parameters and an in-
parameters. Therefore, data belonging to years tercept). All possible combinations in-
2020 and 2021 have been left for testing the de- cluding 4 modeling parameters out of 10
veloped models and data between 2005-2019 have modeling parameters have been consid-
been used in optimization. In this equation, y i ered. This results in 210 combinations
represents the real installed RE capacity (MW) (N C = 210).
for year i and the estimated renewable energy
Then, the optimization problem is solved for each
capacity (MW), ˆy, is:
of the combinations in each case. The order of se-
lection for modeling parameters (MPs) is shown
in Table 2 for the first 12 combinations of each
N
X
ˆ y i (b) = b 0 + b k x ki (3) case. In this table, the numbers stated for each
modeling parameter (from 1 to 10) are the ones
k=1
mentioned in section 4.
Hence for Case 1, the optimization problem has
been solved 45 times by choosing two different
where x ik is the value of the modeling parame-
modeling parameters in each time the optimiza-
ter k in year i, b 0 is the intercept, and b k is the
tion was performed. Similarly for Case 2, 120 op-
weighting factor related with modeling parame-
timization problems have been solved in order to
ter k. b k ’s (b 0 , b 1 , ...b N ) are the decision variables
determine the best combination of the three mod-
of the designed optimization problem. Hence, the
eling parameters. Finally in Case 3, 210 optimiza-
optimization problem has N+1 decision variables.
tion problems have been solved consisting of four
The constraint in equation (2) is to ensure that a
different modeling parameters in each one. These
feasible solution results in no error (hence the es-
optimizations are done by setting an initial point
timated value is the same with the real value) for
consisting of N+1 ones (1s), for each combination.
each year. This constraint enforces the optimiza-
This initial point represents the N+1 optimization
tion algorithm to try to decrease each individual
variables; the weighting factors related with the
error to 0. Although the algorithm may converge
modeling parameters (N variables) and the inter-
into an infeasible solution, that solution would be
cept (1 variable). Hence, there are three, four and
the result of an effort towards satisfying all the
five optimization variables in Case 1, Case 2 and
constraints. The designed optimization problem
Case 3 respectively. Each time the optimization
is solved by using Matlab, as follows: The related
algorithm is run (45 times in Case 1, 120 times
objective function (eqn. (1)) and the constraints
in Case 2 and 210 times in Case 3), it uses the
(eqns. in (2)) have been created and the fmincon same initial point (consisting of 1’s) to start its
function is used in order to solve each problem. search and finally converges to a point that min-
SQP (Sequential quadratic programming) is cho- imizes the objective function (namely MAE) for
sen for fmincon, as it aims to satisfy bounds at that combination.
all iterations. This method is called MAEOPT
(Mean Absolute Error based OPTimization) in 4.3. Comparison with a benchmark
this study. Based on MAEOPT, the problem is
In order to provide a benchmark for the developed
solved for all the combinations of the modeling
models, their Multiple Linear Regression (MLR)
parameters. Three cases have been considered:
versions are also developed. In order to do this,
• Case 1: Including 3 optimization vari- Matlab’s regress function is used (instead of solv-
ables (2 modeling parameters and an in- ing the optimization problem stated in 1 and 2)
tercept). All possible combinations in- for each of the combinations in each case. The
cluding 2 modeling parameters out of 10 regress function returns a vector b of coefficient
143