Improving the computational efficiency of stochastic programs using automated algorithm configuration: an application to decentralized energy systems

Abstract

The optimization of decentralized energy systems is an important practical problem that can be modeled using stochastic programs and solved via their large-scale, deterministic-equivalent formulations. Unfortunately, using this approach, even when leveraging a high degree of parallelism on large high-performance computing systems, finding close-to-optimal solutions still requires substantial computational effort. In this work, we present a procedure to reduce this computational effort substantially, using a state-of-the-art automated algorithm configuration method. We apply this procedure to a well-known example of a residential quarter with photovoltaic systems and storage units, modeled as a two-stage stochastic mixed-integer linear program. We demonstrate that the computing time and costs can be substantially reduced by up to 50% by use of our procedure. Our methodology can be applied to other, similarly-modeled energy systems.

Appendices

Appendix A

The entire two-stage stochastic MILP of the residential quarter is shown in the following (nomenclature is listed in Table 2):Objective function:

$$ \begin{aligned} costs &= \mathop {min}\limits_{{c_{g,i} ,e_{\omega ,t}^{grid} ,e_{\omega ,t}^{fi} }} ANF \cdot \mathop \sum \limits_{g = 1}^{4} \mathop \sum \limits_{i = 1}^{{k_{1} }} cost_{i} \cdot c_{g,i} \\ & \quad \quad \frac{ + 1}{N} \cdot \mathop \sum \limits_{\omega = 1}^{N} \mathop \sum \limits_{t = 1}^{T} \left( {p^{grid} \cdot e_{\omega ,t}^{grid} - p^{fi} \cdot e_{\omega ,t}^{fi} } \right), \\ \end{aligned} $$

(A.1)

Table 2 Nomenclature of the residential quarter modeled as a two-stage stochastic program

• the installed PV capacity of the quarter: $\mathop \sum \limits_{g = 1}^{4} c_{g,i = PV} = 240$,

• the number of heat pumps for SH within a building group: $c_{{g,i = HP_{SH} }} = 1$,

• the number of heat pumps for DHW within a building group: $c_{{g,i = HP_{DHW} }} = 1,$

• the number of heating elements for the SH storage: $c_{{g,i = HE_{SH} }} = 4,$

• the number of heating elements for the DHW storage: $c_{{g,i = HE_{DHW} }} = 4$.

Additionally, electrical supply and demand have to be balanced:

$$ e_{\omega ,t}^{pv} + e_{\omega ,t}^{grid} = d_{\omega ,t}^{ee} + \mathop \sum \limits_{g = 1}^{4} \mathop \sum \limits_{u = 1}^{2} \left( {d_{\omega ,g,u,t}^{hp} + d_{\omega ,g,u,t}^{he} } \right) + e_{\omega ,t}^{fi} \quad \forall \omega \forall t, $$

(A.2)

with supplied PV energy $e_{\omega ,t}^{pv} = \mathop \sum \limits_{g = 1}^{4} e_{\omega ,t}^{pv,kwp} \cdot c_{g,i = PV}$ and balanced thermal supply and demand:

$$ \begin{aligned} & COP_{\omega ,u,t} \cdot d_{\omega ,g,u,t}^{hp} + \eta \cdot d_{\omega ,g,u,t}^{he} + \left( {1 - l_{u} } \right) \cdot s_{\omega ,g,u,t} + q_{\omega ,g,u,t} \\ & \quad = d_{\omega ,g,u,t}^{th} + L_{\omega ,g,u,t} + s_{\omega ,g,u,t + 1} + pos_{\omega ,g,u,t} \cdot r_{u} \quad \forall \omega ,\forall g,\forall u,\forall t, \\ \end{aligned} $$

(A.3)

with the storage heat losses $l_{u = SH} = 0.003$ and $l_{u = DHW} = 0.006$ and ramp-up losses $r_{u} = 0.05$.

The storage possibility is restricted by:

$$ s_{g,u}^{min} \le s_{\omega ,g,u,t} \le c_{{g,i = S_{u} }}\quad \forall \omega ,\forall g,\forall u,\forall t, $$

(A.4)

where $s_{g,u}^{min} = 0$. Load changes are taken into account by:

$$ z_{\omega ,g,u,t + 1} - z_{\omega ,g,u,t} = pos_{\omega ,g,u,t} - neg_{\omega ,g,u,t} \quad \forall \omega ,\forall g,\forall u,\forall t. $$

(A.5)

The heating element supply for each building group is given by:

$$ \eta \cdot d_{\omega ,g,u,t}^{he} \le c_{{g,i = HE_{u} }} \cdot d^{he,max}\quad \forall \omega ,\forall g,\forall u,\forall t, $$

(A.6)

and the heat pump supply by:

$$ COP_{\omega ,u,t} \cdot d_{\omega ,g,u,t}^{hp} = \frac{1}{m} \cdot d_{\omega ,t}^{hp,max} \cdot z_{\omega ,g,u,t} \quad \forall \omega ,\forall g,\forall u,\forall t, $$

(A.7)

$$ z_{\omega ,g,u = DHW,t} \le m \cdot c_{{g,i = HP_{DHW} }} \quad \forall \omega \forall g\forall t, $$

(A.8)

$$ \mathop \sum \limits_{u = 1}^{2} z_{\omega ,g,u,t} \le m \cdot \mathop \sum \limits_{u = 1}^{2} c_{{g,i = HP_{u} }} \quad \forall \omega ,\forall g,\forall t. $$

(A.9)

If heat pumps run only at idle, half or full load, then $m = 2$ with $z_{\omega ,g,u = SH,t} \in \left\{ {0,1,2,3,4} \right\}$ and $z_{\omega ,g,u = DHW,t} \in \left\{ {0,1,2} \right\}$, otherwise $z_{\omega ,g,u = SH,t} ,z_{\omega ,g,u = DHW,t} \in R_{ + }$. The following constraints equal the element of the first and last time step $t$:

$$ s_{\omega ,g,u,t = T} = s_{\omega ,g,u,t = 1} \quad \forall \,\omega ,\forall g,\forall u, $$

(A.10)

$$ z_{\omega ,g,u,t = T} = z_{\omega ,g,u,t = 1} \quad \forall\, \omega ,\forall g,\forall u. $$

(A.11)

All presented variables need to be positive:

$$\begin{aligned} &c_{g,i} ,e_{\omega ,t}^{grid} ,e_{\omega ,t}^{fi} ,q_{\omega ,g,u,t} ,d_{\omega ,g,u,t}^{hp} ,d_{\omega ,g,u,t}^{he} ,s_{\omega ,g,u,t} ,L_{\omega ,g,u,t} ,pos_{\omega ,g,u,t} ,\neg_{\omega ,g,u,t} ,z_{\omega ,g,u,t}\\ &\ge 0\,\forall \omega ,\forall g,\forall i,\forall u,\forall t.\end{aligned} $$

(A.12)

Appendix B

Table 3 lists the three parameters that have the largest effect on performance and their mean wall-clock time reduction, itemized in detail for all 27 partitions.

Table 3 Results of the ablation analysis listing the three parameters that have the largest effect on performance altered from the default to the partition-specific optimized configuration of CPLEX. The parameters are determined by SMAC on nine instances for the given 27 partitions. The values in the brackets show the mean wall-clock time of the nine instances to achieve a MILP gap of at most 0.6%, whereby unsuccessful runs that could not be solved within the cutoff-time of 1800 s are counted as having taken 18,000 s