# Optimised Heat Pump Management for Increasing Photovoltaic Penetration into the Electricity Grid

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## Abstract

**:**

`intlinprog`, it is shown that the HCA provides more accurate heat pump control with an increase of up to 9% in system Operating Expense OPEX and a decrease of 8% in self-consumption values.

## 1. Introduction

`GlobalSearch`and

`fmincon`and the MATLAB LP solver

`intlinprog`. Simulations are carried out over short and long periods (48 h and one month respectively) in the summer and winter using real consumption data from a typical Swiss house as well as calculated PV production. Details on the model, data used and simulation framework are given in the next section.

## 2. Methodology and Simulation Framework

#### 2.1. System Components and Modelling

#### 2.1.1. Inverter

#### 2.1.2. Heat Pump

#### 2.1.3. Domestic Hot Water Tank

#### 2.2. HP Operation Optimisation

#### 2.2.1. Data Input

#### 2.2.2. Parameters of Optimal Control Problem

#### 2.2.3. Nonlinear Optimisation Problem

#### 2.2.4. Mixed-Integer Linear Optimisation Problem

#### 2.2.5. Heuristic Optimal Control Problem

#### 2.2.6. Optimisation Solvers

`fmincon`and

`GlobalSearch`. However, their large computational burden and their dependency on the starting point (${x}_{0}$) raise the interest to linearize the original OCP into a simpler version shown in Section 2.2.4, for which the fast MILP MATLAB solver

`intlinprog`was selected. Moreover, the heuristic OCP was fed into the novel HCA developed in this work. A flowchart of this HCA algorithm can be found in Figure 3. Other variables included in this flowchart are listed in Table A1 in the Appendix A.

#### 2.2.7. Instantaneous Control Algorithm (ICA)

#### 2.3. Evaluation Metrics

`intlinprog`solvers in particular (case i = no superscript), as shown in Section 3.2. Equation (18) shows the OPEX function, which includes the cost of buying and selling electricity as well as the HP running and switching costs:

## 3. Results and Discussion

`intlinprog`,

`fmincon`and

`GlobalSearch`are used as benchmarks. Firstly, the HCA’s effectiveness is evaluated in 48 h horizon short-term simulations in winter and summer for a system size of 1 kW HP, 600 l TES and 3 kWp PV. This size is chosen to reproduce a typical household configuration with a large DHW tank to enhance PV self-consumption. The second section restricts the analysis of long-term system operation to cases controlled by the HCA and

`intlinprog`using the same system sizing applied to short-term simulations in winter and summer.

`fmincon`and

`GlobalSearch`are excluded from this case since they could not always find a reasonable local optimum or did not converge at all. Those issues arise as a consequence of the relatively flat search path experienced by NLP solvers when computing the values of the objective function defined in the nonlinear OCP (Section 2.2.3) throughout the feasible domain determined by the applicable set of constraints. The CPU used for simulations comprises an Intel Core i5-3230 (2.60 GHz) processor with 8 GB RAM on a 64-bits operating system.

#### 3.1. Short-Term Simulations

`fmincon`and

`GlobalSearch`was given by the result of

`intlinprog`. This procedure avoids using a non-optimal initial point ${x}_{0}$ and reduces these solvers’ running times. For the sake of clarity, this section is focused solely on comparing the performance of the different optimisation solvers. The performance of the ICA based on optimal HP energy management signals will be analysed in Section 3.2.

`fmincon`decides to operate the HP at mid-power levels intermittently over most of the PV production period to enhance the part-load efficiency values, keeping supply temperatures low. This results in the highest $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ values of all the solvers while the PV self-consumption is increased. Nevertheless, this power profile implies continuous HP switching, which would reduce its lifespan rapidly. For

`GlobalSearch`, the solver yields a HP behaviour similar to that of

`fmincon`, except that it runs the HP during some periods at very low power values with $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ values lower than 1. The reason for both deficiencies relies on the inability of derivative-based NLP solvers to utilize discrete variables. On the one hand, this means that

`fmincon`and

`GlobalSearch`can accept ${P}_{hp}^{opt}\left(t\right)$ only as a continuous decision variable. Hence, although there is a penalty in efficiency, these routines switch the HP on at very low power values as low power ranges are allowed and this inefficient operation incurs no cost in their objective function (Equation (7)). On the other hand, this limitation makes it difficult to assign a cost to the HP switching and running time, leading to detrimental intermittent HP power profiles.

`intlinprog`also follows a similar profile as the HCA’s. However, the definition of a linear $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ with a constant supply temperature has as consequences that the solver considers neither the reduction of $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ values when increasing the tank temperature nor the decrease of $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ when the HP is working at full load due to part-load ratio. All these factors lead to an overestimated $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ profile, which contributes to reducing the HP running periods with respect to those generated by the HCA.

`fmincon`and

`GlobalSearch`lead to 96% and 69% more expensive optimal control strategies, respectively). These results are only improved by the 33% cheaper

`intlinprog`operation. In winter, NLP solvers present the same behaviour as for the summer period, achieving the highest $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ values due to a combination of low supply temperatures and HP operation at optimal part-load efficiency ranges. For this case, they achieve local consumption of almost 100% of the PV generation. Moreover, the HCA and

`intlinprog`attain the lowest OPEX values since they can run the HP more smoothly than the NLP solvers while promoting the self-consumption of locally generated PV electricity. Note that, despite the fact that

`intlinprog`provides the cheapest operation strategy for summer and winter short-term simulations, those results are based on overestimated $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ values. Section 3.2 will present some of the impacts that this $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ inaccuracy produces in the real-time control of the system.

`intlinprog`can effectively deliver control trajectories in a computation time smaller than the simulation time-step, i.e., enabling real-time control. Note that all results gathered in Table 3 are totally deterministic, except running time values, which depend partly on external factors related to computer processing hierarchy, and GlobalSearch output, which may vary according to the set of trial starting points chosen randomly by a scatter search algorithm integrated in this solver [40]. As a consequence of the first exception, short-term winter and summer simulations were repeated twenty times for the HCA,

`intlinprog`and

`fmincon`while they were repeated five times for

`GlobalSearch`in order to obtain a mean run time value and its uncertainty range shown in Table 3. Due to the second exception, the

`GlobalSearch`${\mathrm{OPEX}}^{opt}$ and $S{C}^{opt}$ values produced in the fastest run out of the five executed for winter and summer periods are shown. The

`GlobalSearch`main variable profiles depicted in Figure 4 correspond to this case.

#### 3.2. Long-Term Simulations

`intlinprog`. In the case of DHW, low and high monthly consumption profiles are provided to simulations in summer and winter conditions, respectively. These long-term simulations also allow us to check the robustness of the algorithm with respect to the time dependent environmental conditions and demand profiles.

`intlinprog`records execution times significantly lower than the HCA. As in Section 3.1, summer and winter long-term simulations are repeated twenty times for both solvers, obtaining for each of them, run time data samples of 1200 and 1240 values that correspond to simulations run during the summer and winter months respectively. Distribution of these data samples is represented through several boxplots shown in Figure 5. Note that each of these run time values is the time the HCA or

`intlinprog`spends to calculate an optimal HP management strategy for a 48-h period, which occurs twice per day during the whole simulation month (the ICA running time is not included in these values). The larger PV generation experienced in summer made

`intlinprog`evaluate an increased number of possibilities to allocate HP power scheduling for self-consumption enhancement. This leads to longer running time values compared to results given for the winter month. Even so,

`intlinprog`can converge in 75% of all cases below three seconds and one second in summer and winter seasons, respectively.

`intlinprog`. The self-consumption rate experienced in the winter simulation controlled by the HCA is just 8% lower than is the case when

`intlinprog`is used. During the summer, although the PV resource increases significantly, this difference reduces to less than 1%.

`intlinprog`, whereas for summer this difference is reduced to 4%.

`intlinprog`’s $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ is overestimated/underestimated (yellow area). Even when temperatures remain close to 60 °C, instantaneous $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}$ values are frequently lower than those of

`intlinprog`due to part-load efficiency reasons (blue area). Consequently, according to 3th protection measure (see Section 2.2.7), the ICA has to react using auxiliary heat pump capacity to provide additional heating when the tank temperature drops unexpectedly below the set point (here on 25 January around 10:30 h (purple area)). Moreover, Figure 7 quantifies the impact of the previous shortcomings over system real-time operation by showing values of the supplementary heat pump energy used by the ICA (additional to the power predicted in the optimisation) as well as the DHW deficit durations (periods when tank temperature is below the set point) for winter and summer months.

`intlinprog`, in addition to the external factors mentioned in the case of the HCA, $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}$ overestimation leads to a deficit of HP thermal energy provision, which contributes to an increase of DHW deficit duration with respect to the HCA case (108 h and 7 h for winter and summer months, respectively). This forces the ICA to use the supplementary HP power to compensate for this deficit, providing instantaneously around 2 kWh of thermal energy for more than one hour for the winter month, while 4 kWh and two hours are registered for the summer month. This higher activity of the ICA in summer compared to winter can be explained from the fact that the HP operates for less time in summer than in winter, which makes it less probable that the ICA can find any scheduled HP power in the next 15 min when trying to restore unexpected temperature drops, being forced to use more supplementary HP power.

`intlinprog`inaccuracies would have a greater misleading effect on real-time system operation.

## 4. Conclusions

`fmincon`,

`GlobalSearch`and

`intlinprog`over different scenarios, considering measures such as the system OPEX, PV self-consumption and energy efficiency as well as solver running time and reliability.

`fmincon`and

`GlobalSearch`. Furthermore, the HCA allows for the definition of a nonlinear and discrete AWHP model, enabling the algorithm to run the HP more smoothly than the fragmented and low-efficiency HP operation produced by NLP solvers. All these benefits make the HCA capable of providing optimised HP management up to 49% cheaper than the nonlinear solvers. Additionally, the HCA achieves the highest PV self-consumption share of all solvers during the summer (44%), and reached a 92% rate over the winter period (7% below the best case).

`intlinprog`show that the HCA’s strategy self-consumes as much PV electricity as

`intlinprog`, registering an increase of up to 9% in system operating costs with respect to this linear solver. Moreover, HCA control with a more precise HP representation leads to fewer ICA interventions. In contrast, the

`intlinprog`control requires more ICA interventions to ensure that the tank temperature stays within the desired range, as the linearisation of the HP model frequently induces $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ miscalculations, overestimating in some cases the HP’s thermal energy provision. A further advantage of the HCA is that it can be easily implemented on lean hardware.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AWHP | Air-to-Water Source Heat Pump |

CAPEX | Capital Expenditures |

COP | Coefficient of Performance |

DHW | Domestic Hot Water |

FiL | Feed-in Power Limit |

FiT | Feed-in Tariff |

HCA | Heuristic Control Algorithm |

HP | Heat Pump |

ICA | Instantaneous Control Algorithm |

MILP | Mixed Integer Linear Programming |

NLP | Nonlinear Programming |

LP | Linear Programming |

OCP | Optimal Control Problem |

OPEX | Operating Expenditures |

PLR | Part-Load Ratio |

PV | Photovoltaics |

RBC | Rule-Based Control |

RE | Renewable Energy |

TES | Thermal Energy Storage |

## Appendix A

Pos | Property | Unit | Description | Value |
---|---|---|---|---|

DHW Tank (dtank) | ||||

Inverter (inv) | ||||

Environment (en) | ||||

1 | ${P}_{\mathrm{PV},\mathrm{DC}}$ | W | Photovoltaic DC-production | ${M}_{525601x1}$ |

2 | ${P}_{\mathrm{load}}$ | W | Household electric consumption profile | ${M}_{525601x1}$ |

Temps (ti) | ||||

3 | ${t}_{0}$ | min | Starting time of simulation | ${t}_{0}\phantom{\rule{0.222222em}{0ex}}\u03f5$ [1, 525,601] |

4 | ${t}_{\mathrm{end}}$ | min | End of simulation time | ${t}_{end}\phantom{\rule{0.222222em}{0ex}}\u03f5$ [1, 525,601] |

5 | ${t}_{\mathrm{step}}$ | min | Simulation time step | 1 |

Heat Pump (hp) | ||||

6 | ${d}_{0}$ | Unitless | Independent $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ coefficient | 5.5930 |

7 | ${d}_{1}$ | 1/°C | ${T}_{\mathrm{amb}}$-dependent $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ coefficient | 0.0569 |

8 | ${d}_{2}$ | 1/°C | ${T}_{\mathrm{tank}}$-dependent $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ coefficient | −0.0661 |

9 | ${a}_{1}$ | Unitless | 1st order part-load efficiency coefficient | 8.3350 |

10 | ${a}_{2}$ | Unitless | 2nd order part-load efficiency coefficient | −38.0747 |

11 | ${a}_{3}$ | Unitless | 3rd order part-load efficiency coefficient | 104.6758 |

12 | ${a}_{4}$ | Unitless | 4th order part-load efficiency coefficient | −159.6927 |

13 | ${a}_{5}$ | Unitless | 5th order part-load efficiency coefficient | 121.4477 |

14 | ${a}_{6}$ | Unitless | 6th order part-load efficiency coefficient | −35.9697 |

Instantaneous Control Algorithm (ins) | ||||

15 | $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ | Unitless | Heat pump coefficient of performance | ${M}_{525601x1}$ |

16 | ${P}_{\mathrm{exp}}$ | W | Exporting grid flux | ${M}_{525601x1}$ |

17 | ${P}_{\mathrm{hp}}$ | W | Heat pump electric power | ${M}_{525601x1}$ |

18 | ${P}_{\mathrm{imp}}$ | W | Importing grid flux | ${M}_{525601x1}$ |

19 | ${P}_{\mathrm{load}}$ | W | Household electricity consumption | ${M}_{525601x1}$ |

20 | ${P}_{\mathrm{loss}}$ | W | Power curtailment due to $\mathit{F}\phantom{\rule{-0.166667em}{0ex}}\mathit{i}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}$ | ${M}_{525601x1}$ |

21 | ${P}_{\mathrm{PV},\mathrm{AC}}$ | W | Photovoltaic-AC production | ${M}_{525601x1}$ |

22 | ${T}_{\mathrm{max}}$ | °C | Maximum heat pump supply temperature | 65 |

23 | ${T}_{\mathrm{min}}$ | °C | Minimum tank temperature | 55 |

Optimisation (opt) | ||||

24 | $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}^{opt}$ | Unitless | Optimised heat pump coefficient of performance | ${M}_{96x1}$ |

25 | $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ | Unitless | $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$ = $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}^{opt}\xb7{f}_{c}^{opt}$ | ${M}_{96x1}$ |

26 | $costvec$ | CHF | Matrix of cost values for all tested heating states | ${M}_{qxj}$ |

27 | ${f}_{c}^{opt}$ | Unitless | Optimised HP part-load correction factor | ${M}_{96x1}$ |

28 | ${G}_{f}^{opt}$ | W | Optimised net grid power flux ${P}_{\mathrm{PV},\mathrm{AC}}^{opt}-{P}_{\mathrm{load}}^{opt}-{P}_{\mathrm{hp}}^{opt}$ | ${M}_{96x1}$ |

29 | ${\widehat{P}}_{hp}^{opt}$ | Unitless | Optimised heat pump power profile normalized to ${P}_{nom}$ | ${M}_{96x1}$ |

30 | ${\widehat{P}}_{{\mathrm{hp}}_{\mathrm{old}}}^{opt}$ | Unitless | ${\widehat{P}}_{\mathrm{hp}}^{opt}$ vector taken from the last decision as reference | ${M}_{96x1}$ |

31 | ${P}_{hp,pv}^{opt}$ | W | Amount of PV electricity consumed in the optimised HP operation | ${M}_{96x1}$ |

32 | ${P}_{\mathrm{load}}^{opt}$ | W | Load electric consumption forecast | ${M}_{96x1}$ |

33 | ${P}_{\mathrm{PV},\mathrm{AC}}^{opt}$ | W | Photovoltaic-AC production Forecast | ${M}_{96x1}$ |

34 | ${P}_{\mathrm{PV},\mathrm{AC}}^{opt}$ | W | ${t}_{\mathrm{step}}^{opt}$-min PV production forecast | ${M}_{96x1}$ |

35 | ${\dot{Q}}_{\mathrm{hp}}^{opt}$ | ${W}_{th}$ | Optimised heat pump thermal power | ${M}_{96x1}$ |

36 | ${T}_{\mathrm{max}}^{opt}$ | °C | Maximum temperature value of ${T}_{\mathrm{tank}}^{opt}$ vector | Var |

**Figure A1.**Sixth order polynomial fitting curve of $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ correction factor (${f}_{c}$) vs. Normalized $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ data. Polynomial fit: ${f}_{c}={a}_{6}\xb7PL{R}^{6}+{a}_{5}\xb7PL{R}^{5}+{a}_{4}\xb7PL{R}^{4}+{a}_{3}\xb7PR{L}^{3}+{a}_{2}\xb7PR{L}^{2}+{a}_{1}\xb7PRL.$

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**Figure 4.**AC photovoltaic production (${P}_{\mathrm{PV},\mathrm{AC}}^{opt}$), domestic electrical load (${P}_{\mathrm{load}}^{opt}$), tank temperature (${T}_{\mathrm{tank}}^{opt}$), domestic hot water consumption (${\dot{m}}_{d}^{opt}$), heat pump electric power (${P}_{\mathrm{hp}}^{opt}$) and modulated coefficient of performance ($\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}^{opt}$) profiles yielded by heuristic control algorithm (HCA),

`intlinprog`,

`fmincon`and

`GlobalSearch`for a 48 h time horizon in the summer.

**Figure 5.**Running time durations of optimisation solvers

`intlinprog`and the HCA registered when repeating winter and summer simulations twenty times. These repetitions are necessary to determine running time variability due to computer processing hierarchy. This study is based on a 1-kW heat pump, 600-L domestic hot water tank and 3-kWp PV system.

**Figure 6.**Tank temperature (${T}_{\mathrm{tank}}$), HP coefficient of performance ($\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}$ considers part-load efficiency, $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ does not), electric and thermal power (${P}_{\mathrm{hp}}$ and ${Q}_{\mathrm{hp}}$) given by

`intlinprog`and the instantaneous control algorithm (ICA). This scenario comprises a 1-kW heat pump, 600-L domestic hot water storage and 3-kWp PV system. Yellow and blue areas highlight $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ and $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{P}}_{m}$ differences between

`intlinprog`and the ICA. The purple area shows that the ICA was forced to use additional HP power not expected by

`intlinprog`to keep the tank temperature over the set point (55 °C). The cause of this is the different DHW profiles used by

`intlinprog`and ICA at 30-min and 1-min time steps as well as the $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ mismatch between algorithms.

**Figure 7.**Values of domestic hot water deficit duration (${t}_{\mathrm{short}}$), heat pump electric and thermal energy supply and running time (${E}_{\mathrm{hp},\mathrm{aux}}$, ${Q}_{\mathrm{hp},\mathrm{aux}}$ and ${t}_{\mathrm{aux}}$) calculated by the instantaneous control algorithm (ICA) to compensate for unexpected real-time temperature drops below the set point, prompted by heavy DHW consumption peaks, as well as time step and $\mathit{C}\phantom{\rule{-0.166667em}{0ex}}\mathit{O}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ differences between simulation and optimisation. Winter and summer simulations are performed using a heuristic-based algorithm (HCA) and linear optimisation (

`intlinprog`) solver. The system consists of a 1-kW heat pump, 600-L domestic hot water tank and 3-kWp PV array.

**Table 1.**Lower ($l{b}_{\mathit{M}\phantom{\rule{-0.166667em}{0ex}}\mathit{I}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}}$) and upper ($u{b}_{\mathit{M}\phantom{\rule{-0.166667em}{0ex}}\mathit{I}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}}$) system boundaries at each time step for the $\mathit{M}\phantom{\rule{-0.166667em}{0ex}}\mathit{I}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}$ optimisation problem.

${\mathit{P}}_{\mathbf{hp}}^{\mathit{opt}}$ | ${\mathit{T}}_{\mathbf{tank}}^{\mathit{opt}}$ | ${\mathit{P}}_{\mathbf{loss}}^{\mathit{opt}}$ | ${\mathit{P}}_{\mathbf{imp}}^{\mathit{opt}}$ | ${\mathit{P}}_{\mathbf{exp}}^{\mathit{opt}}$ | ${\mathit{S}}^{\mathit{opt}}$ | ${\mathit{R}}^{\mathit{opt}}$ | ${\mathit{e}}^{\mathit{opt}}$ | $\mathit{O}\phantom{\rule{-0.166667em}{0ex}}{\mathit{B}}^{\mathit{opt}}$ | |
---|---|---|---|---|---|---|---|---|---|

$l{b}_{\mathit{M}\phantom{\rule{-0.166667em}{0ex}}\mathit{I}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}}$ | 0 | 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

$u{b}_{\mathit{M}\phantom{\rule{-0.166667em}{0ex}}\mathit{I}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}\phantom{\rule{-0.166667em}{0ex}}\mathit{P}}$ | ${P}_{\mathrm{nom}}$ | 65 | ∞ | ∞ | $\mathit{F}\phantom{\rule{-0.166667em}{0ex}}\mathit{i}\phantom{\rule{-0.166667em}{0ex}}\mathit{L}$ | 1 | 1 | ∞ | 1 |

**Table 2.**Benefits, limitations and optimal control problems (OCP) fed into the different optimisation solvers. ${x}_{0}$ refers to the starting point provided to solvers

`fmincon`and

`GlobalSearch`.

Solver | OCP | Benefits | Limitations |
---|---|---|---|

fmincon | NLP | Detailed model | Only continuous Equations |

Global Optimisation | Largest computational time | ||

Optimality not guaranteed | |||

High ${x}_{0}$ dependency | |||

GlobalSearch | NLP | Detailed model | Only continuous Equations |

Global Optimisation | Largest computational time | ||

Low ${x}_{0}$ dependency | Optimality not guaranteed | ||

intlinprog | MILP | Simplest model | Accuracy |

Lowest computational time | |||

Continuous/Discrete Equations | |||

Optimality guaranteed | |||

Global Optimisation | |||

HCA | NLP | Detailed model | Optimality not guaranteed |

+ Heuristics | Low computational time | Sequential optimisation | |

Continuous/Discrete Equations |

**Table 3.**OPEX ($OPE{X}^{opt}$), PV self-consumption ($S{C}^{opt}$) and mean run time (${\overline{t}}_{run}$) results of the heuristic control algorithm (HCA),

`intlinprog`,

`fmincon`and

`GlobalSearch`solvers for a single 48-h optimisation for winter and summer. The self-consumption rate compared to a reference case ($S{C}_{ref}^{opt}$) is shown, considering that the entire HP load is met only by importing grid electricity. Upper/lower uncertainty limits ($\Delta {t}_{run,max}^{+}$ and $\Delta {t}_{run,max}^{-}$, respectively) denote, with respect to its mean value, the maximum run time variability observed after repeating each optimisation twenty times, except for GlobalSearch, which was run only five times.

Winter | Summer | |||||||
---|---|---|---|---|---|---|---|---|

HCA | int. | fmin. | GS | HCA | int. | fmin. | GS | |

$S{C}_{ref}^{opt}$ (%) | 66 | 21 | ||||||

$S{C}^{opt}$ (%) | 92 | 96 | 99 | 99 | 44 | 39 | 39 | 39 |

$OPE{X}^{opt}$ | 7.10 | 6.28 | 7.87 | 7.54 | 1.45 | 0.97 | 2.84 | 2.45 |

${\overline{t}}_{run}$ (s) | 80 | <1 | 1047 | 76,891 | 57 | 2 | 920 | 62,448 |

$\Delta {t}_{run,max}^{+}$ (s) | 4 | <1 | 18 | 13,676 | 2 | <1 | 9 | 8215 |

$\Delta {t}_{run,max}^{-}$ (s) | 3 | <1 | 8 | 15,425 | 3 | <1 | 7 | 11,360 |

**Table 4.**The system OPEX and self-consumption rate (SC) achieved when using the HCA and

`intlinprog`solvers in winter and summer simulations. ($SC\mathrm{ref}$) denotes the self-consumption rate considering a reference case in which the entire HP load is met only by importing grid electricity.

Winter | Summer | |||
---|---|---|---|---|

HCA | Intlinprog | HCA | Intlinprog | |

SC_{ref} (%) | 64 | 28 | ||

SC (%) | 88 | 96 | 46 | 47 |

OPEX (CHF) | 166.66 | 153.18 | 37.78 | 36.27 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sánchez, C.; Bloch, L.; Holweger, J.; Ballif, C.; Wyrsch, N. Optimised Heat Pump Management for Increasing Photovoltaic Penetration into the Electricity Grid. *Energies* **2019**, *12*, 1571.
https://doi.org/10.3390/en12081571

**AMA Style**

Sánchez C, Bloch L, Holweger J, Ballif C, Wyrsch N. Optimised Heat Pump Management for Increasing Photovoltaic Penetration into the Electricity Grid. *Energies*. 2019; 12(8):1571.
https://doi.org/10.3390/en12081571

**Chicago/Turabian Style**

Sánchez, Cristian, Lionel Bloch, Jordan Holweger, Christophe Ballif, and Nicolas Wyrsch. 2019. "Optimised Heat Pump Management for Increasing Photovoltaic Penetration into the Electricity Grid" *Energies* 12, no. 8: 1571.
https://doi.org/10.3390/en12081571