**Energy Efficiency, Private Residence**

Fig. 1. Private house on Samso. Built in 1906 as asmall farm. The heated floor space is 273 square metres (the two larger buildings). |

In one of our case studies (Ground Heat, Private Residence), a model estimates the energy consumption of a private house depending on outdoor temperatures. Given a record of the actual number of degree days summed over a week, the model provides an estimate to be compared with the actual energy consumption for that week.

This is useful for at least two purposes: 1) a large deviation is a sign of an irregularity, and 2) if the actual consumption is lower than the estimate, the inhabitants of the house have saved energy.

It is a way to motivate energy efficient behaviour, because the deviation from the model acts as an *econOmeter* that shows every week if the actual consumption is above or below the expected values.

### Table of contents

## Annual consumption

The private home in question is a family dwelling consisting of a main building (Figure 1, right), a smaller annex (Figure 1, left), and an unheated garage (Figure 1, front). The main heat source is a ground source heat pump (VĂ¸lund) providing heating for both houses. A solar collector (Batec) on the roof of the annex provides supplementary hot water during spring, summer, and autumn. Hot water flows through radiators in the two buildings to heat the rooms, and each radiator has a valve (Danfoss) with a thermostat that keeps the room temperature steady. The ground heat installation operates on electricity; therefore the house is electrically heated, and it is relatively easy to record the energy consumption.

Figure 2 shows the annual electricitry consumption over a range of eight years. The electricity supplier uses those numbers to estimate the quarterly installments of the bill. At the end of the heating season the supplier asks for the reading of the electricity meter, and then he adjusts the electricity bill to reflect the exact consumption. The annual energy consumption depends on the weather, however, specifically the number of *heating degree days* for that year, so it can be difficult to predict; therefore the final payment may deviate somewhat from the pre-paid installments.

Fig. 2. Annual energy consumption. There was an increase in 2003, and two years were more expensive. Then the energy consumption dropped. |

In order to compare the energy consumption year-by-year, we calculate the key figure *average energy per degree day*,

Figure 3 is a plot of the energy per degree day for the same period of years. The energy per degree day is relatively steady, it stays within roughly +/- 15 percent of the mean value, except for the last year which was exceptional. Most years lie in the interval 3.0 - 3.5 kWh/HDD. Apparently this home requires that amount of energy — given its state of insulation, leaks, and the number of windows — to keep it warm enough to be comfortable. That is the *observed energy demand* of the house depending on degree days. But the energy demand also depends on the inhabitants' behaviour. To be energy efficient, a *target demand* is chosen in the lower end of the interval,

We wish to be below this figure at the end of the year. The value is low, but it seems possible to achieve by energy efficient behaviour according to Figure 3. In any case it acts as a reference for further analysis. All other things equal, the demand per HDD is in principle a characteristic constant that reflects the energy losses through the envelope of the house as well as the behaviour of the inhabitants. The actual losses also depend on wind and solar irradiation, however, and these have been neglected.

Fig. 3. Energy per degree day. The two periods from 2006 to 2008 were the least efficient. The period 2008 - 2009 was good, but that is because the house was empty for six weeks, and the heat pump was off. |

## Energy characteristic

A dedicated electricity meter measures the electricity consumption due to the heat pump and auxiliary equipment (pumps, electronics). The meter is read off manually every week, if possible.

Figure 4 is a plot of the historical measurements of energy consumption against heating degree days, HDD, that are derived from outdoor temperature measurements. The set of data points in Figure 4 corresponds to more than seven years of operation.

It shows that the energy consumption increases with the number of degree days, but it is not quite a linear dependency. On top of the data points is a line depicting the target demand per HDD.

Fig. 4. Target efficiency. The data points are simultaneous measurements of energy and degree days. There is variation, especially in the high end. The straight line represents the target demand 3 kWh/HDD. Data points falling below the line are from weeks with a good efficiency. |

The data points are distributed around an imaginary characteristic function that determines the energy demand as a function of the outdoor temperature (degree days).

The data points vary considerably due to unmeasured effects, such as wind and sun, and also changes of behaviour (visitors, adjustments to radiator valves, open windows). There is also a solar panel in the installation that affects the energy measurement.

Apparently there is an electricity consumption even when the degree days are zero. That is because the heat pump must produce hot water for the bathroom and kitchen even when it does not produce space heating. This is the *degree day independent consumption*, and it is present all year round. It can be read off the vertical axis at zero degree days.

In general the data points tend to curve upward. In fact, a second order model *M2* of the form

^{2}+ bx + c

provides a more accurate fit to the measured data points. Here *x* is the HDD value, and *y* is the model's estimate of the kWh value. The constants {a, b, c} must be adjusted so that the curve fits the data best. This can be done conveniently in a spreadsheet program, see Figure 5. The upward curvature shows that increasingly more energy is needed in colder weather. One reason is that the ground pipes collect less heat from the ground when it is cold. A second reason is the low solar irradiation on the house during the three winter months while at the same time winds are stronger than usual. A third reason is the diminishing contribution from solar panel as we move toward the winter season.

The model cuts through the observed data points (Figure 5), such that the sum of the squared vertical distances from all points to the line is minimal.

Fig. 5. Energy characteristic (solid line). The line is the model M2 of the heat production. Assuming the model has the form of a second degree polynomial, the line is the best fit to the data points. |

## Simulation

The model can estimate the energy consumption given a degree day value. Figure 6 shows the model estimate together with the actual energy consumption. By observation, there is good agreement in the last 2/3 of the period, while the actual consumption in the leading 1/3 is somewhat higher than the model. The deviation is most likely due to an occupant, who required a higher setting of the radiator thermostats than usual.

The model only considers degree days. A windy, cold week will generate a measurement higher than the estimate. By the same reasoning, high solar irradiation will result in an energy measurement lower than the model estimate. These unmodelled effects are especially pronounced in the winter time, where they make more difference.

It is also possible to detect irregularities or failures. If the actual consumption is much larger than the estimate, for no known reason, it could be a sign of a technical failure somewhere in the heating installation.

Fig. 6. Actual energy measurements and model. Given the actual degree days, the model estimates the energy consumption. Disagreements are due to unmodelled influences, such as wind, sun, visitors, or change of settings on the heat installation. |

## EconOmeter

By changing behaviour, or by simply turning the heat down, the inhabitants can influence the energy consumption. Thus, if the houseowner manages to stay below the black line in Figure 5, energy will be saved.

From the viewpoint of Figure 6, the goal is to stay below the model estimate, the dotted line in the plot. Figure 7 shows the distance between the model and the actual energy consumption. The diagram is renewed every week. It acts as an *econOmeter* in the sense that an actual consumption higher than the model estimate, alerts the owner to take action to compensate for the loss. For example, the owner could decide to save energy the following week, either by turning the radiators down or by making adjustments to the settings of the installation.

Fig. 7. EconOmeter. The actual consumption was less than the model, and energy was saved. |

## External links

- Degree Days.net: HDD and CDD for locations worldwide
- Wikipedia: Heating degree day