HOT WATER SUPPLY APPARATUS AND CONTROL METHOD FOR HOT WATER SUPPLY APPARATUS

Abstract

The hot water supply apparatus includes a heat pump unit operating using electrical energy, a hot water storage tank storing hot water heated by the heat pump unit, and an auxiliary heat source device burning fuel and heating hot water when a stored hot water heat amount is insufficient for a heat amount used for hot water supply. A control device calculates a predicted heat amount for each unit time period based on a past usage record of hot water supply; estimates an error distribution of an actual heat amount relative to the predicted heat amount; calculates an expected value of loss in a target index using the estimated error distribution; obtains an optimal stored hot water heat amount minimizing the expected value; and controls the heat pump unit so that the hot water storage tank stores the optimal stored hot water heat amount.

Description

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the priority benefits of Japanese application no. 2022-183878, filed on Nov. 17, 2022. The entirety of the above-mentioned patent application is hereby incorporated by reference herein and made a part of this specification.

BACKGROUND

Technical Field

The disclosure relates to a hot water supply apparatus and a control method for the hot water supply apparatus.

Description of Related Art

Japanese Patent No. 6086014 (Patent Document 1) discloses a heat pump water heater which includes a heat pump-type hot water storage device for preheating water for hot water supply or filling a bathtub and storing hot water in a hot water storage tank, and a combustion-type auxiliary heat source device for heating water for hot water supply.

In Patent Document 1, the heat amount used for hot water supply, that is required for each unit time period, is predicted based on past usage records of hot water supply. When performing a hot water storage operation using the heat pump-type hot water storage device, the hot water storage operation is configured to be performed in an operation pattern that maximizes energy saving, among multiple operation patterns for hot water storage operation that are set based on a predicted heat amount used for hot water supply.

However, it is extremely difficult to make the predicted heat amount used for hot water supply, which is determined from past usage records of hot water supply, completely match the actual heat amount used for hot water supply. There are quite a lot of errors between the predicted heat amount and the actual heat amount used for hot water supply.

Therefore, when the hot water storage operation is performed according to the predicted heat amount used for hot water supply, the stored hot water heat amount of the hot water storage tank may be insufficient compared to the actual heat amount used for hot water supply, or the stored hot water heat amount of the hot water storage tank may exceed the actual heat amount used for hot water supply, resulting in excessive hot water storage. Such insufficient hot water storage and excessive hot water storage lead to loss in cost associated with the operations of the heat pump unit and the auxiliary heat source device. Thus, there is room for improvement in the efficiency of the hot water supply apparatus.

The disclosure improves the efficiency of a hot water supply apparatus that includes a heat pump unit and an auxiliary heat source device.

SUMMARY

One aspect of the disclosure provides a hot water supply apparatus. The hot water supply apparatus includes a heat pump unit that operates with electrical energy as a driving source; a hot water storage tank that stores hot water heated by the heat pump unit; a hot water supply circuit that supplies hot water from the hot water storage tank; an auxiliary heat source device for burning fuel and heating hot water to be supplied in response to a stored hot water heat amount of the hot water storage tank being insufficient for a heat amount used for hot water supply; and a control device that controls the heat pump unit and the auxiliary heat source device. The control device calculates a predicted heat amount used for hot water supply for each unit time period based on a past usage record of hot water supply, and estimates an error distribution of an actual heat amount used for hot water supply with respect to the predicted heat amount used for hot water supply. The control device calculates an expected value of loss in a target index, caused by a hot water storage operation of the heat pump unit and a combustion operation of the auxiliary heat source device in response to generating the actual heat amount used for hot water supply, using the estimated error distribution. The control device obtains an optimal stored hot water heat amount that minimizes the expected value of loss, and controls the heat pump unit so that the hot water storage tank stores the optimal stored hot water heat amount.

Another aspect of the disclosure provides a control method for controlling a hot water storage operation in a hot water supply apparatus. The hot water supply apparatus includes a heat pump unit that operates with electrical energy as a driving source; a hot water storage tank that stores hot water heated by the heat pump unit; a hot water supply circuit that supplies hot water from the hot water storage tank; and an auxiliary heat source device for burning fuel and heating hot water to be supplied in response to a stored hot water heat amount of the hot water storage tank being insufficient for a heat amount used for hot water supply. The control method includes calculating a predicted heat amount used for hot water supply for each unit time period based on a past usage record of hot water supply; estimating an error distribution of an actual heat amount used for hot water supply with respect to the predicted heat amount used for hot water supply; calculating an expected value of loss in a target index, caused by a hot water storage operation of the heat pump unit and a combustion operation of the auxiliary heat source device in response to generating the actual heat amount used for hot water supply, using the estimated error distribution; calculating an optimal stored hot water heat amount that minimizes the expected value of loss; and controlling the heat pump unit so that the hot water storage tank stores the optimal stored hot water heat amount.

The disclosure is capable of improving the efficiency of a hot water supply apparatus that includes a heat pump unit and an auxiliary heat source device.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing an example of the overall configuration of the hot water supply apparatus according to an embodiment.

FIG. 2 is a block diagram showing an example of the hardware configuration of the controller.

FIG. 3 is a flowchart illustrating the processing procedure for hot water storage control in the hot water supply apparatus according to this embodiment.

FIG. 4 is a diagram showing an example of the probability density function of the normal distribution.

FIG. 5 is a diagram for illustrating the cost lost due to generation of the actual heat amount used for hot water supply.

FIG. 6 is a diagram for illustrating the expected value of the lost cost.

FIG. 7 is a diagram showing an example of the linear approximation formula.

DESCRIPTION OF THE EMBODIMENTS

Embodiments of the disclosure will be described in detail below with reference to the drawings. Hereinafter, the same or corresponding parts in the drawings will be denoted by the same reference numerals, and the description thereof will not be repeated in principle.

Overall Configuration of the Hot Water Supply Apparatus

FIG. 1 is a block diagram showing an example of the overall configuration of the hot water supply apparatus according to this embodiment.

As shown in FIG. 1, the hot water supply apparatus 5 according to this embodiment is a hybrid hot water supply apparatus, and includes a hot water storage and supply unit 10, a heat pump unit 60, and a remote controller (hereinafter referred to as “remote control”) 200. The hot water storage and supply unit 10 has a built-in controller 100 that is the main controller of the hot water supply apparatus 5, and the controller 100 and the remote control 200 are connected through a communication line 17.

The hot water storage and supply unit 10 includes a fuel gas combustion-type auxiliary heat source device 31, a hot water storage tank 30, inflow water pipes 32 and 33, a hot water delivery pipe 34, circulation pipes 35 and 36, a high-temperature water pipe 37, pipes 38 and 39 for the auxiliary heat source device 31, a circulation pump 40, a proportional valve 42, a mixing valve 44, and check valves 45 and 46.

The inflow water pipe 33 is connected between a water inlet 20 and the bottom of the hot water storage tank 30. The check valve 46 is inserted and connected to the inflow water pipe 33. Low-temperature water is introduced into the hot water storage tank 30 through the inflow water pipe 33.

The circulation pipes 35 and 36 are arranged to form a circulation path between the heat pump unit 60 and the hot water storage tank 30 when the circulation pump 40 is in operation. In a hot water storage operation of storing hot water in the hot water storage tank 30, control is performed by adjusting the rotation speed of the circulation pump 40, etc. so that a return temperature for the hot water storage tank 30 detected by a temperature sensor 53 arranged in the circulation pipe 36 reaches a target hot water storage temperature.

The heat pump unit 60 uses electrical energy as a driving source and operates as a main heat source for supplying the energy required for the heat load (hot water supply and space heating) consumed in houses. During the hot water storage operation, the heat pump unit 60 heats hot water with heat absorbed from the outside air and stores the hot water in the hot water storage tank 30. The hot water stored in the hot water storage tank 30 during the hot water storage operation is used for supplying hot water and filling a bathtub.

In addition to the heat pump unit 60, it is also possible to employ a fuel cell unit or a power generation unit that generates power with heat generation such as an engine power generation unit having an engine and a generator, as the main heat source. The water stored in the hot water storage tank 30 may be heated and returned to the hot water storage tank 30 by recovering exhaust heat that accompanies cooling of the power generation unit.

The heat pump unit 60 is configured by connecting a compressor 61, a condensing heat exchanger 62, an expansion valve 63, and an evaporative heat exchanger 64 through a refrigerant circuit 65. The heat pump unit 60 compresses a refrigerant sealed in the refrigerant circuit 65 to a high temperature with the compressor 61, and heats hot water flowing through the circulation pipes 35 and 36 by the circulation pump 40 through heat exchange with the high-temperature refrigerant in the condensing heat exchanger 62. After heat exchange, the refrigerant is expanded by the expansion valve 63 to have a lower temperature than the outside air, absorbs heat from the outside air in the evaporative heat exchanger 64, and is then introduced into the compressor 61 again.

Although not shown, the evaporative heat exchanger 64 includes an outside air temperature sensor that detects the outside air temperature and a blower. The heat pump unit 60 includes an auxiliary controller 66 that controls the compressor 61, the expansion valve 63, the blower, etc. The auxiliary controller 66 is communicably connected to the controller 100 and controls the heat pump unit 60 according to commands from the controller 100.

The high-temperature water pipe 37 is connected between the upper part of the hot water storage tank 30 and a high-temperature side port of the mixing valve 44. Further, the inflow water pipe 32 to which the check valve 45 is inserted and connected is connected between a low-temperature side port of the mixing valve 44 and the water inlet 20. Furthermore, the hot water delivery pipe 34 is connected between a hot water delivery port of the mixing valve 44 and a hot water outlet 25. Furthermore, an electromagnetic valve 48 is arranged between the inflow water pipe 32 and the hot water delivery pipe 34 to bypass the mixing valve 44 and connect the inflow water pipe 32 and the hot water delivery pipe 34.

When a hot water tap (not shown) connected to the tip of the hot water outlet 25 is opened, low-temperature water is introduced into the hot water storage tank 30 via the inflow water pipe 33 according to the inflow water pressure entering the water inlet 20, and high-temperature water is output from the hot water storage tank 30 to the mixing valve 44 (high-temperature side port) via the high-temperature water pipe 37. Furthermore, low-temperature water is input to the mixing valve 44 (low-temperature side port) via the inflow water pipe 32. Thus, high-temperature water and low-temperature water are mixed to output hot water at an appropriate temperature according to a hot water supply setting temperature from the output port of the mixing valve 44 to the hot water outlet 25 via the hot water delivery pipe 34.

The auxiliary heat source device 31 is an instant water heater that heats hot water with the combustion heat of fuel gas. The auxiliary heat source device 31 is operated by a command from the controller 100 to heat hot water introduced from the hot water storage tank 30 via the pipe 38 and output the hot water to the pipe 39. The pipe 39 is provided with the proportional valve 42 for controlling the flow rate of the auxiliary heat source device 31.

When the temperature of the hot water stored in the hot water storage tank 30 is high, the controller 100 stops the auxiliary heat source device 31 and fully closes the proportional valve 42, and supplies the high-temperature water stored in the hot water storage tank 30 to the mixing valve 44 (high-temperature side port), so as to be able to supply hot water.

In addition, when the temperature of the hot water stored in the hot water storage tank 30 is low, the controller 100 sets the proportional valve 42 to an open state and operates the auxiliary heat source device 31, and supplies the high-temperature water heated by the auxiliary heat source device 31 to the mixing valve 44 (high-temperature side port), so as to be able to supply hot water. If the opening degree of the proportional valve 42 is set between fully open and fully closed, it is also possible to supply high-temperature water from both the hot water storage tank 30 and the auxiliary heat source device 31.

The hot water storage tank 30 is provided with a plurality of temperature sensors 51 for detecting the temperature of the stored water. The plurality of temperature sensors 51 are attached to a surface of the hot water storage tank 30 at different heights in the vertical direction. The plurality of temperature sensors 51 and the hot water storage tank 30 are covered with a heat insulating material (not shown) to reduce heat radiation from the stored hot water.

The values detected by the plurality of temperature sensors 51 are used when calculating the stored hot water heat amount in the hot water storage tank 30. In this specification, the “stored hot water heat amount” refers to the amount of hot water stored in the hot water storage tank 30 or the heat amount (heat storage amount) obtained by adding a hot water temperature to the hot water.

The circulation pipes 35 and 36 are provided with temperature sensors 52 and 53, respectively, for detecting the outgoing temperature and return temperature for the heat pump unit 60. Further, a temperature sensor 54 arranged in the inflow water pipe 32 detects the inflow water temperature, and a temperature sensor 55 arranged in the hot water delivery pipe 34 detects the hot water temperature. A temperature sensor 58 is arranged in the high-temperature water pipe 37 to detect the temperature of the high-temperature water input to the mixing valve 44. Furthermore, temperature sensors 56 and 57 for detecting the input temperature and output temperature of the auxiliary heat source device 31 are arranged in the pipes 38 and 39, respectively.

The remote control 200 is an input device for operating the hot water supply apparatus 5, which is placed in the kitchen, bathroom, or the like. The remote control 200 includes a display part for outputting information in a form visible to the user, an operation switch for operating on/off of the hot water supply apparatus 5, and an operation part for accepting an input setting operation performed by the user. The display part is typically configured with a liquid crystal panel. The operation part is typically configured with a push button or a touch button, and is configured to be capable of accepting a setting operation for the hot water supply apparatus 5, which is represented by the hot water supply setting temperature.

The controller 100 controls the operation of each component of the hot water supply apparatus 5 using the values detected by a sensor group including the aforementioned sensors so that the hot water supply apparatus 5 operates according to the user instructions input to the remote control 200. Part of the state data indicating the operating state of the hot water supply apparatus 5, including the values detected by various sensors and input to the controller 100, is stored inside the controller 100.

As an example, the controller 100 controls the operation and stop of the auxiliary heat source device 31 and the opening degree of the proportional valve 42, as described above, based on the stored heat amount of the hot water storage tank 30 calculated from the value detected by the temperature sensor 51. Alternatively, the controller 100 controls the operation of the heat pump unit 60 and the operation and stop of the circulation pump 40 based on the stored hot water heat amount of the hot water storage tank 30.

Further, the controller 100 is capable of controlling the mixing ratio of low-temperature water and high-temperature water in the mixing valve 44 based on the hot water temperature detected by the temperature sensor 55. In addition, when the hot water temperature rises excessively, the controller 100 is capable of opening the electromagnetic valve 48 to lower the hot water temperature.

Furthermore, the controller 100 predicts a future heat amount used for hot water supply based on usage records of the hot water supply apparatus 5 that have been stored. The controller 100 controls the hot water storage operation so as to operate the heat pump unit 60 to store a heat amount, corresponding to the predicted heat amount used for hot water supply, in the hot water storage tank 30 before using the heat amount for hot water supply. As will be described later, the controller 100 executes hot water storage control for setting hot water storage operating conditions, including the predicted heat amount used for hot water supply, hot water storage time, target stored hot water heat amount, etc., based on the past usage records of hot water supply. The controller 100 corresponds to an embodiment of “control device.”

Hardware Configuration of the Controller

Next, an example of the hardware configuration of the controller 100 will be described with reference to FIG. 2.

As shown in FIG. 2, the controller 100 includes a CPU (Central Processing Unit) 70, a RAM (Random Access Memory) 72, a ROM (Read Only Memory) 74, a I/F (Interface) device 76, and a storage device 78. The CPU 70, the RAM 72, the ROM 74, the I/F device 76, and the storage device 78 exchange various data via a communication bus 80.

The CPU 70 expands a program stored in the ROM 74 into the RAM 72 and executes the program. The program stored in the ROM 74 describes processing to be executed by the controller 100.

The I/F device 76 is an input/output device for exchanging signals and data with the remote control 200 and the heat pump unit 60. The I/F device 76 receives the user instructions from the remote control 200.

The storage device 78 is a storage that stores various information, and stores information on the heat pump unit 60, information on the hot water storage and supply unit 10, information on the past usage records of hot water supply, etc. The storage device 78 is, for example, a hard disk drive (HDD) or a solid state drive (SSD).

Hot Water Storage Control

Next, the hot water storage control of the hot water supply apparatus 5 performed by the controller 100 will be described with reference to FIG. 3. In the hot water supply apparatus 5, each time hot water is used, data regarding the date, time, and heat amount used for hot water supply is stored in the storage device 78 of the controller 100 and accumulated as usage records of hot water usage, in conjunction with the hot water storage control.

FIG. 3 is a flowchart illustrating the processing procedure for hot water storage control in the hot water supply apparatus 5 according to this embodiment. The series of processes shown in this flowchart are repeatedly executed by the controller 100 in each pre-divided unit time period (for example, one hour). The symbol Si (i=10, 20, . . . ) in the flowchart indicates each step.

As shown in FIG. 3, the hot water storage control according to this embodiment mainly includes a step of calculating a predicted heat amount used for hot water supply (S10), a step of estimating a prediction error distribution (S20), a step of calculating the cost lost (S30), a step of calculating an expected value of the cost lost (S40), a step of calculating an optimal stored hot water heat amount (S50), and a step of controlling the hot water storage operation based on the optimal stored hot water heat amount (S60).

(1) Step of Calculating the Predicted Heat Amount Used for Hot Water Supply (S10)

In S10, the heat amount x^f used for hot water supply is predicted based on the past usage records of hot water supply of the hot water supply apparatus 5. Specifically, the controller 100 accumulates time-series data for one week as the usage records of hot water supply while updating the time-series data that records the hot water consumption, driving statuses of the heat pump unit 60 and the auxiliary heat source device 31, inflow water temperature, outside air temperature, etc. for each pre-divided time period (for example, 1 hour) of the day. The controller 100 predicts the hot water consumption in each time period of the day based on the accumulated time-series data, and calculates the predicted heat amount x^f used for hot water supply corresponding to the predicted hot water consumption.

The period for accumulating the time-series data such as the heat amount used for hot water supply in the past is not limited to one day and may be set as long as, for example, one month. Alternatively, the data accumulation period may be changed as appropriate to suit the user's lifestyle pattern, such as time-series data that aggregates the heat amount used for hot water supply on each day of the week, and time-series data that aggregates the heat amount used for hot water supply on weekdays and holidays.

Further, since the time-series data accumulated immediately after the hot water supply apparatus 5 starts to be used is insufficient, the predicted heat amount x^f used for hot water supply is calculated using a preset initial value.

(2) Step of Estimating the Prediction Error Distribution (S20)

In S20, a distribution of the error x-x^f between the predicted heat amount x^f used for hot water supply in each time period, which is calculated in S10, and an actual value x of the heat amount used for hot water supply in this time period (hereinafter referred to as “actual heat amount x used for hot water supply”) is estimated.

In this embodiment, it is assumed that the distribution of the error x-x^f between the predicted heat amount x^f used for hot water supply and the actual heat amount x used for hot water supply follows a normal distribution N(0, σ²) with an average of 0 and a variance of σ². According to this, the actual heat amount x used for hot water supply in a certain time period follows a normal distribution N(x^f, σ²) with respect to the predicted heat amount x^f used for hot water supply in the time period. FIG. 4 shows an example of a probability density function φ(x) of the normal distribution N(x^f, σ²). The normal distribution N(x^f, σ²) is symmetrical with respect to x=x^f. When the probability density function φ(x) is integrated over the entire interval, it becomes 1.

(3) Step of Calculating the Cost Lost (S30)

In S30, the cost f(x) lost due to generation of the actual heat amount x used for hot water supply is calculated.

(Cost Lost)

First, the basic concept of the cost lost will be described with reference to FIG. 5.

FIG. 5 is a diagram for illustrating the cost lost due to generation of the actual heat amount x used for hot water supply. FIG. 5 shows a graph representing loss of the cost f(x) superimposed on the prediction error distribution (normal distribution N(x^f, σ²) estimated in S20.

In a conventional hot water supply apparatus, the hot water storage operation is performed based on the predicted heat amount x^f used for hot water supply in each time period. Specifically, the operation of the heat pump unit 60 and the operation and stop of the circulation pump 40 are controlled so as to store a heat amount corresponding to the predicted heat amount x^f used for hot water supply in a time period in the hot water storage tank 30 by the time of start of the time period when hot water is predicted to be used.

However, in principle, it is impossible for the predicted heat amount x^f used for hot water supply and the actual heat amount x used for hot water supply to completely match, and as shown in FIG. 4, the error x-x^f occurs between the predicted heat amount x^f used for hot water supply and the actual heat amount x used for hot water supply. This error x-x^f appears as loss in the cost associated with the operations of the heat pump unit 60 and the auxiliary heat source device 31.

This cost is necessary to generate the actual heat amount x used for hot water supply, and includes concepts other than money. As described below, the cost may include, for example, the consumption of various types of energy (electricity, fuel gas, etc.), the purchase cost of various types of energy, the amount of carbon dioxide (CO₂) generated by various types of energy, etc.

In FIG. 5, it is assumed that the predicted heat amount x^f used for hot water supply and the stored hot water heat amount of the hot water storage tank 30 match each other (x^f=a). In such a case, if the actual heat amount x used for hot water supply is less than the predicted heat amount x^f used for hot water supply (x<x^f), the hot water supply apparatus 5 enters a state of “excessive hot water storage” in which the stored hot water heat amount in the hot water storage tank 30 is excessive for the actual heat amount x used for hot water supply. In the case of excessive hot water storage, the difference a-x between the stored hot water heat amount a and the actual heat amount x used for hot water supply corresponds to a surplus heat amount of the hot water storage tank 30 generated by the operation of the heat pump unit 60.

While part of the surplus heat amount a-x is reused, the rest is lost due to heat radiation from the hot water storage tank 30. In this embodiment, (a-x)×(1−β) is reused according to the reuse efficiency β(0≤β≤1) of the surplus heat amount, and (a-x)×(1−β) is assumed to be heat radiation loss.

Thus, in the case where x<x^f, the hot water supply apparatus 5 stores excessive hot water, and there may be loss in the cost associated with the operation of the heat pump unit 60. As shown in FIG. 5, the cost of the heat pump unit 60 that is lost due to excessive hot water storage increases as the surplus heat amount a-x increases.

On the other hand, if the actual heat amount x used for hot water supply is greater than the predicted heat amount x^f used for hot water supply (x>x^f), the hot water supply apparatus 5 enters a state of “insufficient hot water storage” in which the stored hot water heat amount a of the hot water storage tank 30 is insufficient for the actual heat amount x used for hot water supply.

When it becomes difficult to adjust the hot water in the hot water storage tank 30 to the hot water supply setting temperature due to insufficient hot water storage, the auxiliary heat source device 31 operates to heat the hot water introduced from the hot water storage tank 30 via the pipe 38 and output the hot water to the pipe 39. At this time, the opening degree of the proportional valve 42 provided in the pipe 39 is controlled. That is, the auxiliary heat source device 31 reheats the hot water in the hot water storage tank 30, and adjusts the temperature to the hot water supply setting temperature to supply hot water. The difference x-a between the actual heat amount x used for hot water supply and the stored hot water heat amount a corresponds to the heat amount provided by the auxiliary heat source device 31.

Thus, in the case where x>x^f, the hot water supply apparatus 5 does not store sufficient hot water, and the cost associated with the operation of the auxiliary heat source device 31 is incurred. As shown in FIG. 5, the cost lost due to the operation of the auxiliary heat source device 31 when hot water storage is insufficient increases as the difference x-a between the actual heat amount x used for hot water supply and the stored hot water heat amount a increases.

Next, a method of calculating the cost lost due to excessive hot water storage and insufficient hot water storage will be described. Here, the cost for operating the heat pump unit 60 to generate heat is set to H, and the cost for operating the auxiliary heat source device 31 is set to G.

In the case where the cost lost due to excessive hot water storage is Ce, Ce can be expressed as the following formula (1) using the stored hot water heat amount a, the actual heat amount x used for hot water supply, the cost H of the heat pump unit 60, and the reuse efficiency β of surplus heat amount in the heat pump unit 60. Ce is a positive value in the case of x≤a, and becomes 0 in the case of x>a.

$\left[\mathrm{Formula}1\right]$ $\begin{array}{cc}\mathrm{Ce}=\{\begin{array}{cc}\frac{a}{H}\times \left(1-\beta \right)& \left(x<0\right)\\ \frac{a-x}{H}\times \left(1-\beta \right)& \left(0\le x\le a\right)\\ 0& \left(x>a\right)\end{array}& \left(1\right)\end{array}$

In the case where the cost lost due to insufficient hot water storage is Cs, Cs can be expressed as the following formula (2) using the stored hot water heat amount a, the actual heat amount x used for hot water supply, the cost H of the heat pump unit 60, and the cost G of the auxiliary heat source device 31. Cs becomes 0 in the case of x≤a, and is a positive value in the case of x>a.

$\left[\mathrm{Formula}2\right]$ $\begin{array}{cc}\mathrm{Cs}=\{\begin{array}{cc}0& \left(x<0\right)\\ 0& \left(0\le x\le a\right)\\ \frac{x-a}{G}-\frac{x-a}{H}& \left(x>a\right)\end{array}& \left(2\right)\end{array}$

Then, from formula (1) and formula (2), the cost f(x) lost due to excessive hot water storage or insufficient hot water storage can be expressed as the following formula (3).

$\left[\mathrm{Formula}3\right]$ $\begin{array}{cc}f\left(x\right)=\mathrm{Ce}+\mathrm{Cs}=\{\begin{array}{cc}\frac{a}{H}\times \left(1-\beta \right)& \left(x<0\right)\\ \frac{\alpha -x}{H}\times \left(1-\beta \right)& \left(0\le x\le a\right)\\ \frac{x-a}{G}-\frac{x-a}{H}& \left(x>a\right)\end{array}& \left(3\right)\end{array}$

(Cost)

The cost H associated with the operation of the heat pump unit 60 and the cost G associated with the operation of the auxiliary heat source device 31 described above correspond to the “target index” for evaluating the hot water storage operation performed by the heat pump unit 60 and the combustion operation in the auxiliary heat source device 31. The cost H and the cost G are necessary to generate the actual heat amount x used for hot water supply, and may be set from the viewpoints of energy saving, economic efficiency, environmental conservation, etc.

For example, the amount of consumption of various types of energy (electricity, fuel gas, etc.) for operating each of the heat pump unit 60 and the auxiliary heat source device 31 may be set as the cost. The consumption of various types of energy is converted into consumption of primary energy for comparison. Primary energy refers to energy obtained from the nature such as fossil fuel, nuclear fuel, hydropower, wind power, and sunlight. In this case, the cost H is set using a coefficient of performance (COP) corresponding to the operating efficiency of the heat pump unit 60. The cost G is set using the operating efficiency E_G specific to the auxiliary heat source device 31.

Alternatively, the purchase cost of various types of energy for operating each of the heat pump unit 60 and the auxiliary heat source device 31 may be set as the cost. In this case, the cost H and the cost G are respectively set using the unit prices of various types of energy in addition to the COP of the heat pump unit 60 and the operating efficiency E_G of the auxiliary heat source device 31.

Alternatively, the amount of CO₂ generated by various types of energy for operating each of the heat pump unit 60 and the auxiliary heat source device 31 may be set as the cost.

[In the Case Where the Cost is the Primary Energy Consumption]

The case where the cost associated with the operations of the heat pump unit 60 and the auxiliary heat source device 31 is the respective primary energy consumption may be divided into the following cases depending on the origin of the electricity used.

Specifically, in the case of using only electricity originating from a power plant, the cost H of the heat pump unit 60 is given by H=0.369×COP. 0.369 is the primary energy conversion coefficient for demand-end power, which is adopted in the Act on the Rational Use of Energy (Energy Saving Act). This value takes into consideration the power generation efficiency of the power plant, power transmission and distribution loss from the power generation end to the power reception end, etc. COP is one of the indexes indicating the performance of the heat pump unit 60. Since the heat pump unit 60 transfers the heat of the outside air to the heating target via the refrigerant, the COP varies depending on the outside air temperature and the inflow water temperature. The COP may be calculated based on the inflow water temperature, the outside air temperature, and the setting temperature for the hot water storage operation.

Further, in the case where a small-scale power generation device such as a solar power generation device is connected to the hot water supply apparatus 5 and only electricity originating from this power generation device is used, the power transmission and distribution loss and power generation loss may be ignored, so the cost H of the heat pump unit 60 is given by H=COP.

The cost G of the auxiliary heat source device 31 is given by G=E_G using the operating efficiency E_G specific to the auxiliary heat source device 31 (for example, E_G=about 0.93).

[In the Case Where the Cost is the Purchase Cost of Various Types of Energy]

In the case where the cost associated with the operations of the heat pump unit 60 and the auxiliary heat source device 31 is the purchase cost of various types of energy, the cost is calculated using the operating efficiency thereof and the energy rate unit price.

Specifically, the controller 100 stores an electricity rate plan and a gas rate plan in advance. In the electricity rate plan and the gas rate plan, monthly electricity or gas consumption is divided into a plurality of levels, and the unit price is set for each consumption category. A basic fee may be set for each consumption category; in this case, the basic fee for the consumption category corresponding to the current monthly consumption is divided by that monthly consumption, and this value is added to the unit price for that consumption category, thereby converting it into a unit price that includes the basic fee. These rate plans may be input and set from the remote control 200.

The electricity rate unit price M_E[yen/kwh] corresponding to the current monthly power consumption is set based on the electricity rate plan. The gas rate unit price M_G[yen/m³] corresponding to the current monthly gas consumption is set based on the gas rate plan.

The cost H of the heat pump unit 60 is given by H=COP×860/M_E[kcal/yen] using COP, an electricity heat equivalent value (1 kwh=860 kcal), and electricity rate unit price M_E[yen/kwh].

The cost G of the auxiliary heat source device 31 is given by G=E_G×Q_G/M_G[kcal/yen] using the operating efficiency E_G of the auxiliary heat source device 31, the unit heat amount Q_G [kcal/m³] set according to the type of gas to be used, and the unit price M_G[yen/m³] of gas to be used.

(4) Step of Calculating the Expected Value of the Cost Lost (S40 )

In S40, the expected value M(a) of the cost f(x) lost due to excessive hot water storage or insufficient hot water storage is calculated using the loss of the cost f(x) obtained in S30 and the probability density function 100 (x) of the normal distribution N(x^f, σ²), which is the prediction error distribution estimated in S20.

The expected value M(a) of the lost cost f(x) is a weighted average of all values of the loss f(x) weighted by probability, and can be expressed as the following formula (4).

$\left[\mathrm{Formula}4\right]$ $\begin{array}{cc}\begin{array}{c}M\left(a\right)={\int }_{-\infty }^{\infty }f\left(x\right)\Phi \left(x\right)\mathrm{dx}\\ ={\int }_{-\infty }^0\frac{a}{H}\Phi \left(x\right)\mathrm{dx}+{\int }_0^a\frac{a-x}{H}\Phi \left(x\right)\mathrm{dx}+{\int }_a^{\infty }\left(\frac{x-a)}{G}-\frac{x-a}{H}\right)\Phi \left(x\right)\mathrm{dx}\\ -{\int }_{-\infty }^0\frac{\beta a}{H}\Phi \left(x\right)\mathrm{dx}-{\int }_0^a\frac{\beta \left(a-x\right)}{H}\Phi \left(x\right)\mathrm{dx}\end{array}& \left(4\right)\end{array}$ $\begin{array}{c}=\frac{a}{H}{\int }_{-\infty }^{\infty }\Phi \left(x\right)\mathrm{dx}-\frac{1}{H}{\int }_0^{\infty }x\Phi \left(x\right)\mathrm{dx}+\frac{1}{G}{\int }_a^{\infty }x\Phi \left(x\right)\mathrm{dx}-\frac{a}{G}{\int }_a^{\infty }\Phi \left(x\right)\mathrm{dx}\\ -\frac{\beta a}{H}{\int }_{-\infty }^a\Phi \left(x\right)\mathrm{dx}+\frac{\beta }{H}{\int }_0^{a}x\Phi \left(x\right)\mathrm{dx}\end{array}$

Here, Ψ(x) is the probability density function when the actual heat amount x used for hot water supply in the normal distribution N(x^f, σ²) is standardized to z=(x-x^f)/σ, and the cumulative distribution function thereof is defined as Ψ(x). In this case, formula (4) can be transformed as the following formula (5).

$\left[\mathrm{Formula}5\right]$ $\begin{array}{cc}M\left(a\right)=\frac{a}{H}-\frac{1}{H}{\int }_{-\frac{x^f}{a}}^{\infty }\left(\sigma z+x^f\right)\psi \left(z\right)\mathrm{dz}+\frac{1}{G}{\int }_{\frac{a-x^f}{\sigma }}^{\infty }\left(\sigma z+x^f\right)\psi \left(z\right)\mathrm{dz}-\frac{a}{G}{\int }_{\frac{a-x^f}{a}}^{\infty }\psi \left(z\right)\mathrm{dz}-\frac{\beta a}{H}{\int }_{-\infty }^{\frac{a-x^f}{a}}\psi \left(z\right)\mathrm{dz}+\frac{\beta }{H}{\int }_{-\frac{x^f}{\sigma }}^{\frac{a-x^f}{\sigma }}\left(\sigma z+x^f\right)\psi \left(z\right)\mathrm{dz}=\left(a-x^f\right)\left(\frac{1}{H}-\frac{1}{G}\right)+\frac{x^f}{H}·\Psi \left(-\frac{x^f}{\sigma }\right)+\frac{a-x^f}{G}·\Psi \left(\frac{a-x^f}{\sigma }\right)-\frac{\sigma }{H}\left[-\psi \left(\infty \right)+\psi \left(-\frac{x^f}{\sigma }\right)\right]+\frac{\sigma }{G}\left[-\psi \left(\infty \right)+\psi \left(\frac{a-x^f}{\sigma }\right)\right]-\frac{\beta a}{H}·\Psi \left(\frac{a-x^f}{\sigma }\right)+\frac{\mathrm{\beta \sigma }}{H}\left[-\psi \left(\frac{a-x^f}{\sigma }\right)+\psi \left(-\frac{x^f}{\sigma }\right)\right]+\frac{\beta x^f}{H}·\Psi \left(\frac{a-x^f}{\sigma }\right)-\frac{\beta x^f}{H}·\Psi \left(-\frac{x^f}{\sigma }\right)& \left(5\right)\end{array}$

FIG. 6 is a diagram for illustrating the expected value M(a) of the lost cost. The upper part of FIG. 6 shows the probability density function φ(x) and the cost f(x) lost when the stored hot water heat amount a is less than the predicted heat amount x^f used for hot water supply. The lower part of FIG. 6 shows a curve indicating the product of the loss f(x) and the probability density function φ(x) shown in the upper part of FIG. 6, and a curve representing the expected value M(a) obtained by integrating the curve. According to this, the expected value M(a) becomes a downwardly convex parabola. In the example of FIG. 6, it can be seen that the expected value M(a) is minimized by increasing the stored hot water heat amount a to be more than the predicted heat amount x^f used for hot water supply.

(5) Step of Calculating the Optimal Stored Hot Water Heat Amount (S50)

In S50, the optimal stored hot water heat amount a_opt is calculated from the expected value M(a) of the lost cost obtained in S40. The optimal stored hot water heat amount a_opt corresponds to the stored hot water heat amount a at the time when the expected value M(a) is the minimum in the parabola shown in the lower part of FIG. 6. As shown in the following formula (6), the optimal stored hot water heat amount a_opt may be calculated by obtaining the stored hot water heat amount a at the time when the value obtained by differentiating the expected value M(a) is 0.

$\left[\mathrm{Formula}6\right]$ $\begin{array}{cc}\frac{\partial M\left(a\right)}{\partial a}=\left(\frac{1}{H}-\frac{1}{G}\right)+\left(\frac{1}{G}-\frac{\beta }{H}\right)·\Psi \left(\frac{a-x^f}{\sigma }\right)=0& \left(6\right)\end{array}$

The following formula (7) is obtained by solving formula (6). From this formula (7), the following formula (8) representing the optimal stored hot water heat amount a_opt is derived.

$\left[\mathrm{Formula}7\right]$ $\begin{array}{cc}\Psi \left(\frac{a_{\mathrm{opt}}-x^f}{\sigma }\right)=1-\frac{\left(1-\beta \right)G}{H-\beta G}& \left(7\right)\end{array}$

$\left[\mathrm{Formula}8\right]$ $\begin{array}{cc}{a}_{\mathrm{opt}}=x^f+\sigma ·{\Psi }^{-1}\left(1-\frac{\left(1-\beta \right)G}{H-\beta G}\right)& \left(8\right)\end{array}$

As shown in formula (8), the optimal stored hot water heat amount a_opt is composed of the predicted heat amount x^f used for hot water supply, the standard deviation σ of the error x-x^f, and the inverse function Ψ⁻¹ of the cumulative distribution function. The inverse function Ψ⁻¹ of the cumulative distribution function becomes a positive value in the case of 1−(1−β)G/(H−βG)>0.5, and becomes a negative value in the case of 1−(1−β)G/(H−βG)<0.5. Accordingly, in the case of 1−(1−β)G/(H−βG)>0.5, a_opt>x^f and the optimal stored hot water heat amount a_opt becomes greater than the predicted heat amount x^f used for hot water supply. On the other hand, in the case of 1−(1−β)G/(H−βG)<0.5, a_opt<x^f and the optimal stored hot water heat amount a_opt becomes less than the predicted heat amount x^f used for hot water supply.

It is possible to adopt an unbiased standard deviation that is calculated from the error between the usage records of hot water supply in the past few days and the predicted heat amount used for hot water supply, or a weighted average value of the unbiased standard deviation, as the standard deviation o of the error x-x^f on the right side of formula (8). As mentioned above, the cost H and the cost G may be calculated using the COP of the heat pump unit 60, the operating efficiency E_G of the auxiliary heat source device 31, and the energy rate unit price, etc. depending on the type of the cost.

To calculate the inverse function Ψ⁻¹ of the cumulative distribution function, a range for obtaining 1−(1−β)G/(H−βG) may be checked in advance, and a linear approximation formula or a table showing the value of the inverse function Ψ⁻¹ may be created. FIG. 7 is a diagram showing an example of the linear approximation formula. The horizontal axis of FIG. 7 is 1−(1−β)G/(H−βG), and the vertical axis is the inverse function Ψ⁻¹ of the cumulative distribution function. An existing tabulated value may be adopted for 1−(1−β)G/(H−βG). In the example of FIG. 7, the linear approximation formula is composed of a plurality of linear functions.

(6) Step of Controlling the Hot Water Storage Operation Based on the Optimal Stored Hot Water Heat Amount a_opt (S60)

In S60, the hot water storage operation is controlled based on the optimal stored hot water heat amount a_opt obtained in S50. Specifically, the controller 100 controls the operation of the heat pump unit 60 and the operation and stop of the circulation pump 40 based on the stored hot water heat amount of the hot water storage tank 30 calculated from the value detected by the temperature sensor 51, so as to store the optimal stored hot water heat amount a_opt for the time period in the hot water storage tank 30 by the time of start of the time period when hot water is predicted to be used.

As described above, in the hot water supply apparatus 5 according to this embodiment, the expected value M(a) of the cost f(x) lost due to generation of the actual heat amount x used for hot water supply in the heat pump unit 60 and the auxiliary heat source device 31 is calculated in consideration of the error distribution of the actual heat amount x used for hot water supply with respect to the predicted heat amount x^f used for hot water supply. Then, the optimal stored hot water heat amount a_opt that minimizes the calculated expected value M(a) is determined, and the hot water storage operation performed by the heat pump unit 60 is controlled based on the optimal stored hot water heat amount a_opt, which minimizes the cost f(x) that is lost due to insufficient hot water storage and excessive hot water storage caused by an error in the actual heat amount x used for hot water supply with respect to the predicted heat amount x^f used for hot water supply. Therefore, the disclosure is capable of improving the efficiency of the hot water supply apparatus 5.

Further, by setting the cost from the viewpoints of the purposes of operation (energy saving, economic efficiency, environmental conservation, etc.) required for the hot water supply apparatus 5, the disclosure is capable of realizing the most efficient hot water storage operation for the purposes of operation.

Other Configuration Examples

(1) Regarding the Prediction Error Distribution

It is assumed in the embodiment described above that the prediction error distribution, which is the distribution of the error x-x^f between the predicted heat amount x^f used for hot water supply and the actual heat amount x used for hot water supply, follows the normal distribution N(0, σ²), but the prediction error distribution is not necessarily a normal distribution. The prediction error distribution may be expressed as a t-distribution or a discrete probability distribution. For example, if the number of samples of time-series data of the past usage records of hot water supply is n, the prediction error distribution may be expressed as a t-distribution with n-1 degrees of freedom.

(2) Regarding the Cost

Although the embodiment described above illustrates the primary energy consumption in the heat pump unit 60 and the auxiliary heat source device 31, the purchase cost of various types of energy, and the amount of CO₂ generated by various types of energy as the cost for generating heat, the disclosure is not limited thereto. Furthermore, the cost may be set by combining two or more of the primary energy consumption, the purchase cost of various types of energy, and the amount of CO₂ generated by various types of energy.

(3) Regarding the Main Body of the Hot Water Storage Control

The embodiment described above illustrates a configuration in which the controller 100 of the hot water storage and supply unit 10 executes the hot water storage control shown in FIG. 3. However, the main body that performs the step of calculating the predicted heat amount used for hot water supply (S10), the step of estimating the prediction error distribution (S20), the step of calculating the cost lost (S30), the step of calculating the expected value of the cost lost (S40), and the step of calculating the optimal stored hot water heat amount (S50) is not limited to the controller 100. For example, a server communicably connected to the controller 100 and remotely managing the hot water supply apparatus 5 may execute S10 to S50 to calculate the optimal stored hot water heat amount a_opt. In this case, the server transmits the calculated optimal stored hot water heat amount a_opt to the controller 100. The controller 100 executes the step (S60) of controlling the hot water storage operation based on the optimal stored hot water heat amount a_opt received from the server. That is, the server and the controller 100 correspond to an embodiment of the “control device.”

(4) Regarding the Hot Water Storage Control

In the hot water storage control shown in FIG. 3, the step of calculating the cost lost (S30) and the step of calculating the expected value of the cost lost (S40) may be omitted from the processing that is repeatedly executed by the controller 100 or the server for each unit time period.

For example, the cost H of the heat pump unit 60, the cost G of the auxiliary heat source device 31, and the reuse efficiency β of surplus heat amount in the heat pump unit 60 may be provided and used to prepare in advance a linear approximation formula or a table expressing the inverse function Ψ⁻¹ of the cumulative distribution function, and the optimal stored hot water heat amount a_opt may be determined from formula (8) in S50 based on the predicted heat amount x^f used for hot water supply calculated in S10 and the standard deviation o of the error x-x^f obtained in S20.

The embodiments disclosed herein are illustrative in all aspects and should not be considered to be restrictive. The scope of the disclosure is defined by the claims rather than the above description, and is intended to cover all the changes within the meaning and scope equivalent to the claims.

Claims

1. A hot water supply apparatus, comprising: a heat pump unit that operates with electrical energy as a driving source;a hot water storage tank that stores hot water heated by the heat pump unit;a hot water supply circuit that supplies hot water from the hot water storage tank;an auxiliary heat source device for burning fuel and heating hot water to be supplied in response to a stored hot water heat amount of the hot water storage tank being insufficient for a heat amount used for hot water supply; anda control device that controls the heat pump unit and the auxiliary heat source device,wherein the control devicecalculates a predicted heat amount used for hot water supply for each unit time period based on a past usage record of hot water supply,estimates an error distribution of an actual heat amount used for hot water supply with respect to the predicted heat amount used for hot water supply,calculates an expected value of loss in a target index, caused by a hot water storage operation of the heat pump unit and a combustion operation of the auxiliary heat source device in response to generating the actual heat amount used for hot water supply, using the estimated error distribution, andobtains an optimal stored hot water heat amount that minimizes the expected value of loss, and controls the heat pump unit so that the hot water storage tank stores the optimal stored hot water heat amount.

2. The hot water supply apparatus according to claim 1, wherein the control device estimates the error distribution on an assumption that an error of the actual heat amount used for hot water supply with respect to the predicted heat amount used for hot water supply follows a normal distribution.

3. The hot water supply apparatus according to claim 1, wherein the control device uses cost associated with an operation of the heat pump unit and an operation of the auxiliary heat source device as the target index.

4. The hot water supply apparatus according to claim 2, wherein the control device uses cost associated with an operation of the heat pump unit and an operation of the auxiliary heat source device as the target index.

5. The hot water supply apparatus according to claim 3, wherein the control device calculates cost that is lost in a case where the stored hot water heat amount of the hot water storage tank is less than the actual heat amount used for hot water supply and in a case where the stored hot water heat amount of the hot water storage tank is greater than the actual heat amount used for hot water supply, andcalculates the expected value of loss using the calculated cost that is lost and the estimated error distribution.

6. The hot water supply apparatus according to claim 4, wherein the control device calculates cost that is lost in a case where the stored hot water heat amount of the hot water storage tank is less than the actual heat amount used for hot water supply and in a case where the stored hot water heat amount of the hot water storage tank is greater than the actual heat amount used for hot water supply, andcalculates the expected value of loss using the calculated cost that is lost and the estimated error distribution.

7. The hot water supply apparatus according to claim 3, wherein the cost associated with the operation of the heat pump unit comprises at least one of primary energy consumption in the heat pump unit, purchase cost of electrical energy, and an amount of CO2 generated by electrical energy, and the cost associated with the operation of the auxiliary heat source device comprises at least one of primary energy consumption in the auxiliary heat source device, purchase cost of fuel, and an amount of CO2 generated by combustion of fuel.

8. The hot water supply apparatus according to claim 4, wherein the cost associated with the operation of the heat pump unit comprises at least one of primary energy consumption in the heat pump unit, purchase cost of electrical energy, and an amount of CO2 generated by electrical energy, and the cost associated with the operation of the auxiliary heat source device comprises at least one of primary energy consumption in the auxiliary heat source device, purchase cost of fuel, and an amount of CO2 generated by combustion of fuel.

9. The hot water supply apparatus according to claim 7, wherein in response to the heat pump unit operating with electrical energy transmitted from a power plant as the driving source, the control device sets the cost associated with the operation of the heat pump unit using primary energy consumption based on power generation efficiency of the power plant, power transmission and distribution loss from the power plant, and a coefficient of performance of the heat pump unit, and in response to the heat pump unit operating with electrical energy generated by a solar power generation device connected to the heat pump unit as the driving source, the control device sets the cost associated with the operation of the heat pump unit using primary energy consumption based on the coefficient of performance of the heat pump unit.

10. The hot water supply apparatus according to claim 8, wherein in response to the heat pump unit operating with electrical energy transmitted from a power plant as the driving source, the control device sets the cost associated with the operation of the heat pump unit using primary energy consumption based on power generation efficiency of the power plant, power transmission and distribution loss from the power plant, and a coefficient of performance of the heat pump unit, and in response to the heat pump unit operating with electrical energy generated by a solar power generation device connected to the heat pump unit as the driving source, the control device sets the cost associated with the operation of the heat pump unit using primary energy consumption based on the coefficient of performance of the heat pump unit.

11. A control method for controlling a hot water storage operation in a hot water supply apparatus which comprises: a heat pump unit that operates with electrical energy as a driving source;a hot water storage tank that stores hot water heated by the heat pump unit;a hot water supply circuit that supplies hot water from the hot water storage tank; andan auxiliary heat source device for burning fuel and heating hot water to be supplied in response to a stored hot water heat amount of the hot water storage tank being insufficient for a heat amount used for hot water supply,the control method comprising:calculating a predicted heat amount used for hot water supply for each unit time period based on a past usage record of hot water supply;estimating an error distribution of an actual heat amount used for hot water supply with respect to the predicted heat amount used for hot water supply;calculating an expected value of loss in a target index, caused by a hot water storage operation of the heat pump unit and a combustion operation of the auxiliary heat source device in response to generating the actual heat amount used for hot water supply, using the estimated error distribution;calculating an optimal stored hot water heat amount that minimizes the expected value of loss; andcontrolling the heat pump unit so that the hot water storage tank stores the optimal stored hot water heat amount.

12. The control method according to claim 11, wherein estimating the error distribution comprises estimating the error distribution on an assumption that an error of the actual heat amount used for hot water supply with respect to the predicted heat amount used for hot water supply follows a normal distribution.

13. The control method according to claim 11, wherein calculating the expected value of loss comprises: calculating cost that is lost due to generation of the actual heat amount used for hot water supply using cost associated with an operation of the heat pump unit and an operation of the auxiliary heat source device as the target index; andcalculating the expected value of loss using the cost that is lost and the error distribution.

14. The control method according to claim 12, wherein calculating the expected value of loss comprises: calculating cost that is lost due to generation of the actual heat amount used for hot water supply using cost associated with an operation of the heat pump unit and an operation of the auxiliary heat source device as the target index; andcalculating the expected value of loss using the cost that is lost and the error distribution.

15. The control method according to claim 13, wherein the cost associated with the operation of the heat pump unit comprises at least one of primary energy consumption in the heat pump unit, purchase cost of electrical energy, and an amount of CO2 generated by electrical energy, and the cost associated with the operation of the auxiliary heat source device comprises at least one of primary energy consumption in the auxiliary heat source device, purchase cost of fuel, and an amount of CO2 generated by combustion of fuel.

16. The control method according to claim 14, wherein the cost associated with the operation of the heat pump unit comprises at least one of primary energy consumption in the heat pump unit, purchase cost of electrical energy, and an amount of CO2 generated by electrical energy, and the cost associated with the operation of the auxiliary heat source device comprises at least one of primary energy consumption in the auxiliary heat source device, purchase cost of fuel, and an amount of CO2 generated by combustion of fuel.