Coordinated local frequency control of refrigerators

For a population of refrigerators, at a given time, their TCa are different according to their inherent diversity. Therefore, they are given different FON and FOFF as calculated by (3.13) and (3.14). As a result, following a deviation in frequency, refrigerators will

50 be triggered in sequence instead of switching ON/OFF simultaneously. The greater the deviation of frequency, the higher the number of refrigerators responds to the frequency deviation.

A refrigerator with the highest TCa is given an FON closest to 50 Hz while a refrigerator with the lowest TCa is given an FON farthest from 50 Hz as indicated by Fig. 3.4.

Therefore, following a rise in frequency, refrigerators are switched ON starting from the one with the highest TCa.

Similarly, a refrigerator with the lowest TCa is given an FOFF closest to 50 Hz while a refrigerator with the highest TCa is given an FOFF farthest from 50 Hz as shown in Fig.

3.4. Therefore, following a drop in frequency, refrigerators are switched OFF starting from the one with the lowest TCa.

However, if there is a sudden and severe drop in frequency, a large number of refrigerators will be switched OFF almost simultaneously.

 It is necessary to switch OFF refrigerators with a similar temperature at different times and at different frequencies. This will ensure that temperature of refrigerators amongst a number of refrigerators maintains to be different.

 It is also necessary to try to minimise the number of refrigerators that will be re-connected to the grid at the same time after the refrigerators were switched OFF in response to the frequency drop.

Methods to address the above two concerns are now explained.

3.3.3.1 To minimise the number of refrigerators with the same TCa to be switched simultaneously for frequency response

In addition to Tlow and Thigh of each refrigerator, a mid-temperature, Tmid, defined by (3.15) is also pre-set. As shown in Fig. 3.5(a), the linear relationship between FON and TCa are sectioned at Tmid. This is achieved by incorporating a random number, HD, at Tmid to calculate FON of a refrigerator at Tmid as shown in (3.16). HD is uniformly distributed between 0 and 1 and is randomly assigned to each refrigerator.

𝑇_𝑚𝑖𝑑 = 𝑇_𝑙𝑜𝑤+ 0.5 × (𝑇_{ℎ𝑖𝑔ℎ}− 𝑇_𝑙𝑜𝑤) (3.15) 𝐹_{𝑂𝑁_𝑚𝑖𝑑_𝑛} = 𝑁𝐹 + 𝐻𝐷 × (𝐻𝐹 − 𝑁𝐹), 𝑛 = 1,2, … (3.16)

51 where n is the refrigerator ID number (Refrigerator 1, Refrigerator 2 shown in Fig.

3.5(a)).

The calculation of FON at Tmid generates the different curves of FON against TCa for different refrigerators as depicted by the two-sectional linear relationship shown in Fig.

3.5(a). The relationship between FON and TCa is written by (3.17). FON and TCa in Fig. 3.4 is still maintained over the population. For the three refrigerators shown in Fig. 3.4(a), if they are with the same TCa, for instance Ti at a given time, their FON is different and hence they will be triggered ON for different levels of frequency rise. As a result, temperature and the ON/OFF state of refrigerators are more different amongst the refrigerator population.

(a) FON of a population of refrigerators (b) F_OFF of a population of refrigerators NF

Fig. 3.5. Assignment of trigger frequencies for a population of refrigerators

Similarly as shown in Fig. 3.5(b), FOFF at Tmid is written by:

𝐹_{𝑂𝐹𝐹_𝑚𝑖𝑑_𝑛} = 𝐿𝐹 + 𝐿𝐷 × (𝑁𝐹 − 𝐿𝐹), 𝑛 = 1,2, … (3.18) where n is the refrigerator ID number. LD is also the incorporated random number following the uniform distribution between 0 and 1 for the calculation of FOFF of each refrigerator at Tmid (FOFF_mid_n). The calculation of FOFF_mid_n gives different curves of

52 FOFF against TCa for different refrigerators as the two-sectional linear relationship shown in Fig. 3.5(b). The relationship between FOFF and TCa is depicted by (3.19).

𝐹_𝑂𝐹𝐹(𝑡) =

3.3.3.2 To minimise the number of refrigerators to revert back simultaneously after the provision of frequency response

To avoid simultaneous connection of refrigerators after a sudden and severe frequency drop, when frequency starts to recover, some of the OFF-state refrigerators are controlled to be switched ON before their TCa reach Thigh. This is referred to as early switching action in this thesis.

When frequency is below 50 Hz, FON is autonomously re-calculated using (3.17) by assigning NF the value of f(t−Δt) rather than the 50 Hz that was used in Fig. 3.5(a). Δt is the sampling time of the grid frequency measurements which was 200 ms. The re-calculation and update of FON is shown in (3.20). As a result, FON of some refrigerators is shifted to be lower than 50 Hz as shown in Fig. 3.6(a). frequency FON1 which is updated to be lower than 50 Hz. Before f recovers to reach 50 Hz, the refrigerator will be switched ON if f becomes higher than FON1. Therefore, when frequency starts to recover, some of the OFF-state refrigerators are switched ON early before their TCa reach Thigh. As the relationship shown in Fig. 3.6(a), f will recover to be higher than FON of a refrigerator with a temperature closest to Thigh first. Hence, following the frequency recovery, power consumption of refrigerators will start to recover gradually. This reduces the number of refrigerators that will be connected to the grid at the same time.

53

(a) FON of OFF-state refrigerators during the recovery from a low frequency event

(b) FOFF of ON-state refrigerators during the recovery from a high frequency event

Fig. 3.6. Update of FON and FOFF during the recovery from frequency events

Similar rules apply to the ON-state refrigerators after a frequency rise event. Some of them are controlled to be switched OFF earlier than they otherwise would be following the frequency recovery from the frequency rise. As shown in Fig. 3.6(b), when frequency is above 50 Hz, FOFF is re-calculated using (3.19) by assigning NF the value of f(t−Δt) instead of 50 Hz. This is written by (3.21). Therefore, the ON-state refrigerators start to be switched OFF earlier before their TCa reach Tlow such as the refrigerator shown in Fig. 3.6(b) which will be switched OFF at T2 rather than at Tlow

when f recovers to be lower than FOFF2. The refrigerator with the lowest TCa will be switched OFF first because f will recover to fall below FOFF of this refrigerator first.

𝐹_𝑂𝐹𝐹(𝑡) =

In the frequency control of a refrigerator, when calculating FON and FOFF, the value of NF in Fig. 3.5 and Fig. 3.6 are combined to cover both the frequency response behaviour and the frequency recovery behaviour. Hence, equation (3.22) and (3.23) are obtained.

NF for calculating FON:

𝑁𝐹 = min(50, 𝑓(𝑡 − ∆𝑡)) (3.22) NF for calculating FOFF:

54 𝑁𝐹 = max(50, 𝑓(𝑡 − ∆𝑡)) (3.23) The calculation and update of FON and FOFF are summarised in Table 3.4.

TABLE 3.4. Dynamic Update of Trigger Frequencies FON and FOFF

f measurement: f(t) NF of calculating F_ON(t) in (3.17) NF of calculating F_OFF(t) in (3.19)

𝑓(𝑡) ≥ 50 𝐻𝑧 NF=50 Hz for frequency rise NF= f(t−Δt) for frequency recovery

𝑓(𝑡) < 50 𝐻𝑧 NF=f(t−Δt) for frequency

recovery NF=50 Hz for frequency drop