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CN109167350B - Construction method of industrial load response model - Google Patents

Construction method of industrial load response model Download PDF

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CN109167350B
CN109167350B CN201811025669.0A CN201811025669A CN109167350B CN 109167350 B CN109167350 B CN 109167350B CN 201811025669 A CN201811025669 A CN 201811025669A CN 109167350 B CN109167350 B CN 109167350B
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time
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demand response
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CN109167350A (en
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徐乾耀
周保荣
姚文峰
洪潮
毛田
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China South Power Grid International Co ltd
China Southern Power Grid Co Ltd
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China Southern Power Grid Co Ltd
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for AC mains or AC distribution networks
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J3/00—Circuit arrangements for AC mains or AC distribution networks
    • H02J3/008—Circuit arrangements for AC mains or AC distribution networks involving trading of energy or energy transmission rights
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J2203/00—Indexing scheme relating to details of circuit arrangements for AC mains or AC distribution networks
    • H02J2203/20—Simulating, e g planning, reliability check, modelling or computer assisted design [CAD]

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Abstract

The invention discloses a method for constructing an industrial load response model, which comprises the following steps: acquiring a user set participating in real-time response of the industrial load and an equipment set corresponding to each user; acquiring the demand response quantity, demand response price and response cost of real-time response of the industrial load; and constructing an objective function with the maximum economic benefit. By adopting the embodiment of the invention, quantitative analysis can be realized on the characteristics of different industrial loads, and the maximization of economic benefit is realized.

Description

Construction method of industrial load response model
Technical Field
The invention relates to the field of power demand management, in particular to a construction method of an industrial load response model.
Background
With the gradual advance of the electric power market reform and the vigorous development of renewable energy sources, the uncertainty in the electric power system is rapidly increasing, and how to effectively improve the flexibility of the electric power system becomes a new research hotspot. At present, relatively few research on relevant models of real-time automatic demand response of a demand side based on price excitation is carried out, or response automation of load equipment is not considered, or actual characteristics of load operation are not considered, or practical feasibility is lacked, so that production requirements, operation characteristics, electricity utilization properties and response characteristics of regional loads such as parks and micro-grids cannot be fully considered, and economic benefit maximization cannot be realized.
Disclosure of Invention
The embodiment of the invention aims to provide a construction method of an industrial load response model, which can realize quantitative analysis on the characteristics of different industrial loads and realize maximization of economic benefit.
In order to achieve the above object, an embodiment of the present invention provides a method for constructing an industrial load response model, including:
acquiring a user set participating in real-time response of the industrial load and an equipment set corresponding to each user;
acquiring the demand response quantity, demand response price and response cost of real-time response of the industrial load;
constructing an objective function with the maximum economic benefit, wherein the objective function is as follows:
Figure BDA0001788458100000011
wherein T is the total time interval; Δ t is the duration of each time interval in min; ΨdrA set of users participating in real-time demand response of industrial loads;
Figure BDA0001788458100000012
a device set for participating in real-time demand response for a user i; dtThe demand response given for time t; pi is the external call demand response price; pii，jResponse cost for user i, device j; di，j，tThe adjustment amount is the demand response of the user i and the equipment j at the moment t.
Compared with the prior art, the construction method of the industrial load response model disclosed by the invention comprises the steps of firstly, acquiring a user set participating in real-time response of the industrial load and an equipment set corresponding to each user; then acquiring the demand response quantity, demand response price and response cost of real-time response of the industrial load; and finally constructing an objective function with the maximum economic benefit. The problem that in the prior art, due to the fact that related model researches of real-time automatic demand response of a demand side based on price excitation are relatively few, economic benefit maximization cannot be achieved is solved, quantitative analysis of characteristics of different industrial loads can be achieved, and economic benefit maximization is achieved.
As an improvement of the above scheme, the objective function satisfies a demand response constraint condition, where the demand response constraint condition is:
Figure BDA0001788458100000021
wherein d isi，tResponding to the adjustment quantity for the demand of the user i at the moment t; gamma rayi，tThe response reliability of the user i at the time t is expressed by percentage and is obtained by the historical performance statistics of the user, and the initial time is set to be 80-90%.
As an improvement of the above scheme, the objective function satisfies an adjustment depth constraint condition, where the adjustment depth constraint condition is:
Figure BDA0001788458100000022
wherein,
Figure BDA0001788458100000023
the original load of the user i and the equipment j at the moment t is obtained;
Figure BDA0001788458100000024
the maximum adjusting depth of the user i and the equipment j at the moment t is the proportion of the maximum adjustable load to the original load;
Figure BDA0001788458100000025
the minimum adjusting depth of the user i and the equipment j at the moment t is the proportion of the minimum adjustable load to the original load; i isi，j，tThe calling state of the user i and the device j at the time t is equal to 1 when being called, and is equal to 0 otherwise.
As an improvement of the above scheme, the device satisfies a forward relevance constraint condition, where the forward relevance constraint condition is:
Figure BDA0001788458100000031
wherein,
Figure BDA0001788458100000032
for positive turn-off of user iA set of connected device pairs; (x, y) is any pair of devices with a forward correlation;
the device satisfies a reverse relevance constraint condition, which is:
Figure BDA0001788458100000033
wherein,
Figure BDA0001788458100000034
a set of reverse association device pairs for user i; (x ', y') is any pair of devices with reverse correlation.
As an improvement of the above scheme, the device satisfies a forward latency constraint condition, where the forward latency constraint condition is:
Figure BDA0001788458100000035
wherein,
Figure BDA0001788458100000036
a set of forward delay device pairs for user i; (a, b) is any device pair with forward delay, namely, after the device a is called for a period of time, the device b is called; t is t(a，b)A delay interval of (a, b);
the equipment meets the constraint condition of reverse time delay, and the constraint condition of reverse time delay is as follows:
Figure BDA0001788458100000037
wherein,
Figure BDA0001788458100000038
a reverse delay device pair set for user i; (a ', b') is any equipment pair with reverse delay, and after the equipment a 'is called for a period of time, the equipment b' stops being called; t is t(a′，b′)The delay interval of (a ', b').
As an improvement of the above scheme, the objective function satisfies a duration constraint condition, where the duration constraint condition is:
Figure BDA0001788458100000039
wherein T is the total time period number of the real-time demand response of the industrial load; t isi，jThe adjustable duration for user i, device j.
As an improvement of the above scheme, the device satisfies a single participation constraint condition, where the single participation constraint condition is:
Figure BDA0001788458100000041
wherein,
Figure BDA0001788458100000042
a variable that is either 0 or 1, and,
Figure BDA0001788458100000043
indicating that the user i and the equipment j are called at the moment t, and otherwise, indicating that the calling is 0;
Figure BDA0001788458100000044
a variable that is either 0 or 1, and,
Figure BDA0001788458100000045
indicating that the user i and the device j are stopped to call at the time t, and otherwise, the value is 0.
As an improvement of the above scheme, the objective function satisfies a climbing constraint condition, where the climbing constraint condition is:
Figure BDA0001788458100000046
wherein,
Figure BDA0001788458100000047
response rates for user i and device j;
Figure BDA0001788458100000048
the recovery rate for user i, device j.
Drawings
FIG. 1 is a flow chart of a method for constructing an industrial load response model according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a user and a device in a method for constructing an industrial load response model according to an embodiment of the present invention;
FIG. 3 is a graph illustrating an original load of a device in a method for constructing an industrial load response model according to an embodiment of the present invention;
FIG. 4 is a graph of the plant demand response adjustment in the method for constructing an industrial load response model according to an embodiment of the present invention;
fig. 5 is a graph of an actual load of a device in a method for constructing an industrial load response model according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1, fig. 1 is a flowchart of a method for constructing an industrial load response model according to an embodiment of the present invention; the method comprises the following steps:
s1, acquiring a user set participating in industrial load real-time response and an equipment set corresponding to each user; wherein each user comprises at least one device;
s2, acquiring the demand response quantity, the demand response price and the response cost of the real-time response of the industrial load; wherein, the demand response quantity is a set target value; the demand response price is the electricity price of the equipment; the response cost is the operation cost for realizing load regulation of unit capacity in unit time, and is unit/(kW & min).
S3, constructing an objective function with the maximum economic benefit, wherein the objective function is as follows:
Figure BDA0001788458100000051
wherein T is the total time interval; Δ t is the duration of each time interval in min; ΨdrA set of users participating in real-time demand response of industrial loads;
Figure BDA0001788458100000052
a device set for participating in real-time demand response for a user i; dtThe demand response given for time t; pi is the external call demand response price; pii，jResponse cost for user i, device j; di，j，tThe adjustment amount is the demand response of the user i and the equipment j at the moment t.
T is the time interval number and is a natural number, for example, the duration time that the electric furnace can participate in the demand response is 30 minutes, and if delta T is 1 minute, T is 30; if Δ T is 5 minutes, T is 6.
It is worth mentioning that, in the embodiment of the present invention, the user participating in the real-time demand response plan can change and set the own electricity demand condition at any time. In the embodiment of the present invention, a demand response amount is preset, for example, a load of 5000KW is reduced when the demand response amount is 14 to 15, and then a demand response adjustment amount of which device in which user is optimally required at which time is calculated according to the objective function. Therefore, the power consumption of each device is reasonably distributed, and at the moment, the users participating in the real-time demand response can change own power consumption demands according to the demand response adjustment quantity, so that the demand response quantity is achieved.
For industrial users, the load generally includes a plurality of branches including a basic electric branch, an auxiliary electric branch and a production electric branch, wherein 80% to 90% of the load is the production electric load, and therefore, most of the load branches participating in demand response of the industrial users are production electric branches, and the branches generally comprise a plurality of devices and have different connection types such as series connection type, parallel connection type and mixed type.
Preferably, the objective function satisfies a demand response constraint condition, where the demand response constraint condition is:
Figure BDA0001788458100000061
wherein d isi，tResponding to the adjustment quantity for the demand of the user i at the moment t; gamma rayi，tThe response reliability of the user i at the time t is expressed by percentage and is obtained by the historical performance statistics of the user, and the initial time is set to be 80-90%.
Specifically, real-time automatic demand response allows a user to opt-out of participation in a response phase within a short time window (0.5-10 min) during design, as electrical loads may have sudden safety, economic or other special problems. In order to ensure the completion of the demand response, a part of the demand response load is usually required to be called as a spare, and the size of the spare is highly related to the response credibility of each user.
Preferably, the objective function satisfies an adjustment depth constraint condition, where the adjustment depth constraint condition is:
Figure BDA0001788458100000062
wherein,
Figure BDA0001788458100000063
the original load of the user i and the equipment j at the moment t is obtained;
Figure BDA0001788458100000064
the maximum adjusting depth of the user i and the equipment j at the moment t is the proportion of the maximum adjustable load to the original load;
Figure BDA0001788458100000065
for minimum adjustment of user i, device j at time tThe minimum adjusting depth is the proportion of the minimum adjustable load to the original load; i isi，j，tThe calling state of the user i and the device j at the time t is equal to 1 when being called, and is equal to 0 otherwise.
Specifically, in order to measure the workload of the user participating in the demand response, it is required to estimate the original load curve if the user does not participate in the demand response, and use the curve as a reference, which is called a demand response reference curve. In the embodiment of the invention, an average value method is adopted, and the average value of the corresponding load values 5 days before the demand response event is used as the original load.
Preferably, the device satisfies a forward relevance constraint condition, where the forward relevance constraint condition is:
Figure BDA0001788458100000071
wherein,
Figure BDA0001788458100000072
associating a set of device pairs for user i in a forward direction; (x, y) is any pair of devices with a forward correlation;
the device satisfies a reverse relevance constraint condition, which is:
Figure BDA0001788458100000073
wherein,
Figure BDA0001788458100000074
a set of reverse association device pairs for user i; (x ', y') is any pair of devices with reverse correlation.
In particular, since an industrial production line generally has a relationship that when a certain device is called, another device of the same production line also needs to be called (or stop being called), it is necessary to fully consider modeling of such a situation. Two devices that need to be called simultaneously and stop being called simultaneously are said to have forward correlation, and two devices that need to be on and off simultaneously are said to have reverse correlation.
Preferably, the device satisfies a forward latency constraint condition, where the forward latency constraint condition is:
Figure BDA0001788458100000075
wherein,
Figure BDA0001788458100000076
a set of forward delay device pairs for user i; (a, b) is any device pair with forward delay, namely, after the device a is called for a period of time, the device b is called; t is t(a，b)A delay interval of (a, b);
the equipment meets the constraint condition of reverse time delay, and the constraint condition of reverse time delay is as follows:
Figure BDA0001788458100000077
wherein,
Figure BDA0001788458100000078
a reverse delay device pair set for user i; (a ', b') is any device pair with reverse delay, namely, after the device a 'is called for a period of time, the device b' stops being called; after the device a 'is stopped to be called for a period of time, the device b' is called; t is t(a′，b′)Delay interval of (a ', b'), for the device pair with reverse delay, t, when the device b 'starts calling after the device a' is stopped calling for a period of time(a′，b′)Is a negative number.
Specifically, part of industrial equipment is located in the same production line and has a relation between a front process and a rear process, under the condition, after the front process is started or stopped for a period of time, the rear process must be started or stopped, otherwise, the conditions of material overstock, disorder, damage and the like in the processes can occur. This strict continuous front-to-back production characteristic is called the delay. If the former procedure is started (stopped) for a period of time, the latter procedure must be started (stopped), and the product is called to have positive time delay; if the previous procedure is started (stopped) for a period of time, the next procedure must be stopped (started), and the next procedure is called to have reverse time delay; the delay duration is called the delay interval.
Preferably, the objective function satisfies a duration constraint condition, where the duration constraint condition is:
Figure BDA0001788458100000081
wherein T is the total time period number of the real-time demand response of the industrial load; t isi，jThe adjustable duration for user i, device j. Specifically, the duration is the maximum duration for completing the specified load adjustment amount, and is set by a user in advance, and the adjustable duration is the maximum duration for completing the specified load adjustment amount, and different adjustable durations can be set according to different adjustment depths.
Preferably, the device satisfies a single participation constraint that:
Figure BDA0001788458100000082
wherein,
Figure BDA0001788458100000083
a variable that is either 0 or 1, and,
Figure BDA0001788458100000084
indicating that the user i and the equipment j are called at the moment t, and otherwise, indicating that the calling is 0;
Figure BDA0001788458100000085
a variable that is either 0 or 1, and,
Figure BDA0001788458100000086
indicating that the user i and the device j are stopped to call at the time t, and otherwise, the value is 0. The single participation constraint ensures eachThe device is invoked only once in a demand response event and is not allowed to be invoked multiple times.
Preferably, the objective function satisfies a climbing constraint condition, where the climbing constraint condition is:
Figure BDA0001788458100000087
Figure BDA0001788458100000095
wherein,
Figure BDA0001788458100000091
response rates for user i and device j;
Figure BDA0001788458100000092
the recovery rate for user i, device j.
Specifically, the response rate is the load adjustment amount which can be realized in unit time, and the unit is kW/min; the recovery rate is the load capacity recovered in unit time after the maximum duration is reached or the demand response is completed, and the unit is kW/min, which is set by a user in advance.
Further, according to the formulas (1) to (10), the formula
Figure BDA0001788458100000093
And constructing an industrial load response model for optimizing the mixed integer linear programming model of the decision variables.
Further, a specific embodiment is specifically described, referring to fig. 2, fig. 2 is a schematic diagram of a user and a device in a method for constructing an industrial load response model according to an embodiment of the present invention; wherein, users are represented by a1 to a13, B1 to B9, and C1 to C6, devices are represented by D1 to D9, for example, an A6 user has two devices D1 and D2, an a10 user has one device D3, an a12 user has one device D4, a C3 user has two devices D5 and D6, a C6 user has one device D7, a B4 user has one device D8, and a B5 user has one device D9 (the other users not showing the number of devices are users not participating in real-time response of industrial load).
The power distribution network adopts double branch power supply, generality is not lost, all branch line parameters are assumed to be consistent, the power distribution network has 28 industrial users in total, 9 large-scale industrial devices or device branches participate in real-time demand response, and the parameters are shown in table 1.
TABLE 1 Equipment parameters participating in demand response
Figure BDA0001788458100000094
Wherein, the devices D1 and D2 have positive relevance and are called at the same time or stop being called at the same time; the devices D5 and D6 have forward latency, and when D5 is called, D6 needs to be called after 15 minutes. The response confidence levels are all set to 90%. In afternoon of a certain day in summer, the industrial park obtains a real-time demand response instruction, the load of 5000kW is reduced when the demand response index is 14 hours to 15 hours, the response price is 1.67 yuan/(kW.min), namely 100 yuan can be obtained when 1kW load continuously responds for 1 hour; the raw load curves for participating demand response devices at 14 through 15 are shown in FIG. 3.
The technical scheme of real-time automatic demand response is realized by adopting a remote measuring remote signaling remote control technology, an intelligent electric meter is installed at an electric shunt gateway for equipment, monitoring data of the intelligent electric meter is collected into a regional data exchanger, and the exchanger transmits data remote signaling to a central cloud platform; the command issuing is reverse to the above process, and the remote control function is realized. The data delay time of the whole information loop is within 400ms, and the time efficiency requirement of real-time automatic demand response is met.
Carrying out optimization solution by using the model formulas (1) to (10) to obtain a demand response regulating quantity curve of each device shown in fig. 4, wherein the total regulating quantity is 5556.3 kilowatt hours, the corresponding load of each device needs to be reduced, and specifically, the reference is made to fig. 4 for how much load is reduced at a certain moment, until 15 hours (corresponding to 60min in fig. 4) are reached, the load of 5556.3 kilowatt hours is reduced altogether, and the objective function value is 40.356 ten thousand yuan, which is the maximum profit of the industrial park through participation in the demand response; wherein 50 ten thousand yuan (external calling demand response price) is obtained from the upper calling platform, namely 9.644 ten thousand yuan (extra profit) can be paid to the industrial enterprise participating in the demand response. It should be noted that, in the calling process, as the response reliability is considered, a certain extra cost is paid to ensure that the demand response index is completed. Fig. 5 is a graph of the actual load of each device after participating in the demand response.
The optimized computing platform provided by the embodiment of the invention is an IBM Power System S812L blade server, the CPU is POWER 83.42GHz, the memory is DDR3-64G, and the optimized time is 6 milliseconds based on C language modeling. Furthermore, the quantity of equipment participating in real-time demand response is measured and calculated, and the optimized calculation time is within the range of 50-150 milliseconds.
In specific implementation, a user set participating in industrial load real-time response and a device set corresponding to each user are obtained; then acquiring the demand response quantity, demand response price and response cost of real-time response of the industrial load; and finally constructing an objective function with the maximum economic benefit. The problem that in the prior art, due to the fact that related model researches of real-time automatic demand response of a demand side based on price excitation are relatively few, economic benefit maximization cannot be achieved is solved, quantitative analysis of characteristics of different industrial loads can be achieved, and economic benefit maximization is achieved.
The construction method of the industrial load response model provided by the embodiment of the invention can deeply analyze the operating characteristics of the industrial load, can realize quantitative analysis on the characteristics of different industrial loads, and provides a real-time demand response optimization decision method of a superior load control master station.
While the foregoing is directed to the preferred embodiment of the present invention, it will be understood by those skilled in the art that various changes and modifications may be made without departing from the spirit and scope of the invention.

Claims (5)

1. A method for constructing an industrial load response model is characterized by comprising the following steps:
acquiring a user set participating in real-time response of the industrial load and an equipment set corresponding to each user;
acquiring the demand response quantity, demand response price and response cost of real-time response of the industrial load;
constructing an objective function with the maximum economic benefit, wherein the objective function is as follows:
Figure FDA0002474593510000011
wherein T is the total time interval; Δ t is the duration of each time interval in min; ΨdrA set of users participating in real-time demand response of industrial loads;
Figure FDA0002474593510000012
a device set for participating in real-time demand response for a user i; dtThe demand response given for time t; pi is the external call demand response price; pii，jResponse cost for user i, device j; di，j，tAdjusting quantity for demand response of a user i and equipment j at a moment t;
the objective function meets the constraint condition of the demand response quantity, and the constraint condition of the demand response quantity is as follows:
Figure FDA0002474593510000013
wherein d isi，tResponding to the adjustment quantity for the demand of the user i at the moment t; gamma rayi，tThe response reliability of the user i at the time t is represented by percentage and is obtained by historical performance statistics of the user, and the initial time is set to be 80-90%;
the objective function meets the constraint condition of adjusting depth, and the constraint condition of adjusting depth is as follows:
Figure FDA0002474593510000014
wherein,
Figure FDA0002474593510000015
the original load of the user i and the equipment j at the moment t is obtained;
Figure FDA0002474593510000016
the maximum adjusting depth of the user i and the equipment j at the moment t is the proportion of the maximum adjustable load to the original load;
Figure FDA0002474593510000017
the minimum adjusting depth of the user i and the equipment j at the moment t is the proportion of the minimum adjustable load to the original load; i isi，j，tThe calling states of the user i and the equipment j at the moment t are equal to 1 when being called, otherwise, the calling states are equal to 0;
the target function meets a climbing constraint condition, and the climbing constraint condition is as follows:
Figure FDA0002474593510000018
Figure FDA0002474593510000021
wherein,
Figure FDA0002474593510000022
response rates for user i and device j;
Figure FDA0002474593510000023
the recovery rate for user i, device j.
2. The method of constructing an industrial load response model of claim 1, wherein the plant satisfies a forward relevance constraint that is:
Figure FDA0002474593510000024
wherein,
Figure FDA0002474593510000025
associating a set of device pairs for user i in a forward direction; (x, y) is any pair of devices with a forward correlation;
the device satisfies a reverse relevance constraint condition, which is:
Figure FDA0002474593510000026
wherein,
Figure FDA0002474593510000027
a set of reverse association device pairs for user i; (x ', y') is any pair of devices with reverse correlation.
3. The method of constructing an industrial load response model of claim 1, wherein the plant satisfies a forward latency constraint, the forward latency constraint being:
Figure FDA0002474593510000028
wherein,
Figure FDA0002474593510000029
a set of forward delay device pairs for user i; (a, b) is any device pair with forward delay, namely, after the device a is called for a period of time, the device b is called; t is t(a，b)A delay interval of (a, b);
the equipment meets the constraint condition of reverse time delay, and the constraint condition of reverse time delay is as follows:
Figure FDA00024745935100000210
wherein,
Figure FDA00024745935100000211
a reverse delay device pair set for user i; (a ', b') is any pair of devices with reverse time delayAfter a 'is called for a period of time, the b' stops being called; t is t(a′，b′)The delay interval of (a ', b').
4. The method of constructing an industrial load response model of claim 1, wherein the objective function satisfies a duration constraint, the duration constraint being:
Figure FDA0002474593510000031
wherein T is the total time period number of the real-time demand response of the industrial load; t isi，jThe adjustable duration for user i, device j.
5. The method of constructing an industrial load response model of claim 1, wherein the plant satisfies a single engagement constraint that:
Figure FDA0002474593510000032
wherein,
Figure FDA0002474593510000033
a variable that is either 0 or 1, and,
Figure FDA0002474593510000034
indicating that the user i and the equipment j are called at the moment t, and otherwise, indicating that the calling is 0;
Figure FDA0002474593510000035
a variable that is either 0 or 1, and,
Figure FDA0002474593510000036
indicating that the user i and the device j are stopped to call at the time t, and otherwise, the value is 0.
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