CTS2M: concurrent task scheduling and storage ...

IET Cyber-Physical Systems: Theory & Applications Research Article

CTS2M: concurrent task scheduling and storage management for residential energy consumers under dynamic energy pricing

ISSN 2398-3396 Received on 20th March 2017 Revised 25th July 2017 Accepted on 17th September 2017 doi: 10.1049/iet-cps.2017.0050 www.ietdl.org

Ji Li1 , Xue Lin2, Shahin Nazarian1, Massoud Pedram1 1Department

of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115, USA E-mail: [email protected]

2Department

Abstract: Dynamic energy pricing policy introduces real-time power-consumption-reflective pricing in the smart grid in order to incentivise energy consumers to schedule electricity-consuming applications (tasks) more prudently to minimise electric bills. This has become a particularly interesting problem with the availability of photovoltaic (PV) power generation facilities and controllable energy storage systems. This study addresses the problem of concurrent task scheduling and storage management for residential energy consumers with PV and storage systems, in order to minimise the electric bill. A general type of dynamic pricing scenario is assumed where the energy price is both time-of-use and power dependent. Tasks are allowed to support suspend-now and resume-later operations. A negotiation-based iterative approach has been proposed. In each iteration, all tasks are ripped-up and rescheduled under a fixed storage charging/discharging scheme, and then the storage control scheme is derived based on the latest task scheduling. The concept of congestion is introduced to gradually adjust the schedule of each task, whereas dynamic programming is used to find the optimal schedule. A near-optimal storage control algorithm is effectively implemented. Experimental results demonstrate that the proposed algorithm can achieve up to 60.95% in the total energy cost reduction compared with various baseline methods.

1 Introduction Today's power systems are faced with numerous challenges, including climate change, environmental sustainability, power supply quality and reliability, and increasing uncertainties of power production and load consumption [1, 2]. To reduce the power system's overall carbon footprint and create a sustainable energy future, a transition to renewable energy generations has taken place with ambitious targets across the world [3–5]. Among various renewable energy sources, the solar photovoltaic (PV), which converts solar radiation into direct current electricity [6], has been growing considerably at an average annual rate of 55% over the past 5 years. Unlike the constant and reliable power sources such as fossil fuels, the PV power generation varies according to the position of the sun in the sky and local weather conditions, which can dramatically change the solar irradiance intensity. Furthermore, the PV power generation is intermittent, i.e. a PV system can generate electricity only during daytime when the solar irradiance is above a certain threshold [7, 8]. An effective solution to smooth out the variability and address the intermittent nature of PV power generation is to incorporate an energy storage system into the offgrid PV power system [8–11]. The proposed energy storage system will ideally store electric energy whenever there is excessive PV generation for the current load demand, and subsequently release the stored energy to the load devices during peak power consumption hours to reduce the level of power that is drawn from the grid. In addition to efforts to incorporate renewables and energy storage systems into the existing power grid, the concept of smart grid has been introduced to refer to the next-generation power grid, which optimises the power generation, transmission and delivery means and enables advanced energy saving techniques and demand response applications [12]. The smart grid may be best understood as the overlaying of a unified control and data communication system on the existing power delivery infrastructure that is supported by the increasing deployment of the smart-meter devices in the automatic metering infrastructure [12–14]. Accordingly, the smart grid enables control protocols to be embedded into the grid so as to allow different entities (e.g. end-use devices, transmission

and distribution system controllers, customers etc.) to take actions based on the information gathered by the smart meters and delivered through the communication channels [12, 15]. As discussed before, a major challenge in the power grid is the timing skew between the peak power generation and load demand. In order to avoid power failures such as black-outs or brown-outs, the amount of generation, transmission and distribution capacities that needs to be provisioned by the utility company is supposed to match the peak demand rather than the average. One effective solution is to apply the dynamic pricing policy that introduces realtime power-consumption-reflective pricing based on the entire supply chain of delivering electricity at a certain location, quantity and period [14, 16, 17]. With the dynamic pricing, customers are incentivised to curtail or shift their power usage to a period when energy prices and supply are most favourable. Consequently, the peak load of the transmission system is reduced while the efficiency and reliability of the grid are enhanced [18]. Considerable research efforts have been devoted to the problem of minimising costumers’ electric bills under dynamic pricing models. A scheduling algorithm is developed in [19] by mapping the problem into the multiple knapsack problem, well known in the computer sciences field. A force-directed algorithm is proposed in [15], which allows consumers to schedule their electricityconsuming applications (tasks) outside preferable time periods but with incurred inconvenience cost. The aforementioned works [15, 19] can solve the task scheduling problem under dynamic pricing policy; however, they have not modelled the off-grid PV power system and energy storage system. Authors in [9] consider PV system and energy storage system but do not provide the control scheme over residential task scheduling. In this paper, we present a concurrent task scheduling and storage management (CTS2M) algorithm to minimise energy users’ electric bills under a dynamic pricing model. We assume that the user is equipped with an off-grid PV system and an energy storage system, and the control scheme in this paper considers the concurrent scheduling of tasks and the charge management of the energy storage system. A realistic energy pricing function is adopted, which is comprised of a time-of-use (TOU) price that depends on the time of the day, and a power-dependent price,

IET Cyber-Phys. Syst., Theory Appl. This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)

1

Fig. 1 Sample schedule of a residential energy user in a day

Fig. 2 Block diagram of the residential energy user of interest

which reflects a penalty when the user's real-time power consumption is high. We also adopt an accurate energy storage model developed in [9] that considers the power loss of the storage. The proposed algorithm is inspired by a routing algorithm developed for field-programmable gate array in [20], which has also been applied to other scheduling problems [21, 22], and utilises the convex optimisation technique of [23]. An iterative approach is adopted such that in each iteration, the scheduler (i) decides the best task scheduling for the unscheduled tasks under the latest energy storage control decision, and based on the cumulative scheduling results from previous iterations, and then (ii) re-derives the energy storage control decision by solving a convex optimisation problem under the updated task scheduling result. The concept of congestion is introduced to guide the scheduler to avoid time periods when power consumption is high and utilise time periods when PV generation is sufficient. The priority of each task is adjusted at the beginning of each iteration. Note that compared with the preliminary versions of this work in [2, 24] which only dealt with non-interruptible tasks, we allow interruptible tasks (i.e. those tasks that allow suspend-now and resume-later operations), and use dynamic programming to specify the optimal start time of non-interruptible tasks, do optimal decomposition of interruptible tasks into subtasks, and specify the optimal start time of each fragment of interruptible tasks. Experimental results demonstrate that the proposed algorithm can achieve significant energy cost reduction up to 60.95% compared with baselines.

2 System model and cost functions 2.1 Time model and task model We adopt a slotted time model [24], where the physical time period of interest is divided into T discrete time slots of equal duration, denoted by D, therefore, all time-related parameters, constraints and decisions are indexed by specific time slots. With no loss of generality, we consider the task scheduling problem in 1 day where the length of each time slot is 1 h, i.e. T = 24 and D = 60 min. Note that the proposed algorithm is applicable to a different time scale such as a week or a month. The user performs a number of electricity-consuming tasks on a daily basis which can either be non-interruptible (without suspension) or interruptible (may encounter multiple suspensions during their execution). Each task is uniquely identified by an index i, and the set of all task indices is denoted by F = {1, …, N}, where N is the total number of tasks for the residential energy user of interest. At the beginning of the day, each task i is associated 2

with an earliest start time si, a deadline ei, a duration di to complete the operation and an interruption flag f i which indicates whether task i is interruptible or not. All the aforementioned timing parameters and flag status are specified ahead of the day by the user and will remain unchanged during the day. For task i, the time window [si, ei] is referred to as the preferred operation time window, and execution of task i outside this time window will be penalised with an inconvenience cost, which quantifies the degree of inconvenience to the user if the task is executed outside its preferred window. The total number of operation fragments of the interruptible task i is denoted by Fi. The scheduled start time and finish time for the kth fragment of interruptible task i are denoted by αi, k and βi, k, respectively, for 1 ≤ k ≤ Fi. The total operation length of all fragments of each interruptible task i should be equal to its duration F specified by the user, i.e. di = Σk i= 1(βi, k − αi, k + 1). The notations for total number of fragments and scheduled start time as well as finish time are also applicable to non-interruptible task i which has only one fragment (i.e. Fi = 1) operating from αi, 1 to βi, 1 and satisfies the condition di = βi, 1 − αi, 1 + 1. The scheduling decision for task i is the set of start and finish times for all fragments of this task (we use ζi to denote the scheduling decision for task i, i.e. ζi = {αi, 1, …, αi, Fi, βi, 1, …, βi, Fi}). With the unique notation of scheduling decision for a single interruptible or non-interruptible task, the final task scheduling results generated by the proposed algorithm is the set of scheduling decisions for all tasks, which is denoted by Ztask, i.e. Ztask = {ζ1, …, ζN }. Fig. 1 illustrates a sample household scheduling in a day, where each bar indicates the scheduled operation time for each task. We denote the power consumption function of task i in time slot t by pi(t). In reality, the execution of an electric-consuming task has several phases with different power profiles, e.g. a task may start with an initialisation phase, then continue with normal operation, sleep mode, high-performance operation, and end with a shuttingdown phase. In this paper, we assume the operating phases and the corresponding power consumption profiles for each noninterruptible task are pre-determined and independent of the scheduled start time. Besides, the power consumption profiles for interruptive tasks are considered to be pre-known and independent of suspensions, since the time for modern residential electric applications to power on or power off is negligible compared to the discrete time interval D used in this paper. Therefore, the power consumption of (non-interruptible or interruptible) task i will follow a known profile (from αi, k to βi, k) independent of its scheduled start time. 2.2 System diagram of the residential energy user Fig. 2 illustrates the block diagram of the residential energy system, which comprises an off-grid PV module, an energy storage system, power buses, power conversion circuitry and the residential AC load, which is in turn due to the electricityconsuming tasks (as discussed in Section 2.1). The PV module and storage system are connected to the DC bus through unidirectional and bi-directional DC–DC converters, respectively. The AC load is connected to the AC bus, and bidirectional AC/DC interfaces are utilised to connect the DC bus and AC bus. The AC bus is further connected to the state-level or nation-level grid, which is outside the scope of the residential energy user's system. In this paper, we consider representative power conversion circuits as found in the market today (i.e. the power conversion efficiency values are 1 V st ⋅ Ist, ref if − 1 ≤

|Pst, in(t)| V st ⋅ Ist, ref

δ2

Pst, in(t) ≤1 V st ⋅ Ist, ref

Pst, in(t) if < −1 V st ⋅ Ist, ref

(4)

where Ist, ref is the reference current of the storage system, which is proportional to the storage's nominal capacity; δ1 and δ2 are coefficients ranging from 0.8 to 0.9 and 1.1 to 1.3, respectively. We use the function Pst(t) = f st(Pst, in(t)) to denote the relationship between Pst(t) and Pst, in(t), which is a concave and monotonically increasing function over the input domain −∞ < Pst(t) < + ∞. Due to the monotonicity property, the inverse function

Pst, in(t) = f st−1(Pst(t)) is also a concave and monotonically increasing function over the input domain −∞ < Pst(t) < + ∞. 2.4 Price function and problem definition A realistic dynamic price function ξ(t, Pgrid(t)) is adopted in this paper, which consists of a TOU price that depends on the time slot t of the day and a power-dependent price that depends on the grid power consumption Pgrid(t). Instead of a single flat rate for electricity price, the TOU price is set to be higher in the peak-hour time slots than the off-peak time slots, which incentivises the user to shift peak-hour loads to offpeak periods so as to minimise the electric bill. Nevertheless, if the majority of energy users shift their loads towards the off-peak hours, the original off-peak hours becomes peak-hours and plants are still supposed to match the huge loads in the worst case scenario. In order to prevent users from overly shaving the loads to off-peak hours, a power-dependent pricing component is introduced, which monotonically increases with respect to the grid power Pgrid(t). By applying the above dynamic pricing policy, the chance to see uncontrolled service disruptions such as power outage is much lower. The inconvenience cost is introduced to penalise scheduling tasks outside their preferred operation time windows. The inconvenience cost for task i given a schedule decision ζi is denoted by Ii(ζi) and is proportional to the number of time slots assigned to task i that fall outside its preferred operation time window [si, ei]. The problem formulation is stated as follows. 2.4.1 Residential energy cost minimisation problem: Find the optimal start time αi, 1 for any non-interruptible task i, the optimal start time αi, k and finish time βi, k for the kth fragment of any interruptible task i for 1 ≤ i ≤ N, and the optimal energy storage output power level Pst(t) for 1 ≤ t ≤ T so as to minimise the total energy price: Total cost =

T

N

t=1

i=1

∑ ξ(t, Pgrid(t)) ⋅ Pgrid(t) + ∑ Ii(ζi)

(5)

max −Pstmax , c ≤ Pst(t) ≤ Pst, d

(6)

subject to:

0 ≤ Estini − Σtt′ = 1Pst, in(t′) ⋅ D ≤ Estmax,

∀t ∈ [1, T]

(7)

The first term in (5) is the aggregated energy cost in a day for the user, whereas the second term in (5) corresponds to the max inconvenience cost. In (6) and (7), Pstmax , c and Pst, d are the maximum allowable amount of power flowing into and out of the storage system during charging and discharging processes, respectively; Estini is the initial amount of energy storage at the beginning of the day and Estmax is the total energy storage capacity. The output of the proposed algorithm is the scheduling decisions Ztask as discussed in Section 2.1 and the storage control policy denoted by Zst = {Pst(t), ∀t ∈ [1, T]}.

3 Concurrent task scheduling management algorithm

and

storage

The cost minimisation problem described in Section 2.4 is an integer non-linear programming problem subject to non-linear constraints. No provably optimal solutions can be derived in polynomial time, nor are they likely to ever be available for this kind of problems [26]. In a cost minimisation problem with a large set of unknowns, it is impractical to find the optimal decisions because of the impractically high computational complexity [15]. Therefore, in this paper, we seek to find a high-quality solution instead of the optimal one.


3

As mentioned in Section 3.1, at the beginning of the jth iteration, all tasks are ripped-up, i.e. the previous scheduling results of each task is cleared, and then the proposed NBDP algorithm reschedules each task based on a priority order that will be explained at the end of this subsection. We first assume the NBDP i, j schedules from the first task to the Nth task. Let Pgrid(t) denote the total grid power consumption in time slot t after the ith task has been scheduled in the jth iteration, where t ∈ [1, T], i ∈ [1, N] and j ∈ [1, K]. When the ith task is being scheduled, we define the cost increase as the net increase in the total energy cost after the ith task has been scheduled. The best scheduling decision for task i in this iteration is the ζi that minimises the cost increase. When scheduling the ith task in the jth iteration, the NBDP first checks whether task i is interruptible. Fig. 3 Algorithm 1: Pseudo code for the CTS2M algorithm

3.1 Concurrent application scheduling and storage control algorithm The CTS2M algorithm is proposed to effectively find a suitable solution for the cost minimisation problem described in Section 2.4 using an iterative manner. In the first part of each iteration, all tasks are ripped-up, i.e. scheduling decision for each task in the previous iteration is cleared, and each task is rescheduled based on the latest storage control policy. Next in the second part of the iteration, the storage control policy is reset and the storage charging/discharging decisions for all time slots are re-decided based on the latest task scheduling results. The negotiation-based dynamic programming (NBDP) algorithm and the negotiation-based storage control (NBSC) algorithm have been devised to find the near-optimal task scheduling in the first part of each iteration and the near-optimal energy storage control scheme in the second part of each iteration, respectively. We set K as the maximum number of iterations to prevent the CTS2M algorithm from infinite loops. The proposed algorithm terminates with a suitable scheduling solution and energy control scheme when the total energy cost calculated by (5) has not decreased for L consecutive iterations, or when the number of iterations exceeds K, whichever condition is met first. The basic structure of the CTS2M algorithm is shown in Algorithm 1 (see Fig. 3). Details about the NBDP and the NBSC are provided in the subsequent subsections. 3.2 Negotiation-based dynamic programming algorithm for task scheduling The optimisation problem in NBDP, namely the optimal task scheduling under fixed storage control, is modelled as follows. 3.2.1 Task scheduling problem for a residential energy user under fixed energy storage control scheme: Given the optimal energy storage output power level Pst(t) for 1 ≤ t ≤ T, and f i for 1 ≤ i ≤ N. Find the optimal number of fragments Fi and the scheduling decisions ζi for all task i where 1 ≤ i ≤ N to minimise total cost =

T

N


t=1

(8)

i=1

CI =

F

di = Σk i= 1(βi, k − αi, k + 1),

∀i ∈ [1, N]

(9)

where (8) is the same as (5) except that the term Pst(t) is known when we calculate the term Pgrid(t), and constraint (9) means the total number of time slots scheduled for the execution of each task should be the same as its duration even when the task is divided into several small fragments (as discussed in Section 2.1).

T

∑

t=1

i, j

i, j

ξ(t, Pgrid(t) ⋅ Pgrid(t)

i − 1, j −ξ(t, Pgrid (t))

⋅

i − 1, j Pgrid (t))

(10) + Ii(ζi)

An important observation is that the time slots that have already been occupied by the previous i − 1 tasks are more likely to incur a higher energy cost because the dynamic energy pricing function is increasing monotonically with respect to the power extracted from the grid in each time slot. If a time slot has already been occupied by several tasks in one iteration, this time slot is considered to be congested, and in order to quantify the degree of congestion in the time slot t, we introduce an intra-iteration congestion term R(t), which is defined as the number of tasks that have already been scheduled to occupy the time slot t within the current iteration. Thus, it is reasonable to avoid those congested time slots with large R(t) so as to minimise the overall energy cost. Another important observation is that the scheduler should try to utilise all the PV power generations. Therefore, we introduce an inter-iteration term H(t) that is defined as the total number of times in the previous j − 1 iterations when the PV power is not fully utilised in time slot t. Therefore, assigning tasks to those time slots with large H(t) should result in more efficient usage of PV power generations in those time slots. To avoid the potential problem that a large number of suitable scheduling solutions may be eliminated when one or more tasks always have high priority to be scheduled and thus they always occupy the same time slots in consecutive iterations, we introduce an inter-iteration congestion term h(i,t) which indicates the total number of times in previous iterations when task i occupies time slot t. The cost becomes larger when a task occupies the same set of time slots as the iteration count increases. Of note, the intraiteration congestion term R(t) is reset at the beginning of each iteration and is updated when a task has been scheduled within the iteration, whereas the inter-iteration terms H(t) and h(i,t) are reset at the start of the NBDP and are updated at the end of one iteration. With the above terms, (10) can be rewritten by CI′ = Ii(ζi) + ×

subject to:

4

3.2.2 NBDP algorithm for non-interruptible tasks: If the task i is non-interruptible (i.e. f i = 0), then set Fi = 1. The cost increase in this case can be calculated by

T

∑ (a ⋅ R(t) − b ⋅ H(t) + 1) ⋅ (c ⋅ h(i, t) + 1)

t=1

i, j ξ(t, Pgrid(t))

⋅

i, j Pgrid(t)

−

i − 1, j ξ(t, Pgrid (t))

⋅

(11)

i − 1, j Pgrid (t)

where a, b and c are positive constants that indicate the weights of R(t), H(t) and h(i,t), respectively (The proposed NBDP can also be applied to a residential energy user without PV power system by setting b=0.). The modified cost increase in (11) directs the scheduler to avoid congested time slots within the current iteration by the high cost increase due to the term a ⋅ R(t), and makes time slots whose PV power are not fully used in previous iterations more attractive by the reduced cost increase resulting from the term


n time slots where n ≤ m. If m is not larger than n but is larger than di, then the cost increase function will also be set to infinity since the task i only needs di time slots to complete its operation. The congestion calculation in (12) is similar to the one in (11). The pseudo code for the proposed NBDP is given in Algorithm 2 (see Fig. 4). The computational complexity for the proposed NBDP algorithm is O(N 2T 2). 3.3 NBSC algorithm Given the latest scheduling results from the NBDP algorithm, the NBSC algorithm is proposed to find the optimal charging/ discharging decisions for all time slots. The optimisation problem in NBSC can be modelled as follows. 3.3.1 Energy storage control problem for a residential energy user under fixed task schedule: Given the latest task scheduling result Ztask. Find the optimal energy storage output power level Pst(t) for 1 ≤ t ≤ T to minimise total cost =

T

N

t=1

i=1


(13)

subject to: max −Pstmax , c ≤ Pst(t) ≤ Pst, d

0 ≤ Estini − Σtt′ = 1Pst, in(t′) ⋅ D ≤ Estmax, Fig. 4 Algorithm 2: Pseudo code for the NBDP algorithm

b ⋅ H(t). Finally, the term c ⋅ h(i, t) adds to the cost increase when task i occupies the time slot t in multiple consecutive iterations. 3.2.3 NBDP algorithm for interruptible tasks: If the task i is interruptible (i.e. f i = 1), then the proposed NBDP algorithm needs to find the optimal Fi as well as the optimal starting time and finish time of each fragment. For an interruptible task i, we define the cost increase function Qi(m, n) to compute the minimum cost increase after optimally scheduling the first m slots of task i (among all the di slots of task i that need to be scheduled) in the first n time slots among all T slots. The cost increase function is written as follows: (see (12)) where all the terms R(t), H(t) and h(i,t) have the same meanings as discussed for (11) and Ii′ is the inconvenience cost that is determined by whether time slot n is inside or outside the preferred execution time window of task i. Equation (12) gives the mathematical definition of cost increase function Qi(m, n) in terms of smaller cost increase functions Qi(m − 1, n − 1) and Qi(m, n − 1). The first term in (12) means the nth time slot is chosen to execute the mth fragment of task i, whereas the second term indicates the nth time slot is not selected. The term with smaller cost increase will be the optimal scheduling choice. The recurrence process begins with Qi(1, 1) and ends when Qi(di, T) is calculated. Based on (12), each recurrence guarantees the optimality of the solution, hence, the final cost increase in Qi(di, T) is the optimal result. We also discuss base cases for the NBDP algorithm for interruptible tasks. If there are no time slots of task i to be scheduled (i.e. m = 0), then there will be no cost increase, i.e. Qi(0, n) = 0. If m is larger than n, then the cost increase function will be set to infinity since it is not possible to schedule m slots into

(14) ∀t ∈ [1, T]

(15)

where (13) is inherited from (5), and constraints (14) and (15) are inherited from constraints (6) and (7), respectively. When calculating (13), the term Pload(t) and inconvenience cost are known given the latest task scheduling result Ztask. To effectively solve the energy storage control problem via convex optimisation techniques [23], we transform the energy storage control problem into a convex optimisation problem based on the following result. Theorem 1: The energy storage control problem becomes a convex optimisation problem with convex objective function and inequality constraints if (i) Pst, in(t) for t ∈ [1, T] are utilised as optimisation variables instead of Pst(t) and (ii) the price function ξ(t, Pgrid(t)) is fixed (and independent of Pgrid(t)) for each time slot t. Based on the above observation, an effective heuristic method is presented, where the energy price ξ(t, Pgrid(t)) in each time slot t is assumed to be fixed during each iteration, and a convex optimisation problem is solved to derive the optimal Pst(t) for t ∈ [1, T]. The energy price ξ(t, Pgrid(t)) will be updated at the end of each iteration based on the Pgrid(t) derived from the optimal Pst, in for t ∈ [1, t]. The heuristic method will converge to an effective near-optimal solution of the original energy storage control problem. The details of the proposed heuristic are provided in Algorithm 3 (see Fig. 5).

4 Experimental results To demonstrate the effectiveness of the proposed approach, the CTS2M algorithm is tested on various cases and compared with different baselines corresponding to the abovementioned price and system models. As mentioned in Section 2.1, we examine the proposed algorithm in one day with 1 h time slot duration, and

Qi(m, n) = min (a ⋅ R(n) − b ⋅ H(n) + 1) ⋅ (c ⋅ h(i, n) + 1) i, j

i, j

i − 1, j

i − 1, j

× ξ t, Pgrid(n) ⋅ Pgrid(n) − ξ t, Pgrid (n) ⋅ Pgrid (n) +I′i + Qi(m − 1, n − 1), Qi(m, n − 1) ,

(12)

0 < m ≤ min(n, di)


5

Table 1 Performance of three baselines (NS, no storage; WS, weak storage) and concurrent task scheduling and storage management algorithm with different numbers of tasks and 16 kWh storage capacity Task Total energy cost, $ Cost reduction with respect to Number NS Greedy WS CTS2M NS, % Greedy, % WS, % 5 10 15 20 25 30 35 40 45 50

0.274 0.330 1.398 4.025 1.864 2.480 3.355 3.421 6.022 7.776

0.149 0.327 0.751 3.366 1.439 3.077 2.614 3.532 7.395 9.965

0.134 0.222 0.618 3.226 0.838 1.946 2.253 2.946 5.508 6.928

0.132 0.215 0.593 2.693 0.728 1.674 2.013 2.185 4.548 6.202

accordingly, the final decisions Ztask and Zst are represented with the granularity of 1 . A realistic residential PV generation system is considered, and we use PV power profiles measured at Duffield, VA, in the year 2007, an example of which is in Fig. 6a. A realistic residential energy storage system is considered, and typical values of conversion efficiency are applied to the user's power conversion circuitries. We assume that at the beginning of the day, the utility company provides TOU price and power-dependent price. The TOU-dependent price part is the base energy price when the grid power consumption is zero, as shown in Fig. 6b, and the power consumption-dependent price part is monotonically increasing with the user's real-time grid power consumption. The user provides task parameters including duration, inconvenience cost, power profile, preferable earliest start time and

Fig. 5 Algorithm 3: Pseudo code for the NBSC algorithm

Fig. 6 Experimental set-up (a) PV output power measured in a day at Duffield, VA, (b) Base (TOU) electricity price provided by the utility company in a day

51.76 34.69 57.57 33.10 60.95 32.53 39.99 36.13 24.48 20.24

11.35 34.27 20.95 20.00 49.39 45.62 22.98 38.14 38.50 37.76

1.15 3.24 3.93 16.54 13.15 14.00 10.64 25.83 17.43 10.48

deadline, and whether the task is interruptible. Our experiments are conducted using the proposed CTS2M algorithm and following baselines: • Task scheduling without storage (no-storage): the NBDP algorithm is applied for user without an energy storage system. This baseline is designed to demonstrate the importance of energy storage systems. • Greedy task scheduling (greedy): under the optimal energy storage control scheme, each task is scheduled to minimise its own energy cost. This baseline is used to demonstrate the effectiveness of the NBDP algorithm. • Task scheduling using an energy storage model without considering the rate capacity effect (weak-storage): where most parts are the same with the proposed algorithm expect that power loss caused by rate capacity effect is ignored. This baseline is designed to demonstrate the importance of an accurate energy storage model. In the first experiment, the energy storage capacity is fixed to 16 kWh, and we arbitrarily generate test cases for task number ranging from 5 to 50. Parameters of user's tasks are randomly generated as well, and each task has 50% possibility to be interruptible. Table 1 concludes the total energy cost among three baselines and the proposed algorithm, and the cost reduction with respect to different baselines is also reported. It can be seen from Table 1 that the proposed CTS2M algorithm consistently achieves lower total energy cost compared to different baselines, with the total energy cost reduction up to 60.95%. One can observe that the weak-storage baseline results in the highest performance among the three baselines because it adopts the NBDP and NBSC algorithms for task scheduling and storage control, respectively (which is the closest to the proposed algorithm). In the second experiment, the task number N is fixed to 50 and we compare the results of the three baselines and the proposed algorithm with the storage capacity ranging from 0 to 80 kWh, which covers most storage systems in the market [27], as shown in Fig. 7. It can be observed that the proposed CTS2M achieves the highest performance consistently under different storage capacity scenarios.

5 Conclusion This paper addresses the problem on integrating residential PV and storage systems into the smart grid for joint task scheduling and energy storage management in order to minimise user's electric bills under dynamic prices. The proposed CTS2M algorithm is featured by the concept of congestion in task scheduling and convex optimisation in storage control. The results are compared to three baselines and demonstrate the effectiveness of the proposed algorithm under different scenarios. Fig. 7 Results of the three baselines and the CTS2M using different storage capacities under a fixed task number 6

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