Distributed Optimal Day-Ahead Scheduling in a Smart Grid : A Trade-Off among Consumers , Power Suppliers , and Transmission Owners

To cope with the challenges due to increasing peak load, an optimal day-ahead scheduling problem for social welfare maximization is proposed, in which not only the comfort level of consumers and costs of power suppliers but also the power losses in transmission and operation costs of transmission owners are taken into account.Then this optimal day-ahead scheduling problem is reformulated and solved via the alternating direction method of multipliers (ADMM), by which fast convergence is guaranteed and the privacy of participants is ensured, in a distributed manner. Specifically, in the proposed distributed optimal day-ahead scheduling, the hourly prices for consumers are divided into hourly supply prices and hourly delivery prices, which will be updated by the independent system operator based on the hourly demand-supply situations and hourly demand-delivery situations, respectively. And the consumers, power suppliers, and transmission owners make their individual optimal day-ahead scheduling based on their individual hourly prices, hourly supply prices, and hourly delivery prices, respectively, until the hourly demand-supply balances and hourly demand-delivery balances are achieved. Effectiveness of the proposed distributed optimal day-ahead scheduling is verified by the cases studied.


Introduction
Recently, increasing peak load due to economic development and replacing fossil fuels with electricity power has resulted in serious challenges to the power grid, such as increasing the operation costs and decreasing the reliability of power grid [1,2].To cope with these challenges, day-ahead scheduling, which is widely recognized as an indirectly and effective approach for costs reducing and peak load shaving, has been intensively discussed [3][4][5][6].
So far, there is plenty of literature about the day-ahead scheduling, minimizing costs of power suppliers [7,8], minimizing energy costs of consumers [9], minimizing costs of utility company and payments of consumers [10], or minimizing the day-ahead operation costs of integrated urban energy system [11].From the social perspective, it is desired to maximize the sum of comfort level of consumers and meanwhile minimize the costs of utility companies.This is also called the social welfare maximization and has attracted much attention [12][13][14][15][16]. Besides, distributed coordination approaches [17,18], which are widely applied in many other fields, have been introduced into the day-ahead scheduling in a smart grid recently.
As well known, power system consists of not only power generation and consumption but also power transmission, and the costs of transmission owners, which includes power losses in transmission and operation costs, play important part in the social welfare.Besides, since improper dayahead scheduling might challenge the operation of power transmission and cause congestions, the capacity limits of tie lines should be considered in the day-ahead scheduling, which makes the day-ahead scheduling more complex but more reasonable.
While most of the mentioned literature [10,12,16] assumes that all the consumers and generators are connected to the same bus, few literatures [13] have considered the 2 Complexity capacity limits of tie lines, but the costs of transmission owners are ignored.
Besides, although many distributed theories [19,20] and decomposition based approaches have been proposed to solve the day-ahead scheduling in distributed manners, the convergence rates of some approaches, such as subgradient projection method, are not fast enough, and they are highly dependent on the choice of step size [16].Moreover, for most decomposition based approaches, in certain cases, their convergence criteria may not hold and modified decomposition method is required [21].
Nowadays, low cost communications technologies enable more cost-reflective price for electricity services, which can finally animate the day-ahead scheduling and facilitate the optimization of social welfare.Then, in this paper, a distributed optimal day-ahead scheduling for social welfare maximization is proposed, in which not only the comfort level of consumers and costs of power suppliers but also the costs of transmission owners are considered.
In the proposed distributed optimal day-ahead scheduling, the participants are consumers, power suppliers, and transmission owners.The day-ahead hourly prices for consumers are divided into hourly supply prices and hourly delivery prices, and each consumer has signed contracts with a power supplier and several transmission owners who govern the tie lines connecting himself to the power supplier chosen by him.Since each power supplier has his individual hourly supply prices which indicate his individual hourly demand-supply situations and each transmission owner has his individual hourly delivery prices which indicate his individual hourly demand-delivery situations, then each consumer has his individual hourly prices.
In the beginning, consumers, power suppliers, and transmission owners submit their individual initial day-ahead schedules to the independent system operator (ISO).Most of the time, there are hourly demand-supply differences for each power supplier and hourly demand-delivery differences for each transmission owner.Then based on experience and their individual hourly differences, ISO proposes hourly supply prices for each power supplier and hourly delivery prices for each transmission owner and broadcasts the hourly demand-supply differences, hourly demand-delivery differences, hourly prices, hourly supply prices, and hourly delivery prices to the corresponding participants.
First, based on the related hourly demand-supply differences, hourly demand-delivery differences, and his individual hourly prices, each consumer makes his optimal day-ahead hourly demands and submits them to the ISO.With the updated day-ahead hourly demands of consumers, the hourly demand-supply differences and hourly demand-delivery differences are half updated by the ISO and broadcast to the corresponding power suppliers and transmission owners, respectively.
Next, each power supplier makes optimal day-ahead hourly supplies based on his individual hourly supply prices and half updated hourly demand-supply differences and submits them to the ISO as well.Similarly, each transmission owner makes optimal day-ahead hourly deliveries based on his individual hourly delivery prices and half updated hourly demand-delivery differences and submits them to the ISO.Then with the updated day-ahead schedules of all the participants, all of the hourly demand-supply differences and hourly demand-delivery differences are fully updated, and all of the hourly supply prices and hourly delivery prices are updated by the ISO.
The newest hourly demand-supply differences, hourly demand-delivery differences, hourly prices, hourly supply prices, and hourly delivery prices will be broadcast to the corresponding participants, and this process will be ended until all the hourly demand-supply balances and hourly demand-delivery balances are achieved.
The contributions of this paper are summarized in the following.First, quite different from the existing literature [10,12,13,16], in which only the comfort level of consumers and fuel costs of power suppliers are considered, in this paper, the power losses in transmission and operation costs of transmission owners are considered as well.Accordingly, for each transmission owner, the hourly delivery prices, which indicate his individual hourly demand-delivery situations, are introduced.
Second, the costs of power suppliers associated with hourly supplies variability, which play an important role in hourly supply prices [22], have been taken into account.That makes the proposed social welfare more comprehensive.
Third, with the aid of the ISO, the proposed distributed optimal day-ahead scheduling can be properly matched with the framework of alternating direction method of multipliers (ADMM), by which a global and fast convergence is guaranteed and the privacy of participants is ensured.
This paper is organized as follows.The proposed distributed optimal day-ahead scheduling problem is formulated in Section 2. The design of the proposed day-ahead scheduling and the corresponding algorithms are presented in Section 3. The cases studied are illustrated in Section 4, and the last section is the conclusion.

Problem Formulation
In this section, topology of the smart grid is illustrated in Figure 1, and a highly coupled day-ahead scheduling problem for social welfare maximization is proposed, in which not only the comfort level of consumers and costs of power suppliers but also the power losses in transmission and operation costs of transmission owners are considered.

Cost Function of Transmission Owner.
In this paper, assume that the voltage in the power grid is constant; denote the set of tie lines by ; as well known, for each tie line , power losses in transmission can be written as where   is the active power,   is the reactive power,   is the nominal voltage,   is the equivalent resistance of tie line .Denoting the line power factor of tie line  by cos , we have Then from (1), define   = 2  /(  cos ) 2 and denote the set of time slots by , we formulate the cost function of tie line  in time slot  as where  , is the active power delivered by tie line  in time slot ,   and   are the parameters related to operation costs of transmission owner .
Define L  = ( ,1 , . . .,  , )  ;  is the number of time slots; then the cost function of tie line  can be formulated as (4)

Cost Function of Power Supplier.
For the fuel costs of power suppliers, we choose the most commonly used quadratic function.Denote the set of suppliers by , G  = ( ,1 , . . .,  , )  ; then, for supplier , his fuel costs in the next day  1 can be formulated as where  , is the active power generated by power supplier  in time slot ;   ,   , and   are positive parameters.
Besides, for supplier , the costs associated with hourly supplies variability are formulated as the following quadratic function: Hence the costs function of power suppliers  can be formulated as 2.3.Utility Function.Denote the set of consumers by , for each consumer  in each time slot , the utility function   , which values his comfort level, can be formulated as follows [12]: where  , is the aggregate demand of consumer  in time slot ,  , is the parameter related to comfort level of consumer  in time slot , a higher  , implies a higher utility value, and  is a predefined parameter related to the electricity costs of consumer.
In essence, with the assumption that power suppliers have quadratic costs functions, for consumer  and the power supplier  chosen by consumer ,  is dependent on the active power  , generated by power supplier  in time slot .Generally, consumer  has a small  for off-peak time slots and a large  for on-peak time slots,  ∈ ,  ∈ ,  ∈ .
In this paper, we can assign different  to different time slots for each consumer  based on general daily demand curve of consumers, such as 0.02, 0.3, and 0.5 for off-peak, mid-peak, and on-peak time slots, respectively [12].But for simplicity, we assign a middle  for all the consumers in all the time slots, and this has no substantial impact on the proposed distributed day-ahead scheduling.

Proposed Optimal Day-Ahead Scheduling Problem.
Define x = (X 1  , . . ., X   )  ,  is the number of consumers, , and V is the number of tie lines.Then the objective function for social welfare maximization is formulated as For day-ahead scheduling in this paper, we assign load curtailment to real-time market [23] and concentrate on load shifting.That means the total demand for each consumer  in the next day is a constant in this scheduling and we have where   is the total demand for consumer  in the next day.And for each  , ,  ∈ ,  ∈ , we have where  , and  , are the lower and upper bounds for demand of consumer  in time slot , respectively.For each power supplier  in time slot , its output  , should equal the sum of demands of consumers who purchase power from him.In this paper, we assume that the consumers in the same aggregation choose the same power supplier and denote the set of consumers who purchase power from power supplier  by   ; then we have and we have where  , and  , are the lower and upper bounds for  , , respectively.Besides, for each transmission owner , his active power delivery should equal the sum of demands of consumers who have signed a transmission contract with him.Similarly, we assume that the consumers in the same aggregation choose the same transmission owners and denote the set of consumers who have signed a transmission contract with transmission owner  by   ; then we have and we have where  , and  , are the lower and upper bounds for  , , respectively.

Distributed Optimal Day-Ahead Scheduling
To achieve a distributed and fast optimal day-ahead scheduling in this paper, the proposed optimal day-ahead scheduling problem is reformulated at first; then it is solved via the ADMM in a distributed manner.

An Overview on ADMM.
As well known, a standard form of ADMM, the details of which can be found in [24], solves the following problem: Assume z ∈ R  , assign a Lagrange multiplier vector  ∈ R  to the equality constraint Ax = z, and we get the following augmented Lagrange function: where  is an arbitrary positive constant.Then the search for a constraint saddle point of the augmented Lagrange function is performed with an alternating procedure that starts from arbitrary initials z(0) and p(0) and iteratively updates entries as follows: x ( + 1) = arg min When  1 and  2 are convex functions, and C 1 is a compact set or else the matrix A  A is invertible, ADMM converges to a unique stable point, which is assured to be a constraint saddle point for , hence the optimal solution.

Design of Distributed Optimal Day-Ahead Scheduling.
First, according to problem (18) solved by ADMM, we reformulate the proposed optimal day-ahead scheduling problem (17) as follows.
Function  1 (x) of the proposed day-ahead scheduling is formulated as which is a strictly convex function.
For each consumer , his local constraints are collected in the following compact set,  ∈ , and then the set C 1 can be denoted by the Cartesian product which is a compact set.Let (Z  1 , Z  2 )  = z; then function  2 (z) for the proposed day-ahead scheduling is formulated as which is a strictly convex function as well.
Assume that consumer  0 purchases power from power supplier  0 , and denote the set of transmission owners, with whom consumer  0 has signed transmission contracts, as   0 .Then based on (17) and (20), the objective function of each consumer  0 can be formulated as min where   0 (X  0 ) is the comfort level of consumer  0 , Lagrange multipliers   0 , () and   0 , () represent the th supply price of power supplier  0 and the th delivery prices of transmission owner  0 in time slot , respectively, in the th round of day-ahead scheduling, and the terms with  represent the penalties set by the ISO for achieving demandsupply and demand-delivery balances.
For the consumer  0 , assume his total demand of the next day is   0 , |  0 | is the cardinality of the set   0 , X  0 () = (  0 ,1 (), . . .,   0 , (), . . .,   0 , ())  .As hourly supply prices   0 , hourly demand-supply differences   0 of  0 , hourly delivery prices   0 , and hourly demand-delivery differences   0 of  0 ,  0 ∈   0 , have been broadcast to  0 by the ISO, define where then according to subproblem (29), each consumer  0 updates its X  0 () in parallel as Algorithm 1 shows.The essence of Algorithm 1 is the equal incremental cost criterion; in other words, for consumer  0 , one more unit of demand in any time slot makes the same increase of his comfort level.
Based on ( 17) and ( 20), the objective function of each transmission owner  0 can be formulated as min where   0 (L  0 ) represents the power losses in transmission and operation costs; the terms with  represent the penalties set by the ISO for achieving demand-delivery balance.
When each power supplier  0 and transmission owner  0 have submitted their updated individual hourly supplies and hourly deliveries to the ISO, the ( + 1)th hourly demandsupply differences and hourly demand-delivery differences can be formulated, respectively, as and the hourly supply prices  , and hourly delivery prices  , will be updated as (40) shows,  ∈ ,  ∈ ,  ∈ . , ( + 1) =  , () +  , ( + 1) ,  , ( + 1) =  , () +  , ( + 1) . ( That is, ISO adjusts the hourly supply prices based on the hourly demand-supply differences, if demands exceed supplies, increases the supply charges, and otherwise decreases the supply charges.And the hourly delivery prices will be adjusted in the same way based on the hourly demanddelivery differences.And this scheduling process between consumers, power suppliers, and transmission owners will continue until both demand-supply and demand-delivery are balanced.
The proposed distributed optimal day-ahead scheduling can be summarized as follows.
Step 1. Consumers, suppliers, and transmission owners submit their individual initial day-ahead schedules to the ISO, and then ISO calculates the hourly differences according to (28) and set the initial hourly prices based on experience.
Step 2. ISO broadcasts the hourly prices and hourly differences to the corresponding participants.
Step 4. ISO broadcasts δ(+1) calculated according to (32) to the corresponding power suppliers and transmission owners.
Step 5. Suppliers and transmission owners update their individual day-ahead schedules in parallel according to Algorithm 2 and (38), respectively, and submit them to the ISO.
Step 6. ISO updates the hourly differences and hourly prices according to (39) and (40), respectively.Step 7. If demand-supply and demand-delivery are balanced, this process is ended; otherwise turn to Step 2.

Case Study
In this section, we assess the convergence performance of the proposed optimal day-ahead scheduling, and some key parameters are discussed.In the cases studied, we consider a smart power grid illustrated in Figure 2, which consists of 100 consumers partitioned into 5 aggregations, 2 power suppliers, and 7 transmission owners.The contractual relationships for consumers aggregations are summarized in Table 1.
For the utility functions of consumers, the parameter , which has been discussed in detail in [12], is set to be 0.3, and  is chosen as 0.004, the number of time slots is 24, parameters  , are selected from 1.5 to 3. Besides, the parameters of cost functions of power suppliers and transmission owners are presented in Table 2.

Convergence Performance and Social
Welfare.The convergence performance of the proposed optimal day-ahead scheduling is presented in Figures 3 and 4. Figure 3 is the trajectories of infinity norm of hourly differences, where  1 is the infinity norm of hourly differences of G1, and Figure 4 is the hourly supply prices trajectories of G1.
It can be seen that the infinity norm of hourly differences of all the participants and hourly supply prices of G1 achieves convergence in 10 iterations.Specifically, the proposed optimal day-ahead scheduling is carried out on a computer based on Intel(R) Core(TM) i3-4170 CPU @3.70 GHz, RAM: 8.00 GB, 64-bit Operating System, ×64-based processor; the total time is 39.76 seconds.That means the proposed algorithm takes approximate 3.976 seconds per iteration.In fact, the hourly delivery prices of transmission owners, which is omitted due to limited textual paragraphs, do the same convergence performance.Then it can be concluded that the proposed distributed optimal day-ahead scheduling solved via ADMM achieves convergence fast.
Figure 5 is the social welfare trajectories of the proposed distributed day-ahead scheduling.Note that the comfort level of consumers is measured in money in this paper.In essence   the proposed social welfare is a trade-off between the comfort level of consumers and the costs of power suppliers and transmission owners.It can be seen that the social welfare is truly increased via the proposed distributed day-ahead scheduling.6 is partial hourly supply prices trajectories of G1; Figure 7 is partial hourly delivery prices trajectories of L2; Figure 8 is the demand shifting of day-ahead scheduling of consumers aggregation U2, where "initial" and "optimal" correspond to the initial hourly demands and optimal hourly demands with  = 0.002 of U2, respectively.

Demand Shifting. Figure
From Figure 6 and Figure 8, it can be seen that the hourly supply prices vary with the demands of consumers.With the same initial value 0.2, the hourly supply prices increase from their initial value when the demands of consumers are high and decrease from their initial value when the demands of consumers are low.From Figure 7 and Figure 8, we have the same conclusion for hourly delivery prices of transmission owners.Then accordingly, consumers will shift their demands from the hours with high electricity prices (high supply prices or high delivery prices or both) to the hours with low electricity prices, which is illustrated in Figure 8.

4.3.
The Effect of  on Day-Ahead Scheduling.Note that  represents the costs of power suppliers associated with hourly supplies variability.The effect of  on the proposed optimal day-ahead scheduling is illustrated in Figure 9 by the optimal day-ahead hourly demands of U2, and the optimal day-ahead scheduling of the other consumers aggregations has similar trends.
From Figure 9, it can be seen that the peak demands of U2 decrease as  increases from 0.000 to 0.006, that means  plays an important role, not only in electricity price but also in reducing the peak demand of consumers.This can be explained by the response of consumers to the supply prices.As  increases, the costs of power suppliers increase, especially the hours with dramatic supply variability which are the hours with peak demand as well, then the supply prices of these hours will be high; correspondingly consumers will shift demands from the hours with high supply prices to the others to minimize their individual costs.

Conclusion
In this paper, a distributed optimal day-ahead scheduling for social welfare maximization is proposed, which will be suitable for a future smart grid.In this day-ahead scheduling, the day-ahead market consists of a day-ahead supply market and a day-ahead delivery market.The aggregations of consumers or large consumers purchase generation capacities from the power suppliers in the day-ahead supply market and transmission capacities from the transmission owners in the day-ahead delivery market, respectively.Besides, the hourly supply prices of each power supplier are regulated by the ISO based on his individual hourly demand-supply situations, and the hourly delivery prices of each transmission owner are regulated by the ISO based on his individual hourly demand-delivery situations.This interaction among consumers, power suppliers, and transmission owners will be continued until all the hourly demand-supply and hourly demand-delivery are balanced.Besides, the proposed optimal day-ahead scheduling can be properly matched with a standard ADMM framework, by which a global and fast convergence is guaranteed and the privacy of participants is ensured.The effectiveness of the proposed day-ahead scheduling is verified by the cases studied.

Figure 1 :
Figure 1: Topology of the smart grid.Note that   is the th consumers aggregations.

Figure 4 :
Figure 4: The hourly supply prices trajectories of G1.

Figure 9 :
Figure9: The effect of  on optimal hourly demands of U2.

Table 1 :
Contractual relationship between the consumers aggregations and power suppliers, transmission, owners.