A Distributed Optimization Method for Optimal Energy Management in Smart Grid

2.1 Dynamic social welfare maximization problem

Consider a smart grid consisting of nconventional generation (CG) units, qrenewable generation (RG) units, and m demand units. Denote P1,Gt,…,Pn,Gtthe powers generated by CG units; P1,Rt,…,Pq,Rtthe powers generated by RG units; and P1,Dt,…,Pm,Dtthe powers consumed by demand units; and Pt≜P1,Gt…Pn,GtP1,Dt…Pm,DtT;P≜P1T…PNTT.

The generation cost for CG unit iis CiPi,Gt=aiPi,G2t+biPi,Gt+ci,with prescribed coefficients ai,bi,ci. The utility function for demand unit j[8, 9, 10], which shows the consumer’s satisfaction with respect to its consumed power, is

where αjand βjare predetermined parameters. Then the DSWM problem in smart transmission grids is as follows:

where PLtis the total power loss in the grid at time slot t, (5) is the power balance constraint, (6)–(7) represent the realistic limits on the output powers of CG units and their ramp rates, and (8) is a constraint for consumed powers of demand units. Next, denote the power loss coefficients of CG unit i, demand unit j, and RG unit lby γi,G≜∂PL∂Pi,G, γj,D≜∂PL∂Pj,D, γl,R≜∂PL∂Pl,R, respectively [10]. Then the power balance constraint (5) can be rewritten as

where PRt≜∑l=1q1−γl,RPl,Rt. On the other hand, for i=1,…,n, (6) and (7) can be conveniently rewritten as

where

for t=2,…,N, and P̂i,Gmin,t≜Pi,Gmin, P̂i,Gmax,t≜Pi,Gmaxfor t=1. In addition, (8) is rewritten as

2.2 Power scheduling with electric vehicle

In this problem, we include battery operation into the microgrid to suppress the high demand at the high-cost time of utility electricity. Furthermore, we consider EV battery instead of the stationary battery storage to reduce the installation cost of the battery system.

Our microgrid model consists of ndiesel generations (DGs), mdemand units, qPV generations, vEVs, and one microgrid operator. This microgrid is connected to the main grid (MG) whose electricity trading price is predescribed as qt. The electricity trading price inside the microgrid, one of decision variables, is denoted by pt.

3.1 Multi-agent system description for smart grid

The SDC-ADMM approach is based on MAS and consensus theory so that it can be run in parallel in all generation and consumption units. Hence, the MAS description for smart grid needs to be introduced first. More specifically, each agent is assigned to a generation or demand unit, and the communication among agents is represented by an undirected graph ℊ. Each node represents a unit in the grid whose variable is xior ξ, and each edge represents the communication between two nodes. For each node i, denote ℵiits neighbor set and ℵithe cardinality of ℵi.

3.2 Reformulation of smart grid optimization problems

To develop the SDC-ADMM approach, (25) is first reformulated into the 2-block form of ADMM. The following closed convex sets are defined corresponding to the global and local constraints:

together with their indicator functions [24]

Hence, (25) can be rewritten as follows:

Because the indicator function of a closed, non-empty convex set is proper, closed, and convex [25], the cost functions in (28) are also proper, closed, and convex with respect to Pand X. Hence, (28) and (29) is exactly in a 2-block ADMM form [24]. Next, an augmented Lagrangian associated with (28) is defined as follows:

where ρ>0is a scalar penalty parameter and uis called the scaled dual variable or scaled Lagrange multiplier [24]. Subsequently, the optimization problem (28) and (29) can be iteratively solved where at each iteration k=1,2,…,the variables P,X,uare updated by [24]:

This algorithm is stopped if the criteria rk2≤εpriand sk2≤εdualare both satisfied, where rk≜Pk−Xkand sk≜Xk−Xk−1are the primal and dual residuals at iteration k; εpri>0and εdual>0are called primal and dual feasibility tolerances that can be chosen by

with εabs>0and εrel>0which are some absolute and relative tolerances suggested to be 10−3or 10−4[24]. In [26], those tolerances and stopping criteria are shown to be computed and verified in distributed manners.

In the following, the variables P,X,uwill be updated in a distributed manner using MAS and consensus theory. It can be observed that uis updated in a decentralized fashion when Pand Xare already updated. Therefore, only the updates of Pand Xare introduced below.

3.3 P-update step

The update of Pin (30) is in fact equivalent to solving the following convex optimization problem:

of which the strong duality holds [25]. Let λk+1be the Lagrange multiplier associated with (34) and λ¯k+1its optimal value, at the iteration k+1of the SDC-ADMM algorithm. Then the Karush-Kuhn-Tucker (KKT) conditions can be used to find Pk+1from the following equations:

Since fiPiare known, Pik+1can be easily determined from (35). To find λ¯k+1, the values of Pik+1computed from (35) are substituted into (36). Usually, fiPiare chosen to be linear or quadratic functions for simplicity. Hence, the first term on the left-hand side of (35) is in the form of a linear function of Pik+1, denoted by a∼iPik+1+b∼i. This leads to

where âi≜μi2a˜i+ρ,b̂i≜μiρ−Xik+uik+μib˜ia˜i+ρ. The optimal value of λk+1shown in (37) is a global variable because the information of all variables are needed. Therefore, in order to calculate it in a distributed manner, consensus theory is employed where each unit in the grid corresponding to a variable is represented by an agent which exchange its parameters âiand b̂iand ξ(if ξ≠0) with a few neighbors in each iteration kof the SDC-ADMM algorithm. This is introduced in the following theorem:

Theorem 1. Given a connected undirected communication graph ℊamong agents, the initial conditions of agents are set to be

Consequently, the following consensus law is utilized:

where wijare Metropolis weights [27] defined by

Then the consensus of all agents is achieved, where the consensus states are

Thus, the optimal Lagrange multiplier can be obtained in a distributed manner by

Proof: Similar to the proof of Theorem 1 in [26], so it is omitted here for brevity. ■

Remark 1: If ξt=0, then zM+1is not needed, and hence, there are only Magents in the consensus problem in Theorem 1. Accordingly,

3.4 X-update step

The update of Xin (31) is obtained by solving a constrained quadratic programming:

Further, (42) and (43) is decentralized because both the cost function (42) and the constraint (43) are individually decomposable. Therefore, each agent (unit) just needs to solve a local scalar-constrained quadratic programming:

which is easily solved by existing methods.

For the DSWM problem, (45) is simply an interval, and Xihas an analytical solution [26]. Similar results are obtained for the PSwEV problem.

3.5 Algorithm summarization

The convergence of the proposed SDC-ADMM algorithm can be proved in a similar manner to that in [26]; hence it is ignored here for brevity. In the following, a summary for the algorithm is provided.

3.6 Pricing mechanism

For several grid optimization problems represented in the form of (1), e.g., economic dispatch or power scheduling, DSWM, PSwEV, etc., an electricity pricing mechanism is needed to drive the electricity trading in the grid. Interestingly, the optimal Lagrange multiplier associated with the equality constraint (2) is often regarded as the market-clearing price. That will be proved under mild assumptions in the following:

Theorem 2. For a sufficiently small but positive ρ, i.e., ρ→0, the optimal energy price in the considering smart grid converges to λ∗, the optimal Lagrange multiplier corresponding to the constraint (34) or equivalently (26), as k→∞.

Proof: For the OEM problems in smart grids, the convex functions fixitin (1) and (25) represent the costs for power generations or for the consumers’ satisfaction. Therefore, we can denote the following functions as the reward functions for agents:

where ptis the energy price at time slot t. In the ADMM reformulation (28) and (29), xitis replaced by Pik+1t. For simplicity, the time index is also dropped hereafter.

The optimal energy price in the system is achieved when the market is cleared, at which the marginal costs of all agents are the same. Hence,

In the proposed SDC-ADMM algorithm, this means

Substituting (48) into (35), we have

If ρ→0, we obtain from (49) that

Thus, as long as the proposed SDC-ADMM algorithm converges, the optimal energy price p∗is equal to the optimal Lagrange multiplier λ∗≜limk→∞λ¯k+1.■

4.1 Test case 1: DSWM problem

For this problem, a modified IEEE 39-bus system (see Figure 1) is used as the test system where it is assumed that there are PV generation units in the system with the total maximum output of 210 MW. The parameters of generators and demand units are taken from [26]. The average PV output curve, which is shown in Figure 2, is suitably scaled from the real PV data collected in New England [28]. Moreover, an average load profile taken from New England Independent System Operator [29] is utilized and properly scaled to obtain the time-varying upper bounds of demand units. This load profile has two demand peaks at 11:00 am and 8:00 pm, and the highest demand is at 8:00 pm, as seen in Figure 3.

Other parameters in the simulation are as follows. The absolute and relative tolerances are set to be εabs=10−4,εrel=10−3, while ρ=0.06. Moreover, the power loss coefficients of generators and consumers are randomly generated within 10%. Then the simulation results are shown in Figures 3–7.

Figures 3 and 4 display the total and individual power generated and consumed in the test system which are obtained by the proposed SDC-ADMM algorithm. Thanks to PV energy, the peak demand is shifted from 8 pm to 1 pm at which the PV output is maximum, as observed in Figure 3.

Then the electricity price is exhibited in Figure 5 showing that it is highest at 8 pm and lowest at 1–2 pm. This explains why the peak demand can be shifted. Next, the total welfare in the grid is shown in Figure 6 where the maximum welfare is attained at 1 pm when the consumers use most energy and the CG units produce least energy. Finally, Figure 7 shows the convergence of the proposed SDC-ADMM algorithm.

4.2 Test case 2: PSwEV problem

In this test case, a microgrid having 2 DGs, 16 EVs, 4 load demands, 4 PV generations, and a microgrid operator is considered with a 3-day duration. The total number of agents is 27, in which the controllable agents are those for 2 DGs, 16 EVs, and the microgrid operator. The total PV curve and the total demand curve are given in Figure 8. The prescribed electricity transaction price qtbetween the microgrid and the utility company is depicted in Figure 9.

The departure and arrival time and required traveling energies of 16 EVs are randomly generated from mix Gaussian distributions of this parameter of an actual set of 1400 EVs. The maximum energy capacity of each EV battery is 17.6 kWh, and the SOC limits of each EV battery are set at 20 and 80% of the maximum energy capacity, respectively.

The absolute and relative tolerances are set to be εabs=10−4and εrel=10−3. Next, ρ=0.06. These parameters are the same with those in Section 4.1.

Consequently, the simulations for hundreds of different scenarios corresponding to the above random times are run with the proposed SDC-ADMM algorithm. The results for one specific scenario are then shown in Figures 10–12.

Figure 10 shows the benefit of utilizing PV generation and EV charging in the microgrid. First, DG output power is reduced around the time period of high PV output, while the load demand is quite high. Second, when PV output is high, the microgrid operator can sell the redundant electricity to the utility at the highest price. Last, even though EVs do not have much correlation to PV output due to the EV traveling times, they still benefit the microgrid with their optimal charging and discharging schedules in which the charging is executed at low-price time periods of utility, and vice versa, the discharging is made at high-price time periods of utility. Moreover, the discharging also provides microgrid electricity for selling to the utility at a high price.

Subsequently, the SOC profiles of 16 EVs are depicted in Figure 11. It can be observed that the EV batteries will be charged in the early morning when the electricity price is low, but not to the maximum allowed SOC, i.e., 80% of maximum battery capacity, because of the using of EV charging/discharging strategy in Algorithm 1 and the realistic EV data that only around 6 kWh is enough for each EV round-trip. Further, the charged/discharged SOCs of EVs are different due to their differences on charging/discharging times and required energy for traveling.

Finally, the convergence of the proposed SDC-ADMM algorithm in the PSwEV problem is shown in Figure 11, and the consensus processes in the distributed algorithm for calculating the optimal electric price inside the microgrid are displayed in Figures 12 and 13.