Decentralized Dynamic Power Management with Local Information

The multi-period version of the optimal power flow can tackle the power dispatch problem of modern distribution system with distributed renewable energy sources and energy storage system. In this paper, a communication-efficient decentralized optimization algorithm (DOA) for the multi-period optimal power flow problem is presented. Firstly, the power management of modern distribution system is modelled as a linear conic optimization problem based on the conically relaxed power flow equations. Secondly, some ancillary variables at the junction bus are introduced to decompose the distribution system into several separate parts. Moreover, the DOA based on the extended version of Alternating Direction Method of Multipliers (ADMM) is proposed. The DOA evolves by partial local message exchanges without central coordination. Finally, the colouring scheme, which is in accord with the network colouring used in communication protocols to avoid packet collisions, is applied in the decentralized algorithm as well. DOI:  10.5755/j01.eie.25.1.22734


I. INTRODUCTION
The optimal power flow (OPF) has been marked as one of the most operational needs for a long time in the management of distribution systems.There are many algorithms used to solve the OPF problem, such as mathematical programming and heuristic optimization [1].The development and utilization of renewable energy sources (RES) [2] brings great effect on modern distribution system.Both wind power and solar power based distributed generators (DGs) are of intermittent nature.On this account, technologies such as energy storage systems (ESS) and flexible demand [3], both of which contribute to increase the penetration of RES connecting to the power system as DGs.
The OPF with storage should consider some intertemporal constraints due to the operation of storage being strongly coupled over the time periods [4].Therefore, the dynamic the DOPF problem of a test radial distribution system.The simulation results demonstrate the effectiveness of the proposed DOA and show the effect of the power management.

II. PRELIMINARY AND NOTATION
Let a directed graph G = (E(G), Ω(G)) be the description of the radial distribution system，where E(G) := {0, 1, …, m} is the set of buses and Ω(G) is the set of edges.The j th branch in the edge set Ω(G) is expressed as ψj := (i, j) which means the line circuit from the parent bus i to the child bus j in the distribution system, where the bus i is on the path from bus 0 (the root of the tree) to bus j.
In order to describe the relationship of edges and nodes in graph G, we construct the induced incidence matrix M which is the {0, ±1}-matrix with rows and columns indexed by the vertices and edges respectively, that is if is the child node of , 1, if is the parent node of , 0, otherwise, where the root bus 0 is not considered in M and M is m × m matrix.
The main variables and assumptions are summarized in Table I.For every branch in the distribution system, the branch model of power flow equations can be build according to the Ohm's law and power balance at each bus [14].Let wj := |Vi| 2 + |Ij| 2 and lj := |Vi| 2 -|Ij| 2 , thus the equations that describes the branch power flow can be rewritten as:     :, , 2 where where V0 denotes the voltage of the bus 0. For a given nodal injected power (pj and qj), the branch power flow equations are the linear combination of the variables (Pj, Qj, vj, lj, wj, ) except for the quadratic equation (5).

Ij(t)
the complex current from bus i to j for ψj at time t Pj(t) + jQj(t) the complex power flowing out from bus i to j at time t Vj (t) the complex voltage on bus j at time t pj(t) + jqj(t) the net injected power on bus j at time t pj G (t) + jqj G (t) the complex power of DGs on bus j at time t pj cS (t), pj dS (t) the charging power (pj cS (t)) and discharging power

Symbols Explanations
(pj dS (t)) of ESS on bus j at time t R, X, C the diagonal matrices in which the diagonal elements are (R)jj = Rj, (X)jj = Xj, (C)jj = (Rj) 2 + (Xj) 2

I, 0
the m × m identity matrix and the m × m zero matrix in which all the elements are equal to zero om the m × 1 column vector in which all the elements are equal to zero the column vectors in which the elements are as follows: the column vectors in which the elements are as follows: (p(t))j = pj(t), (q(t))j = qj(t), (p D (t))j = pj D (t), (p G (t))j = pj G (t), (q D (t))j = qj D (t), (q G (t))j = qj G (t), (p cS (t))j = pj cS (t), (p dS (t))j = pj dS (t)

III. PROBLEM FORMULATION
The proposed formulation of power management in the intelligent distribution system is a DOPF problem over a period T, where T:= {1, 2, …, tend}.It minimizes the power losses and the deviation of the voltage across the whole time horizon subject to power flow constraints and system security constraints.

A. Objective Function
The objective function of the proposed DOPT includes two parts: one part

 
L j ft denotes the active power losses; and ft describes the deviation of the voltage.
Then the objective of the DOPF problem is minimize where Here, vjm is the mean voltage of j th bus during the time horizon.The objective function is a continuously differentiable convex function of the optimization variables for all j ∈ Ω and t ∈ T.

B. Constraints 1) Conically relaxed power flow constraints
Considering the branch power flow equalities mentioned above in (2)-( 5), these equalities can be formulated by the intersection of the affine set and the second order cone  P ,Q ,v ,w ,l .
where t  T , equality constraint ( 8) is the description of power flow equations ( 2)-( 4), M is the induced incidence matrix in (1).And Kj(t) is the second order cone obtained by the conic relaxation of equations (5), that is 2) Security constraints For the first branch that points from bus 0 to bus 1, bus 0 denotes the substation bus that has a fixed voltage V0 where t  T .Branch capacity requires that the branch current will not be allowed to go above a threshold where t  T , Imax is the upper bound of the branch current.The voltage magnitudes must be maintained in tight ranges where t  T , Vmin and Vmax are the lower and upper bound of the bus voltage.

3) Storage system constraints
The generated power is determined by the generated power by the renewable energy based DGs and the state of charge of ESS.For the jth bus where t  T .Suppose that the output power of the solar-based DGs follow the point from maximum-power point tracking.The active power of DGs is suggested to be made maximal use of, that is, p G (t) can't be controlled.However, the ESS can be considered as either a generator when it is discharged or a load when it is charged.Then the charging and discharging power of ESS, p cS (t) and p dS (t), can be controlled and optimized.
At jth bus in the distribution system, let Ej(t) denote the amount of energy storage at each time t ∈T.The amount of storage at time t depends on the residual energy at the last time t-1 and the rate of charge/discharge of energy at time t [15].That is where  T ηc and ηd denote the charge and discharge efficiency respectively, Δt denotes the length of time step from t-1 to t, and denotes the initial condition of energy level in ESS at the beginning.The ESS is not an energy source but an assistant storage device, so the power generated by the ESS is the power that has been accumulated in the ESS for a time.Since a time horizon (e.g.tend = 24 h) is considered for optimization, energy level in the storage device at the final time point should be returned to the initial state [16]     0, where . Moreover, the amount of energy in the storage device should be bounded as

 
,min ,max , where  T , (Ej,min, Ej,max) denotes the lower and upper bounds on energy level at jth bus.Here, it is assumed that they are 20 % and 90 % of the installed capacity of the storage units, respectively.Considering the effect on the cycle life of storage device, the charge/discharge power should be capped by the battery's nominal charge/discharge rate by: where where

4) Reactive power constraints
The penetration of solar-based DGs can do inverter-based reactive power control [18].So the reactive power of DGs can be controllable and optimized: where t  T , q Gmin (t) and q Gmax (t) denote the lower and upper bounds of the injected reactive powers of DGs at time t.

C. Dynamic Optimal Power Flow Problem in Centralized Formulation
Given where Clearly, this problem is belonging to the conic programming who's the feasible set is the intersection of affine set and second order cone.And the optimal solution can give a true statement when it is substituted into (2)-( 6).

A. Decomposition
It can be seen from Fig. 1 that the decomposition is happened in junction bus nk.Before decomposition, the graph G is consist of two sub-graph Ga and Gb, where the edge sets of the sub-graphs satisfy Ω(Ga)∪Ω(Gb) = Ω(G), Ω(Ga)∩Ω(Gb) = ∅ and the node sets satisfy E(Ga)∪E(Gb) = E(G), E(Ga)∩E(Gb) = k.To separate the network, the dummy buses nk+ and nk-are introduced such that the distribution system is separated into the receiving and sending parts.are obtained by replacing the common node k with k+ and kin E(Ga) and E(Gb) respectively, are used to describe the receiving and sending parts [19].
In the optimization problem DOPF-c, the variable y falls into two categories: the core variables of the separate parts, and their common variables.Suppose that the decomposition is happened in the junction bus nk, the core variables ya and yb are given by: The common variables consist of the voltage in junction bus nk and the branch power in the branch ψh = (k, h), thus the common variables are described by yab = (vk(t), Ph(t), Qh(t), t∈T).
Then the optimal variable y = (ya, yab, yb) can be obtained by the following: where .
The objective function f0(y) of problem DOPF-c is independent of the common variable yab, so f0 = f0a + f0b.The constraints couple the core and common variables together.
For the sending part, Ph(t) + jQh(t) is regarded as the dummy load power at the dummy node nk+.So the net active and reactive power at the node nk+ is equal to the original net power at node nk plus the dummy load power, that is: where t  T , for the receiving part, the dummy node nk+ can be considered as a dummy substation whose voltage vk-(t) = vk(t).In addition, the voltage of dummy substation satisfies where t  T .For the sake of decomposition, the auxiliary variables ( qt  ) can be regard as the copy of the common variables yab.

 
The introduction of the auxiliary variables, that is to say, copies for common variables are given to the separate parts respectively.So with help of the auxiliary variables, the common variable yab is decomposed into two variables: .
The problem DOPF-c can be reformulated by: where the equality constraint (34) is to make sure that the copes are equivalent.Two sub-feasible sets Da and Db are with respect to the part G ' a and G ' b, they are given by: ~( 17),( 18),( ~( 17), (20) .
The problem DOPF-cd with the coupling constraint (34) can be decoupled into two sub-problems based on dual decomposition.By relaxing the coupling constraint as: where μ is the Lagrange multiplier corresponding to the constraint (34).

B. Decentralized Algorithm
Supposing that the distribution system G is divided into parts {G ' g = (Eg, Ωg), g = 1, …, n.}, each part is considered as a communication node in the corresponding CN is Gc = (Ec, Ωc), where Ec = {c1, …, cn} is the set of nodes and Ωc  Ec × Ec is the set of edges.Thus the problem DOPF-c can be rewritten as: where g = 1, …, n. where and Dg is the g-th feasible set of part G ' g.
The above problem DOPF-cdn is now separable except for the coupled equality constraints (40).It should be note that all the feasible sets {Dg, g = 1, …, n} are convex.As in the method of multipliers, the augmented Lagrangian of problem DOPF-cdn is formed where the is the set of the dual variables, and ρ > 0.
The decentralized optimization algorithm of ADMM includes the iterations: , The proposed algorithm is shown in Algorithm 1. Firstly the initialization of the auxiliary variables includes: randomly choosing the auxiliary voltages, the auxiliary powers and the dual variables.Significantly, the value of the dual variables must satisfy ,, In steps 2-6, the communication nodes work according to their colours, and the nodes with the same colour solve the optimization subsystem in parallel.In step 4, the solution of the optimization subsystem  

11
, , 2 where Esg is the set of sending subsystems of subsystem g and Erg is the set of receiving subsystems of subsystem g.
Then the dual variables are updated in steps 7-9.Here the function sn() is the sign function, defined as sn(x) = 1 if x > 0, and sn(x) = -1 if x < 0.
Finally, the error implies that the algorithm is convergent when   converges to zero.It can be easily proved that the proposed algorithm DOA is convergent since the {Dg, g = 1, …, n} are convex and the communication network Gc is connected [21], [22].

V. CASE STUDY
The distribution system considered for the case study is the PG&E 69-buses benchmark examples.As shown in Fig. 1, it is a 12.66 kV distribution system with a peak load of 3802.19 + j2694.60 kVA.The main substation at bus 0 is used to feed an area.Moreover, the data of the system are given in [23].
In order to improve the life and efficiency of the storage device, the current flowing through the ESS must be limited within a certain rage.Then the maximum absorbable and available power must be considered, the suggested limit is 1C in the charging phase.

Sr jb pC  (44)
where Cb denotes the capacity of ESS in kWh.And the initial amount of energy in the ESS is set to be sixty percent of capacity, E(0) = 0.6 Cb.
Let us discretize the day into 24 uniform time intervals, each of which is equal to 1 hour (Δt =1 hour).Figure 2 shows the average power demand and the average solar power of a day.The hourly power data for the system under study is as a percentage of the annual peak value.It can be seen that the output power of solar-based DG reaches the peak at midday, nevertheless, there is no solar power when the sun goes down.The energy storage system is necessary that it can provide power and store the surplus power as well.II.Using the proposed DOA to optimal dispatch active power of ESS and reactive power of DGs.

A. Implementation
The dynamic power flow optimization model in distribution system have been programmed in MATLAB R2010b; the proposed distributed algorithm DOA has been implemented by MOSEK's optimization solver.The experiments are conducted on a 64-bit PC with Intel Core 17 CPU at 1.73 GHz and 4GB RAM.

B. Results
For this 69-buses distribution system, the developed communication-efficient DOA can give the optimal active and reactive power management at the bus which has DGs and ESS in fully distributed manner.The minimum bus voltage of the system for three scenarios is shown in Fig. 3.With the help of the power generated by solar-based DGs, the voltage drop is avoided when the load reaches peak level at midday.In addition, the charging and discharging management of installed ESSs rise the voltage in the evening, and smooth the voltage profile as well.In addition, reactive power of DGs improve the voltage in the evening.Note that smother voltage profiles are obtained when energy storage systems are enabled in the third scenario (shown by the blue solid line).Figure 4 shows the hourly power losses for different scenarios.It can be observed that the power loss of the distribution system is significantly decreased at noon and in the evening in the third scenario.The total power losses during a day for different scenarios are summarized in Table II.The power loss is 3777.99 kW without the DG and ESS, the addition of DGs can decrease the power loss by 1189.07 kW (31.48 %).In terms of the proposed DOA, optimal management of ESSs' active power and DGs' reactive power can further reduce the power loss by 758.48 kW (20.08 %).Fig. 4. The hourly power losses of the system for different scenarios.Accordingly, it can smooth the voltage profile and decrease the power losses if the intelligent distribution system optimally manages the power of DGs and ESSs.The ESS devices are located at the same bus as DGs and is able to directly manage the fluctuated renewable energy.The operating schedule for the ESS devices on different buses are shown in Fig. 5.The ESSs are in the charging process when the curves are above the zero level.Otherwise, the ESSs are in the discharging process when the curves are below the zero level.It can be seen from the figure that the ESS devices can absorb the surplus energy when the power generated by the solar-based DG reached peak level at midday.However, during the periods when there is no solar power, the ESS devices generate power.Figure 6 shows the relative error ||(X k -X * )/X * ||∞ versus the number of iterations.Here, X k is the solution of DAO-algorithm at the end of kth iteration, and X * is the optimal solution of DOPF-c, computed in a centralized way.The figure shows that the proposed DAO-algorithm requires about 40 iterations to achieve any relative error bellow 10 -3 .Moreover, subsystems only talk to their neighbours for the information, and the exchange information is the part that is related to the neighbours.It can be concluded that the DAO-algorithm is convergent and efficient.

VI. CONCLUSIONS
The main contribution of this paper contains the following aspects.First of all, in order to optimize the reactive and active power in the distribution system with renewable energy and storage system, a dynamic optimal power flow problem is modelled by considering the conic relaxation of power flow equalities.And then, the model is decomposed after the introduction of the auxiliary variables about the junction buses.In the model, the convexity of feasible set enables the convergence of the decentralized algorithm.Finally, the decentralized algorithm based on the extended ADMM is proposed, such that the connected communication nodes exchange only the information they are interested in.The communication nodes work according their colours; thus an optimal solution can be obtained by use a minimal amount of communication.Simulations based on the 69-buses benchmark distribution system illustrate the effectiveness of the proposed decentralized optimization algorithm.Moreover, the effect of data packet dropout on the decentralized algorithm will be considered in our future work.
It should be noted that the feasible set of the DOPF problem, D:={(y)| y satisfies (8)-(18)}, is convex.Consider the DOPF in centralized formulation: Thus the generated graph G ' a and G ' b, of which edge sets are Ω(G ' a) = Ω(Ga), Ω(G ' b) = Ω(Gb) and node sets (E(G ' a) and E(G ' b)) into the sub-problem of the sending part G ' a.Meanwhile, the auxiliary variable vk-(t) is introduced into the sub-problem of the receiving part G ' b.The auxiliary variables (vk-(t), of the auxiliary variables in sub-graph G ' g, where vector   voltage and branch power) to be transferred from communication node cg to ch via the communication link (cg, ch) ∈ Ωc.Let yg be the core variable of the separate part G ' g.
scheme[20] is used to synchronize the order of the nodes.Each communication node is assigned a colour in C = {1, …, C}, where C :=|C| is the total number of colours, and C(ck) denotes the colour of node ck.The colour scheme requires C(ck) ≠ C(ch) for all (ck, ch) ∈ Ωc.Up on that each iteration is divided into C steps.In each step, all the communication nodes with the same colour perform the optimizations in parallel.The proposed algorithm is shown in Algorithm 1. Firstly the initialization of the auxiliary variables includes: randomly choosing the auxiliary voltages, the auxiliary powers and the dual variables.Significantly, the value of the dual variables

Fig. 2 .
Fig. 2. Hourly average load and power generated by solar based-DG of a day.The following scenarios are used to illustrate the effectiveness of the DOA algorithm and the DOPF model.1.No DG and ESS installed in the distribution system.No other distributed energy devices are considered.2. As for scenario 1 + DG.The location and capacity of solar-based DGs are shown in the Table II.No reactive power generated by DGs. 3. As for scenario 2 + ESS.The location and capacity of ESSs are given in the Table II.Using the proposed DOA to optimal dispatch active power of ESS and reactive power of DGs.

Fig. 5 .
Fig. 5.The hourly output power of ESS on corresponding buses.

TABLE I .
NOTATIONS.

TABLE II .
THE TOTAL POWER LOSSES DURING A DAY FOR DIFFERENT SCENARIOS.