Computational Techniques for Autonomous Microgrid Load Flow Analysis

This paper attempts at developing simple, efficient, and fast converging load flow analysis techniques tailored to autonomous microgrids. Two modified backward forward sweep techniques have been developed in this work where the largest generator is chosen as slack generator, in the first method and all generator buses are modeled as slack buses in the second method. The second method incorporates the concept of distributed slack bus to update the real and reactive power generations in the microgrid. This paper has details on the development of these two methodologies and the efficacy of these methods is compared with the conventional Newton Raphson load flow method. The standard 33-bus distribution system has been transformed into an autonomous microgrid and used for evaluation of the proposed load flow methodologies. Matlab coding has been developed for validating the results.


Introduction
Aggregation of generating units and loads, at medium and low voltage levels, forms small power islands called microgrids.Most researchers concentrate on the design and control aspects of these microgrids with respect to the resource availability and dispatchability of power to the loads.Design issues, generation planning, and economic dispatch in an autonomous microgrid need dedicated and robust power flow computations.Load flow analysis of an autonomous microgrid is necessary for ascertaining the adequacy of supply from DGs without compromising the voltage profile and to determine the state of the system.
Different load flow techniques adopted in the literature are classified into three categories, namely, direct methods, Newton Raphson (NR) based methods, and backward forward sweep based methods.Direct methods involve impedance matrix where the numbering of nodes and lines decides the efficacy and convergence criteria.Chen et al. proposed a rigid power flow method based on series impedance model [1] and Carpaneto et al. suggested a loss allocation technique, based on decomposition of the branch currents [2].Tedious computation is the basic drawback in these methods.
NR method was used in certain methodologies to determine the bus voltages and power flows in distribution networks.A modified Newton method had been discussed by Zhang and Cheng [3] and as an extension Teng and Chang [4] suggested a novel fast three phase load flow analysis for unbalanced radial distribution systems.Bijwe and Kelapure [5] proposed a nondivergent load flow analysis based on NR method.NR-based method was extended to unbalanced systems by Zimmerman and Chiang [6] and further improvements of the computational efficiency of NR-based algorithms were also attempted in the literature [7].Garcia et al. [8] suggested a load flow algorithm based on current injection technique in which generator connected buses were modelled as either PV or PQ buses.Most NR-based methods present a high convergence rate but fail to exploit the system topology.
In this context, researchers started exploiting the radial topology of distribution systems, resulting in the development of backward forward sweep based load flow analysis [9].Breaking of loops and application of the equivalent current injection (ECI) method to the break points was adopted by Cespedes [10] and was further extended to three-phase radial distribution systems by Cheng and Shirmohammadi [11].Modified backward forward sweep method was introduced by Chang et al. [12].Later, data structure and object oriented approaches were incorporated with the modified backward forward sweep method to formulate a load flow analysis suitable to both radial and weakly meshed systems [13,14].All these works focused on the distribution systems without any DGs and hence adopted the substation feeder as the slack bus.Basu et al. [15] attempted an NR-based load flow for microgrids, where the largest generator is assigned as the slack generator but did not exploit the system topology explicitly.
In all the earlier works, a single bus has been considered as the slack bus where the single slack generator is considered to share the total losses in the system thereby getting overloaded.Though analytically this consideration is acceptable, in a practically competitive environment with many owners, it becomes inevitable to clearly identify the contribution of each generator towards load, power flows, and losses.These generators' contributions would vary depending upon their respective locations and network parameters.Kirschen et al. and Strbac et al. [16,17] introduced the concept of "domains" and "commons" of individual generators in a microgrid.Yan [18] suggested the concept of modified slack bus based on participation factor of generators.Distributed slack bus model has been extended with Newton iterative method and the effect of participation of generators with respect to the loads and losses were discussed by Tong and Miu [19][20][21].All these works focus mainly on the impact of participation factor but not on maintaining the slack bus voltage constant.
The authors have therefore developed two load flow methodologies dedicated for autonomous microgrids, based on backward forward sweep algorithm.The first method adopts a single slack bus (bus to which largest generator is connected) with all the other DG connected buses modelled as PQ injection buses.The second method incorporates the distributed slack bus technique in the backward forward sweep algorithm, where all generator buses are modelled as slack buses.Contributions of the generators towards total system losses and loads are utilized to update the real and reactive power generations, in the second method.The efficacy of the proposed techniques has been investigated using the standard 33-bus distribution system [22,23] transformed to an autonomous microgrid.The results are also compared with that obtained using NR method.
The paper is organized as follows: introduction followed by modified backward forward sweep based load flow technique with single slack bus in Section 2 and distributed slack bus based backward forward sweep load flow technique in Section 3. A case study illustrating the proposed methodologies is dealt in Section 4 and a detailed comparison of different load flow methods adopted is shown in Section 5.The paper concludes with Section 6.

Modified Backward-Forward Sweep Based Load Flow Analysis
The basic backward forward sweep technique has been modified to suit the load flow analysis of a sustainable autonomous microgrid.
Figure 1: Steady state representation of a branch "" between th and th buses of a microgrid.

Backward Sweep Technique.
The steady state equivalent circuit representation of a branch "" between the buses "" and "" of microgrid is considered as shown in Figure 1 and the currents are computed using the following: where All branch and node currents are computed using (1)-( 3), respectively, in each of the iterations.The effect of introducing generators in distributed systems is incorporated in (2), where positive sign is assigned to the injected generator current and negative sign for the current components drawn by the load and compensating devices.This polarity assignment incorporates the effect of DGs penetrated in the system; namely, the net current injected at any th bus with a DG attains a positive sign and without DG attains a negative sign.This is automatically reflected in bus voltages computed in the forward sweep.Modifications incorporated in the conventional backward sweep technique is reflected in (2).

Forward Sweep Technique.
Since the effect of addition of DGs has been incorporated in the backward sweep, there is no modification required in the forward sweep.Hence, the basic forward sweep is performed, as shown in the following: The polarity assignment adopted for the current injections facilitates appropriate bus voltage computations.This includes the effect of voltage modifications automatically on forward sweep computations.The bus to which the largest generator is connected is considered the slack bus and hence its voltage alone is maintained at 1 + 0.All other bus voltages are computed using (4).(i) System data, including the connected load and network parameters, are taken.

Algorithm for
(ii) A flat voltage profile of 1 + 0 is adopted initially for all the buses.The tolerance value is defined and the convergence criterion is fixed as where  denotes the iteration count.
(iii) The largest generator is assigned as the slack generator.
(v) Backward sweep is performed and all the node and branch currents of the microgrid are determined using (1)-( 3).
(vi) Voltage at the slack bus is fixed at 1 + 0 and the forward sweep is performed to compute the voltage of all the other buses starting from the slack bus using (4).
(vii) The iterations are continued till the convergence criterion ( 5) is satisfied.
(viii) On satisfying the convergence criterion, system losses are computed.

Significance and
Limitations of the Method.The proposed load flow analysis is found to own certain significant features as well as limitations in comparison with the NR method.

Significance of Modified Backward Forward Sweep Load Flow Method
(i) The radial topology of the system is exploited in determining the bus voltages and the branch currents.
(ii) Computational procedure is very simple and deals with only KCL and KVL avoiding complex equations and matrix manipulations.
(iii) The convergence rate is very high such that the algorithm converges in two to three iterations for any size of the system.
(iv) The algorithm does not require bus impedance or admittance matrices.

Limitations of Modified Backward Forward Sweep Load Flow
Method.An initial guess of the real and reactive power injections is required at all the generator buses and hence this is obtained by performing NR method in this work.This is essential to calculate the current injections during the backward sweep.

Distributed Slack Bus Based Load Flow Analysis
A new load flow methodology based on distributed slack bus technique has also been proposed for radial microgrids in this paper.

Distributed Slack Bus Technique.
In single slack bus model, the generator connected to the slack bus alone is considered to take up the complete losses in the system, in contrary to the actual practice.In this method, all the generators capable of supplying real and reactive powers are considered to be slack generators with constant voltage magnitude and angle, with varying real and reactive powers.Moreover, these generator buses, modeled as slack buses, are considered to share the system demand and distribution losses in different proportions with respect to real and reactive powers.

Disadvantages of Single Slack Bus Model.
The following are the noticeable limitations of single slack bus model in load flow analysis.
(i) In a radial system, the real power losses in the system would be higher if a single generator is committed to compensate the total system losses.This is suitable for a nonautonomous microgrid as the substation feeder also takes a share of the total losses but in an autonomous microgrid a single generator would be unnecessarily taxed to supply the total losses.
(ii) As the largest generator alone is considered as the slack bus and all other generator injections are fixed, there is always a chance for over voltage at the generator buses due to fixed real and reactive power injections by the generators in view of satisfying the demand constraint alone.
(iii) On blackouts or line outages, intentional islanding of the microgrids to supply reliable and uninterrupted power supply to select customers becomes difficult as the demand constraint would not be satisfied when the slack generator gets isolated.

Domains and Commons of Generators.
The distributed slack bus technique aims at distributing both the system loads and losses to all the generators.Hence, to determine the real and reactive power generations of the generators, the actual contributions of each generator towards the real and reactive power flows, loads, and losses in the system are determined using the "domains" and "commons" of the generators.
The "domain" of a generator helps to identify which buses and hence loads are supplied from a generator and also the farthest point till which the power injected by a generator reaches.The "domain" of a generator is defined as the set of buses and branches supplied by the generator in the literature.The "domain" of a generator is determined after identifying the positive power flow direction on the microgrid.For two directly connected buses,  and , (i) if Real(  ×  *  ) − Real(  ×  *  ) ≥ 0, then positive real power flows from bus  to bus ; (ii) if Imag(  ×  *  ) − Imag(  ×  *  ) ≥ 0, then positive reactive power flows from bus  to bus .
The positive real and reactive power flows are decided by the loads and the network parameters and are used to update real and reactive power generations.Further, certain loads would be supplied power by more than one generator in a microgrid.Thus, "domains" of different generators often intersect and they have branches or loads in common.A "common" is defined as a set of contiguous buses supplied by the same generators [19][20][21].Based on the principle of proportionality [20], that is, the proportion of loss and loads supplied by different generators to a "common" is the same as the proportion of positive real power injected by the generators to this "common, " the proportion of loads and losses of a "common" is assigned to the corresponding generator "domain." Thus, all loads and losses in a network are assigned to individual generators using a directed graph.These commons are interconnected through "links" connecting two or more "commons" and there can be more than one link connecting two "commons."

Contribution and Participation Factor of Generators.
An oriented state graph is drawn identifying the generator "domains" and "commons" to compute how much a generator contributes to the loads and flows in the "commons" and "links" which are located "downstream." The inflow of a "common" is defined as the amount of power flowing into a "common" either from generators connected to buses of this "common" or across links from other "commons." Similarly, the outflow of a "common" is the amount of power flowing across links into other "commons" or consumed by loads connected to buses of the "common." The Proportionality Principle States.For a given "common, " if the proportion of inflow which can be traced to generator "" is ", " then the proportion of the load and outflow of this "common" which can be traced to generator "" is also "." First the root nodes or root "commons" are identified ("commons" in which the generators are present) and the contribution by a generator to its encircled "common" is considered 100%.As a next step the contribution of any generator "" to a "common" "" is computed as shown in the following: where After computing contributions of the generators to the "commons, " the participation factor of a generator towards real and reactive power loads and losses in every "common" is determined using the following: These contributions and participation factors are used for updating the real and reactive power generations of the generators.

Algorithm for Distributed Slack Bus Based Load Flow Technique
(1) System data, location, and rating of DGs in the autonomous microgrid, load data, and convergence limit are determined.
(2) A preliminary load flow analysis (NR technique) is performed.
(3) Initially a flat voltage profile of 1 N 0 ∘ is considered for all buses.
(4) One complete backward sweep is performed starting from the terminal buses of the system, with the generator buses operated at 1 N 0 ∘ using (1)-( (5) Slack bus voltages are kept constant (1 N 0 ∘ ) and the forward sweep is performed starting from every generator bus and proceeding towards each terminal bus using (4).
(6) There are as many forward sweeps as that of number of generators and the reversal of current direction on a distributor/lateral is an indication for starting a fresh sweep.
(7) Power flows on the distributors and the total real and reactive power losses in the system are determined.
(8) "Domains" and "commons" of the generators are determined separately by accounting positive real and reactive power flows on laterals.
(9) As per the proportionality principle, the contribution of real and reactive power of each generator to each "common" is determined using ( 6)-( 7).
(10) After determining the contributions, the participation of each generator to share the loads and losses in a "common" are determined using ( 8).
(11) The actual real and reactive powers to be generated by each generator according to the participation factors determined are computed as follows: where (12) The demand constraints as shown in the following are checked: (13) The iterative procedure (steps 4 to 12) is repeated till the tolerance limit ( 12) is satisfied: where  indicates the iteration count.(14) The final updated real and reactive powers to be generated by each of the generators is computed as the solution of the load flow analysis.The voltage profile and the losses corresponding to the final solution are tabulated.

Significance and Limitations of the Proposed Method.
The proposed load flow analysis is found to possess certain significant features as well as limitations in comparison with the standard NR method of load flow analysis.

Significance of Distributed Slack Bus Based Load Flow Method
(i) The proposed method exploits the system topology to determine bus voltages and power generations.
(ii) No power factor controller is required.The real and reactive powers, generated by the DGs, are allowed to vary (practically feasible).
(iii) The method deals with only KCL and KVL avoiding complex equations and matrix manipulations.
(iv) The domain margin of each generator is obtained which would help in pricing issues of autonomous microgrids.
(v) The algorithm does not require the formation of bus impedance or admittance matrix.

Limitations of Distributed Slack Bus Based Load Flow Method
(i) Initial real and reactive power injections are required at all the generator buses to calculate the current injections during the backward sweep.Hence, any other standard load flow analysis is required at the initial stage.
(ii) Forward sweep for a system with many generators needs more computations since it has as many numbers of sweeps as that of the generator buses.
(iii) Real and reactive power contributions need to be computed to determine the updated real and reactive power generations by the DGs.This increases the computational time.

Deliverables of the Proposed Load Flow Methodology.
The proposed distributed slack bus based load flow analysis for an autonomous microgrid is suitable for the following applications: (i) to realize the actual scenario of load and loss shared among different generators in an autonomous microgrid; (ii) to schedule the different generators in a microgrid for selected customers; (iii) to decide the tariff in an autonomous microgrid, where the different generators in the microgrid are owned by different companies; (iv) to avoid pessimistic conclusions on the size of the largest generator unlike in the case of single slack bus based analysis; (v) this methodology can be used as a base case load flow analysis for sizing of DGs in an autonomous microgrid.

Case Study
The standard 33-bus distribution system [22,23], with a demand of 3.715 MW and 4.456 MW in summer and winter, respectively, has been considered for the validation of the proposed methodologies.This distribution system has been transformed into a sustainable autonomous microgrid on inclusion of optimally sized DGs at optimal locations, as shown in Figure 2. The sizing details of the generators are given in Table 1.A base power of 100 MVA and base voltage of 12.66 kV are adopted, respectively.The load flow results for the 33-bus microgrid, obtained by implementing the proposed load flow techniques, namely, modified backward forward sweep and distributed slack bus based load methodologies, are compared with that of the standard NR method of load flow.The load flow has been performed for both the summer and winter demands of the system and the voltage variations are tabulated in Table 2.The existing backward forward sweep techniques are suitable for radial distribution systems with a feeder node serving as a reference.However, in this work since the feeder is absent the existing method is modified with the node having the largest capacity as the slack (reference) node and the load flow is performed.
Table 2 shows evidently that the voltage at all the buses is found to be in close proximity with that obtained by NR methodology.Similarly the losses are compared and presented in Table 3.
The real and reactive power losses are found to be comparable to that of the standard NR method.However, losses are found to be slightly more in distributed slack bus based method than the single slack bus model, as the siting and sizing of DGs connected in the system have been determined with single slack bus model.Real power "domains" and "commons, " for the winter demand, are shown in Figures 3 and 4 and a similar analysis is also done for reactive power flows.The details of the contributions and participation factors of the generators for real and reactive power flows are tabulated in Table 4 for the winter demand of the system.
Table 4 shows clearly the procedure of determining the real and reactive power generations in the microgrid using the distributed slack bus based load flow analysis.

Comparison of Voltage Profile.
The voltage profile for the 33-bus microgrid (shown in Figure 1) on implementation of the proposed methodologies (Table 2) is compared with that of the standard NR method results for realizing the efficacy of the proposed algorithms.This comparison, for summer and winter demands, is shown in Figures 5 and 6, respectively.
Figure 5 depicts that the voltage profiles obtained by NR method and modified backward forward sweep methods are in close agreement, since the generators are modeled as PQ injection machines.Since the real and reactive power generations at all the generators are fixed, the voltage at 31st bus (generator bus) is found to rise above 1 p.u.On the other hand, the voltage profile obtained by distributed slack bus based method is found to be almost flat and the maximum voltage does not exceed 1 p.u. at any bus.

Comparison of Distribution
Losses.The distribution losses for the 33-bus microgrid (shown in Figure 1) obtained  from the proposed methodologies (Table 3) are compared with the results of standard NR method.This comparison of real and reactive power losses for summer and winter demands is shown in Figures 7 and 8. Since the comparison is made for the generators sized using single slack bus model, the losses are found to be slightly high for the distributed slack bus based load flow analysis.

Conclusions
This paper has suggested two backward/forward sweep based load flow methodologies dedicated for autonomous microgrids.One method focuses on single slack bus model whereas the other method incorporates distributed slack bus technique into the sweep algorithm.Both the proposed methodologies exploit the radial structure of microgrids and the possibility of operating synchronous generators to supply varying real and reactive power outputs at constant  Resistance of the distributor between buses "" and "" (p.u.)  load  :

List of Symbols
Apparent load power at th bus (p.u.)   : V o l t a g ea tth bus (p.u.)   : Reactance of the distributor between buses "" and "" (p.u.).
Modified Backward/Forward Sweep Technique.Modified backward forward sweep based load flow analysis is formulated as follows.

Real power distribution pertaining to the winter demand of the
of G1 is 1870 kW, G2 is 950 kW, and G3 is 1690 kW, respectively Reactive power distribution pertaining to the winter demand of the system of G1 is 1179 kVAR, G2 is 583 kVAR, and G3 is 11040 kVAR, respectively

Figure 5 :
Figure 5: Comparison of voltage profile for summer demand of the 33-bus microgrid.

Figure 6 :
Figure 6: Comparison of voltage profile for winter demand of the 33-bus microgrid.

Figure 7 :Figure 8 :
Figure 7: Comparison of losses for summer demand of the 33-bus microgrid.

Table 1 :
Domains of the generators w.r.t.real power flows in the microgrid for winter demand.Optimal size of the distributed generators in the 33-bus autonomous microgrid.

Table 2 :
Comparison of bus voltages for different operating conditions of the 33-bus microgrid.

Table 3 :
Comparison of real and reactive power losses in the 33-bus microgrid.

Table 4 :
Contribution and participation factors of the generators.
Figure 4: "Commons" of the generators in the microgrid w.r.t the real power flows for winter demand.
: Contributionofth generator towards th common (p.u.) Com load  : Real power load in th common (p.u.) Com loss  : Real power loss in th common (p.u.) Com load  : Reactive power load in th common (p.u.) Com loss  : Reactive power loss in th common (p.u.)   : Contributionofth generator through th link to th common (p.u.)   : Power flow contributed by pth link to th common (p.u.)  br  : Branchcurrentforthebranch"" (p.u.)    : Generator injected current at th bus (p.u.)   : Net current injected at th bus (p.u.)  load  : Load current drawn from the th bus (p.u.)   : T o t a lp o w e ri n fl o wi n t oth common (p.u.)  sh  : Shunt compensation current injected at th bus (p.u.)  load  : Participation factor of th generator for real power load in th common  loss  : Participation factor of th generator for real power loss in th common  load : Participation factor of th generator for reactive power load in th common  loss  : Participation factor of th generator for reactive power loss in th common nc: