Mathematical Modelling for Optimal Electrical Energy Generation and Distribution

in Remote Micro-Grids

Pranav R Deshpande1,2, Kaustubh Karnataki2, Ganesh Shankar2 1Electrical and Electronics Department, SRM University, Chennai, Tamilnadu, India

2FluxGen Engineering Technologies Pvt. Ltd., Bangalore, India

pranav.deshpande18@gmail.com, kaustubh@fluxgentech.com, ganesh@fluxgentech.com

Abstract- In remote and energy deprived areas where

conventional electric grid penetration is not feasible, alternative

means of satisfying its demands becomes necessary. Considering

the concept of decentralized energy generation and distribution

using renewable energy sources, this paper focuses on optimal

designing of an electrical energy micro-grid. A mathematical

approach is employed which involves optimization of

distribution network and distributed power sources using

Prim’s algorithm for connected graphs. Furthermore fractional

load assessment is incorporated for determining optimized

placement of renewable energy sources both from energy and

economic perspective. A sample case study of an AC solar

photovoltaic (PV) micro-grid implemented inside a reserve

forest area of Karnataka, India is considered. This work is

expected to be a tool for micro-grid practitioners involved in

electrifying geographically isolated areas scattered around a

specific region using optimal resources.

Keywords-renewable energy; micro-grids; distribution

system optimization

I. INTRODUCTION

Technology and energy has become a vital part of everybody’s life. It has become a less stated, but fairly obvious fact that energy supply has become a dependency for mankind. But not all regions have experienced the gift of electricity. Rural regions of India are one such bracket. Almost 2/3rd of the Indian population resides in these rural regions, making electricity accessible a humungous yet indispensable task. Also, penetrative grids into remote location are technically and economically non-feasible. Moreover terrain and climate issues cause hindrance for maintenance.

Micro-grids lead the way as a solution to these constellations of problems. Furthermore micro-grids assume a cluster of loads, which is exactly how houses are situated in villages, and micro-sources (<100KW) operating as a single controllable system [1]. Villages primarily consists of small areas with houses distributed in form of small clusters called hamlets and micro-grids can be tailor made to meet the needs of such arrangements. Micro-grids are being tested and have been implemented in many regions of India and have proven to be magnificent [2]-[3].The fact that micro-grids use renewable energy sources adds to its competency.

The implementation of such a golden system calls for a method to make it more efficient and economic. This paper discusses an algorithm which assists in efficient implementation of micro-grids in rural areas by reducing installation costs and energy losses.

II. MOTIVATION AND PROBLEM FORMATION

FluxGen Engineering Technologies, as a renewable

energy company has been working on sustainable solutions

for electrical energy based on renewable energy sources in

collaboration with the Divecha center for Climate Change,

Indian Institute of Science, Bangalore, a 2kW solar

photovoltaic micro-grid was deployed in Mendil, a village

situated inside a reserve forest region in Belgaum district of

Karnataka state India. It is presently powering one of the nine

hamlets present in Mendil. The objective of this pilot system

implementation is to perform operational analysis on fields.

It includes energy metering of houses, monitoring the solar

PV parameters and local weather conditions. These functions

are carried out using an embedded controller interfaced with

smart meters at individual houses, and a local weather station.

Currently the monitoring systems is logging vital information

necessary to design future micro-grids depending on the

villagers load usage pattern, variants of load used, local

weather conditions and possibility of energy sharing with the

future micro-grids in order to electrify the remaining

randomly scattered hamlets in a radius of about few

kilometers. In order to setup a technically feasible and

economically viable micro-grid network in Mendil, the paper

provides guidelines on optimal electrical network distribution

based on distributed energy sources in order to electrify

scattered load points over a specific area. This work is

expected to be the basis for designing future micro-grids at

Mendil which is further discussed in the paper as a case study.

III. BACKGROUND AND PREVIOUS WORKS IN RELATION

An electrical power system is a very complex system

consisting of concoction of linear and non-linear variables.

These variables, some being dependent and some

independent, pose a puzzling problem for optimization. An

electrical power system can be thought of to consist of three

constituent systems: generation system, transmission system

and distribution system. Several amazing works have been

innovated to optimize each of these systems. Particle Swarm

Optimization (PSO) is one such method. AlRashidi et al. [4]

showcased the versatility of PSO, by portraying its use in

various optimization problems of the electric power system

such as economic dispatch, reactive power control, power

loss reduction, optimal power flow, power system controller

design and neural network training. Yoshida et al. [5], [6], [7]

and Fukuyama et al. [8] took a stab at the power loss problem

by introducing use of PSO. Furthermore, the use of PSO in

Optimal Power Flow (OPF) problem has also been seen [9].

2015 IEEE European Modelling Symposium

978-1-5090-0206-1/15 $31.00 © 2015 IEEE

DOI 10.1109/EMS.2015.87

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2015 IEEE European Modelling Symposium

978-1-5090-0206-1/15 $31.00 © 2015 IEEE

DOI 10.1109/EMS.2015.87

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Another extensively discussed issue is that of Transmission

Network Expansion Planning (TNEP). The TNEP problem

consists of where and when new circuits are needed and

should be installed to serve in an optimal way, subject to a set

of electrical, economic and environmental constraints. An

Improved Genetic Algorithm can be applied for such

problems of dynamic nature to obtain lowest cost of

expansion [10]. Another interesting approach utilized is the

Ant Colony System (ACS) algorithm [11], which is used to

solve the problem power distribution network, which is a

meta-heuristic approach. Reconfiguration of distribution

lines is also done to reduce losses, an old but applied method

of compensation based power flow was pivotal in [12] to

reduce line losses. A very riveting line of action for

optimizing both real power loss and Voltage Stability Limit

is the use of Bacteria Foraging algorithm [13], which solves

the multi-objective and multi-variable problem for

optimizing location and control of unified power flow

controller (UPFC) along with transformer taps to reduce real

power loss and voltage stability limit.

Although there are a lot of distressing problems involved

in a power system, Micro-grids in remote location, as

discussed in this paper, are a rather simple and selective

marriage of distribution line and a renewable energy

generation unit. So the complexity of the problem is

substantially reduced to a few variables. This paper deals

with the amalgamation of cost i.e. economy and distribution

line loss optimization. A mathematical approach is taken to

solve this problem by making use of fractional load

assessment, as discussed ahead, and graph theory’s Prim’s

algorithm.

IV. CALCULATIONS AND PROPOSED APPROACH

A. Assumptions

The proposed algorithm provides a simple step-wise assistance in designing the distribution system and placement of generation system of a micro-grid. The algorithm considers some necessary general assumptions. The micro-grid is assumed to be located in a remote rural region of India. The distribution of the houses is laid out in form of groups called hamlets. Further it is assumed that the total demand of the village is constant. Another important assumption is regarding the renewable energy source considered for the supply. Although here only solar is considered for simplicity, any other renewable energy source can be used over same.

The cost involved in the solar energy system is in per Watt which is a rough estimate inclusive of inverter and battery bank cost for a 48V PV panel system which is converted and distributed via a 230V single phase distribution system. The distribution line is a 6 mm2 3-core armoured copper power cable consisting of phase, neutral and earthing. This estimate as per Indian standard pricing quotes involved in the micro-grid set up by FluxGen Engineering Technologies in Mendil [2]. The cost of the concrete poles and the wire is also as per these quotes. Furthermore the micro-grid is used to provide

lighting for 6 hours per day for every house. Another assumption is the number of solar hours, this factor affects the rating of the system, the solar hours here is assumed to be 4hrs. Any other source factor might be needed to calculate fixed costs that may be involved as in the case of wind energy etc.

B. Total Cost

There are two factors which affect the efficiency and economy of the system- cost and losses. The cost factor can be resolved into two components: fixed cost and variable cost. Since energy is to be supplied to the entire village, the total demand of the village can be taken as required demand, as this cannot be changed or reduced it can be assumed that the cost of setting up the renewable energy source must be at least sufficient to satisfy this demand and the rating of the energy source, which in this case is the AC solar panel is constant for the whole load, be it single source or multiple distributed sources as the total rating will be same. The minor variations in the cost for multiple sources have been taken care in the approximate cost. This covers the fixed cost component. The other cost involved in the distribution line. This involves cost of conductor and concrete poles to support them. As cost of the conductor and the number of poles needed to support them depend on the variable of conductor length, they can be considered as variable cost. Hence, total cost can be written as

𝐶 = 𝐹𝑖𝑥𝑒𝑑 𝑐𝑜𝑠𝑡 + 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑐𝑜𝑠𝑡 (1) To determine the exact dependency of cost on various

variables, consider the load of the entire village, the total demand is constant. Consider a location where the micro-grid is to be setup, let the region consist of h houses distributed among n hamlets. As per our general assumptions, the houses consist of lighting, so consider 4 LED bulbs of 5W rating.

Demand/house = 4 × 5 W = 20 W Since there are 6 hours of lighting, Demand/house = 20 × 6 W=120 WH There are h houses in the village, hence Total Demand= 120 × ℎ WH (2) As per the assumption of 4 solar hours,

PV panel output =120×ℎ

4W = 30. ℎ W (3)

Now this demand is satisfied by the means of AC Solar Panels as per our assumptions, the cost of setting the AC Solar system is given as

Cost per Watt = Rs.130 ≈ 2.2 $ (as per Indian Standards) Therefore, Constant Cost = 2.2 × 30 × ℎ $ = 66 × ℎ $ (4) The variable cost factor consists of the cost of distribution

line. This cost includes the cost of the concrete poles required to hold the line and the cost of the conductor. Based on standard costs in the Indian market,

Cost of a concrete pole = 85 $ As per the field survey done in the referred regions and

other domestic regions it is proposed that every 50 meters of requires a concrete pole. Assuming one pole is required at the start and end point,

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For a length of l meters the number of poles is given by

(𝑙

50+ 1) (5)

Hence, cost of poles for l meters of distribution line is

(𝑙

50+ 1) × 85$ (6)

Now, cost of distribution line = 2.5 $ /meter For a l meters of distribution line, Cost =2.5 × 𝑙 $ (7) Therefore, total cost of distribution line is equal to

{(𝑙

50+ 1) × 85 + 2.5 × 𝑙} $ (8)

Total variable cost= (4.2𝑙 + 85)$ (9) =>𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑐𝑜𝑠𝑡 ∝ 𝑙 (10) Summing things together, under the given assumptions, Form (1) we know,

𝐶 = 𝐹𝑖𝑥𝑒𝑑 𝑐𝑜𝑠𝑡 + 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑐𝑜𝑠𝑡

Referring to (4) and (9) we arrive at

C = {(66 × ℎ) + (4.2𝑙 + 85)}$ (11)

C. Losses

The losses in a distribution line depend on the impedance of the line, given by

Z = (R + jX ) Ω (12) Where Z is the line impedance, R is the line resistance and

X is the line reactance. The Reactance of the line can be represented as X = (XL − XC) Ω (13) Where XL is the inductive reactance, XC is the capacitive

reactance of the line. Line inductive reactance is given by

XL = j(2πfl) × (μ0

π× ln (

D

GMR)) Ω (14)

Where f is the frequency of the system, l is the length of the line, D is the distance between centres of the cables, GMR geometric mean radius, μ0 is permeability of free space.

Since the distribution line is shorter the 50Km, consider the short line model of transmission line. Therefore the capacitive effects of the line can be neglected.

Hence, X = XL Ω (15) The resistance of the line can be represented as

R =ρl

A Ω (16)

Where ρ is resistivity of the material, l is the length of the line, A is the area of cross-section of the line. The resistivity of the conductor changes along with the temperature

ρ(T) = ρ0(1 + α(T − T0))Ω∙ m (17)

Where T is the temperature, α is the temperature coefficient of resistivity and T0 is the reference temperature.

From (12) and (13), we know impedance is given by Z = R + jXL Ω

Substituting R and XL from (14) and (16),

Z = {ρl

A+ j(2πfl) × (

μ0

π× ln (

D

GMR))} Ω (18)

=> 𝑍 = 𝑙 {ρ

A+ j(2πf) × (

μ0

π× ln (

D

GMR))}Ω

=> 𝑍 ∝ 𝑙 (19) Line losses are given by 𝐿𝑖𝑛𝑒 𝐿𝑜𝑠𝑠 = 𝑖2𝑅 (20) Where i is the current in the line, R is resistance of the line. From (16) it is inferred that

𝑅 ∝ 𝑙 =>𝑙𝑖𝑛𝑒 𝑙𝑜𝑠𝑠 ∝ 𝑙 (21)

D. The Approach

As seen the total cost, losses and impedance vary directly in relation to the length of the distribution line. The increase in length causes an increase in resistance and hence losses. The total cost which is the determined by fixed and variable cost is also increased by increase in line length as the variable costs would leap up drastically. At the same time it also causes an increase in the impedance of the system thus causing a reduction in the power factor of the system.

V. PROPOSED ALGORITHM

The algorithm is discussed stepwise, where each step is

intricately discussed in individual sections starting with an

overview of the algorithm.

A. Overview of The Algorithm

As discussed above the main function of the algorithm is to result in an efficient and economic distribution system. The process begins with the graphic diagram of the system. Every hamlet is considered as a node. A complete graph is then drawn of the said nodes to form a network. The next step is the determination of source nodes and demand nodes. A source node is where a part of the power generating system may be placed, in this case being the solar power system, and a demand node is one where there is load. This is a smaller scale application of distributed generation, which allows flexibility, easier maintenance and future expansion are among the important advantages [14]. Another advantage of distributed sources in the network is reduced current. Consider a simple example to demonstrate this. Assume a node consists of 10 loads and another load of 5 loads as shown in Fig. 1, considering each load draws a current of 1A, we can observe 3 cases:

Figure 1. Representation of the nodes and the current in the line (yellow color represents the source).

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Case1: When the source is placed at the node with 10 loads then the current on the transmission line connecting to the 5 load node would be of only that of 5 loads.

Case2: When it is placed in between then the transmission lines will carry the current by branching out to the two nodes and carry currents of both 10 and 5 loads equalling to 15 load current.

Case3: When placed at the 5 load node where it will have a branch connecting to the 10 load node and carry a current of 10 loads. As seen the loss is least when current and distance is least. Since the distance is fixed, the current is least when the source is at the 10 load node.

Following the same principle the algorithm is used to determine which nodes can be used as a location to place a solar power system near it. This is economically feasible as it doesn’t affect the fixed cost, which is calculated on the basis of total demand and also as it reduces current in the line, thus reducing losses.

Now to reduce the total length of the lines, the graph is reduced by making use of Prim’s algorithm. This will give us the Minimum Spanning Tree (MST). The source nodes are marked on the MST. The branches which connect the demand nodes to more than one source nodes are considered. Let a demand node be connected to more than one source node, now eliminate all branches connecting the demand node to the source nodes except the shortest branch connecting to a source node. Follow the same for all the demand nodes. The end of the process will give the most cost effective and electrically efficient network. The final network can be disconnected graph.The steps of the algorithm are now described in detail stepwise.

B. Step I:Network Graph

For efficient application of the proposed algorithm, proper formation of the network graph is important. If an area consists of h houses distributed among n hamlets, represent each hamlet as a node i.e. a vertex of the network graph. For the sake of simplicity consider Fig. 2(a). Every vertex is joined to every other vertex to form a complete graph. Fig. 2(b) shows the complete graph

The edges have to be assigned with weights. This is done by assigned weights based on length of the edge which is equivalent to the distance between the hamlets connected by the said edge. This leads to a weighted graph, shown in Fig. 2(c). Assuming that Fig. 2(c) represents a regular pentagon,

the sides are equal so they have equal weight and the diagonals are also equal leading to equal weight again.

C. Step II: Source Nodes and Demand Nodes

We must determine the source and demand nodes. For this, consider the function which determines the net fractional load

of a node, let us call it load factor γ. The load factor γ is given by,

γ =Pn

P+ α (22)

where, Pn is the load of the node, P is the total load of the entire system andαis the distributable load factor, which is defined below.

Assume a node n with k houses, since the load per house is same, say w,

load at node n, Pn = k × w (23) Since the network consist of h houses, then total load, P = ℎ × w (24) Let us consider the distributable load factor α,

α = ∑Pi

P

N0 (25)

where, i is a node which lies at a distance less than 250m from the source node n.

Effectively this adds the load fractions of all the nodes which lie at a distance less than 250m from the nth node which is being considered for the grade of source node.

Consider Fig. 3, according to Govt. of India, any cluster of houses in a rural or remote area, called hamlets here, is considered as a separate habitation unless it is located at a distance of more than 200meters from any other hamlet. So the algorithm considers supplying to the clusters coming under the same habitant hamlets with an extended radius of 50meters leading to a distance factor of 250m.

For any node if 𝛾 ≥ 0.3, when approximated to 1 decimal place, it can be considered as a source node, where the solar system can be installed. This effectively applies the distributed generation principle by locating the source of generation at locations that consists of a sub-network which

demands at least 1

3rd of the total demand. All nodes having

𝛾 ≥ 0.3 can be marked as source nodes on the graph. The nodes which are not source nodes are demand nodes.

Figure 2. (a) Representation of hamlets as nodes, (b) formation of

complete graph and (c) a weighted graph based on the length of the

edges.

Position of hamlets

to form nodes i.e.

vertices of the graph

Complete Graph

(a) (b) (c)

Figure 3. Habitation limit of 250m around a node.

n th node

Node included for α

Node not considered

for α

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D. Step III: Branch Elimination

To optimize the cost and losses, the complete graph is

reduced to Minimum Spanning Tree (MST). This done by

either using Prim’s algorithm. With the MST in hand, the

graph has been reduced to minimum weight MST, meaning

least distance. The final steps of optimizing are done by

eliminating unnecessary branches. There are essentially two

steps in this process, first eliminate any branch connecting

two source nodes and finally eliminating repetitive branches,

if a demand node is connected to more than one source node,

then it has repetitive branches, eliminate all the repetitive

branches except the shortest one i.e. consider all the branches

that connect the demand node to the source nodes, keeping

the shortest branch which connects the said demand node to

a source node eliminate all other branches that connect the

demand node to the source nodes leaving behind only the

shortest branch to a source node. Another scenario would be

when two nodes have the same value for load factor, in this

case the node with greater ratio of Pn/P is chosen as the source

node.

After the branch elimination stage the optimized network

is obtained. The micro-grid can now be setup with the source

nodes serving as the location for the solar system. The rating

of the solar power system to be set up at a source node is

given by sum of demands of all the demand branches

connected to the source node and the demand of the source

node itself.

Solar PV system rating at nth node= ∑ {(demand nodes

connected to n) + demand of nth node} (26)

VI. CASE STUDY AND RESULT

To verify the efficacy of the proposed algorithm, a micro-

grid system for Mendil was tested. In this test the total load

was taken on the basis of the general assumptions of lighting

and the number of houses are accurately located and

numbered as per Google Maps and field survey conducted

during the pilot installation of the micro-grid by FluxGen

Engineering Technologies. This case includes an additional

source at node 1 (Fig. 4), which was the pilot installation of

the micro grid. This case study also shows how useful this

algorithm is when such complexities are added.

A. Transmission Network

The algorithm is directly applied to the village. In this

case the network graph is formed by deriving locations of the

hamlets by making use of Google Maps. In this case a pilot

implementation of source has already been done and is

indicated by the yellow color. Further the complete graph is

made, shown in Fig. 5, and then weighted as per the distance

between the hamlets. This is then reduced to MST by

applying Prim’s algorithm, shown in Fig. 6(a). This MST by

prim’s algorithm gives least weight i.e. least total line length

as the edges are weighted in order of their lengths. Once the Minimum spanning tree is formed. The

algorithm follows through with the calculation for determination of source nodes and demand nodes.

As per the calculations the results are shown in Table I. The source nodes are marked and the rest of the algorithm is followed through. See Fig. 6(b), for branch elimination.

Once the branches have been eliminated the final network is obtained. This is optimized network from both electrical and economic stand point. Fig. 6(c) represents the fully optimized network.

Figure 5. Complete graph of the hamlet

network. Figure 4. Representation of hamlets as nodes with

reference to actual locations.

Arrangement of hamlets with

no. of houses

Figure 6. (a) Minimum Spanning Tree after applying Prim’s

algorithm (b) Source nodes and Demand nodes are formed along with elimination of branches (c) Final optimized network

(a) (b) (c)

Cancelled graph

Demand node

Source node

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TABLE I. LOAD FACTOR AND DISTRIBUTABLE LOAD FACTOR.

Node

no.

Pn/P α(distributable

load factor)

γ (load

factor) 1 0.114 0.085 0.2

2 0.085 0.057 0.1

3 0.057 0.256 0.3

4 0.256 0.114 0.4

5 0.029 0.171 0.2

6 0.085 0.171 0.3

7 0.171 0.085 0.3

8 0.085 0.085 0.2

9 0.085 0.114 0.2

B. Comparitive Analysis..

A comparison between different possible network

configurations and the proposed configuration is done by

comparing the cost and losses involved in the implementation

of the network. Let us now consider the arrangement 1 and arrangement2

as shown in Figure 7. We consider the total distribution line length which as per calculations determines the cost as well as losses. The distances between the nodes have been measured here by making use of Google Maps. The total distances for each arrangement is shown in the table 2.

TABLE II. TOTAL DISTANCE OF DISTRIBUTION LINE

Arrangement Distribution line length

Proposed 616.5m

Arrangement 1 1191.4m

Arrangement 2 1835.4m

As seen in the algorithm, the cost of the source is fixed the

variable cost determines its feasibility. The variable cost and the distribution line losses both depend upon the length of the line. Table II clearly shows that the proposed arrangement produces a more superior result compared to the other arrangements by providing minimum distribution line length. With minimum distribution line length it produces a configuration with minimum losses and cost.

VII. CONCLUSION

As Table II portrays how effective the proposed algorithm

is in reducing the distribution line length and hence in

reducing overall cost and losses of the system, it is fair to say

that this algorithm can act as a tool for those who wish to

implement micro grids in rural areas. Furthermore the energy

sharing concept can be deployed which ensures higher

efficient utilization of renewable energy generations systems

and thus forms the future prospects of this work.

As the demand for smart grids and micro grids increase in

the domestic zone, it is safe to say that with certain

modifications one can implement a form of this algorithm to

densely populated areas like the urban areas.

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Figure 7. Proposed arrangement with Arrangements 1 and 2.

Proposed

Arrangement

Arrangement -2

Arrangement -1

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