Power loss minimization in the distribution system is becoming very important. Distribution generators are optimally placed in the system to mitigate losses. But the cost incurred in the installation of extra generators can also increase the total cost that may be more than cost due to losses. So a trade-off must be there in installing the extra generator so that losses decreases, as well as cost, also reduces.
If extra generators are installed on each bus in the system, it will definitely increase the cost and losses cost will be less than that. So it is necessary to choose some potential buses at which generators can be placed. For this purpose, the sensitivity index is calculated in our work. It will arrange all buses in the decreasing order of power losses. Then as per the cost minimization and losses minimization, a number of buses which art at top of the list can be chosen and generators will be installed only on that. The detailed mathematical formulation of sensitivity index is discussed in section 1. after locating the potential buses which are having high loss, the problem of placing the right size generator is rising face. Optimal sizing of the generator reduces cost and losses significantly. For this purpose, we have used an objective function which considers the losses minimized as well as cost minimized for the purposed case. It has been divided into two part: technical constraint, economical constraint. The description of it is provided in section 2. This objective function is minimized using firefly optimization and compared with particle swarm optimization (PSO). The theoretical approach of these algorithms is must be studied in their research papers.
When there was no noise on the process yields, the performance of both algorithms PSO and FFA seem to be not so different to approach to the optimum. FFA tends to be better, especially on the functions having multi-peaks. Complexity or difficulty level of the functions had no effect to the FFA as expected except on the Camelback function. However, execute time in each replication is dramatically higher when they are compared, especially on the functions having a curved ridge or mixed curved ridge and multi-peak. PSO seems to be better in terms of speed of convergence. This might be due to the effect of generating completely different random numbers to be used in the iterative procedures of the algorithm. This implies that the FFA is potentially more powerful in solving noisy non-linear optimization problems. The FFA seems to be a favorable optimization tool in part due to the effect of the attractiveness function which is unique to the firefly behavior. The FFA not only includes the self improving process with the current space, but it also includes the improvement among its own space from the previous stages. Also, Firefly is better than PSO in terms of the time taken for the optimum or near optimum value to be generated provided a certain high level of noise
So we here used firefly optimization for power loss minimization which is good for multi objective optimization. Care has to be taken that same capacity DG can’t be connected on a single node. However multiple DG can be connected on a single node but these should be of different capacity. The flow chart of the proposed work is given in section 3.
1 Loss Sensitivity Analysis
Sensitivity factor method is based on the principle of linearization of original nonlinear equation around the initial operating point, which helps to reduce number of solution space. Loss sensitivity factor method is mainly used to solve the capacitor allocation problem. Its application in DG location is new in this field and has been reported in . The real power loss in the system is given by “exact loss” formula. The sensitivity factor of real power loss with respect to real power injection is obtained by differentiating exact loss formula with respect to real power injection at bus Pi which is given by:
Sensitivity factors are evaluated at each bus, firstly by using the values obtained at base case load flows. The buses are ranked in descending order of the values of sensitivity factors to form a priority list. The total power loss against injected power is a parabolic function and at minimum of losses, the rate of change of real power loss with respect to real power injection becomes zero.
which follows that,
where Pi represents the real power injection at node i, which is the difference between real power generation and real power demand at that node.
where PDGi is the real power injection from DG placed at node i, PDi is the load demand at node i, combining above equations we get
The above equation determines the size of the DG at which the losses are minimum. By arranging the list in ascending order, the bus stood in the top is ranked as the first location of DG and further the process is repeated by placing the concerned size of DG at that particular location which generates the next location of DG. The process is said to be terminated when it determines the same location.
1.1 Bus Voltage Sensitivity Analysis
Another method for reducing the search space is bus voltage sensitivity analysis. In this case each bus is penetrated at a time. After putting DG at each node its voltage sensitivity index can be calculated by equation. When DG is connected at bus I, voltage sensitivity index for bus i is given by:
where Vk is the voltage at kth node and N is the number of nodes. The node with the least BVSI will be chosen for DG placement.
2 Objective function
2.1 Technical constraint of Objective function
The objective of the optimal size and location of DG problem to minimize the total power loss and voltage profile can be expressed as:
The objective function is subjected to the following constraints.
- Bus voltage limits:
It is well known that a small change in nodal voltage affects the flow of reactive power whereas active power practically does not change. Further, the operating voltage at each node must be in safety range as given below.
Vi = voltage at ith node.
Nb = number of buses.
- Feeder capacity limits:
Power flow in each branch must be less than or equal to its maximum capacity as given below.
- Power flow equations:
Total active power generation must be equal to the sum of total active power losses and total active load. Similarly, total reactive power generation must be equal to the sum of total reactive power losses and total reactive load as given by the following equations.
PL = Total active power loss.
QL = Total reactive power loss.
2.3 Economical Constraint of Objective function
With a technical description of the objective function, the cost incurred in the installation is also considered in the form of economical constraint. The objective function for it is given as:
Combining the objective functions described in above two sections, we minimize the losses and cost too. The final objective function in our case is
The flow chart of the process is shown as
This code is for the IEEE14 bus system, though with minor modification it will work for almost all benchmark IEEE system. The power loss minimization in radial bus system needs a different approach than this. We have the MATLAB code to reduce the power losses by optimizing the tie switch positions and optimal positions of DGs in IEEE33 radial bus system. The details and code for this can be checked here: