ADDITIVE AND DIVISIVE CLUSTER ALGORITHM FOR COST REDUCTION IN UNIT COMMITMENT

This paper presents a novel technique to solve the problem of unit commitment through sorting of units into different clusters. This sorting is implemented in order to decrease the overall operating cost and to assure the various constraints that involve minimum up/down. Unit commitment problem is an important optimization task in daily operational planning of power systems which can be mathematically formulated as a large scale nonlinear mixed-integer minimization problem. A new methodology employing the concept of cluster algorithm called as additive and divisive hierarchical clustering has been employed in order to carry out the technique of unit commitment. Proposed methodology involves two individual algorithms. While the load is increasing, additive cluster algorithm has been employed while divisive cluster algorithm is used when the load is decreasing. The proposed technique is tested on a system with generating units in range of 10-100 and the performance of the proposed technique has been reported through simulation results.


INTRODUCTION
The technique of Unit Commitment (UC) involves the calculation of levels of generation pertaining to generating units and their commitment for a certain interval of time in order to reduce the operational cost.It is actually a significant issue with very good impact on the economics.The generalized problem of UC is well known in the electric industry and its efficient solution definitely has the ability to save huge money each year with respect to costs of fuel and other expenses.It is basically an area that involves the scheduling of production which relates to the calculation of status of the units that are generating power for various intervals of time in order to meet the load and other requirements which are actually dependent on the environmental issues, equipment and the system.Generally the problem of UC is a process that involves making complex decisions and it is tough to design optimization techniques that are capable of solving the system in real time.The various multiple constraints also need to be considered while determining the correct commitment schedule [1].
The most often discussed deterministic mathematical programming techniques include: Branch-and-Bound [2], Dynamic Programming [5,6], Priority List [3,4], Lagrange Relaxation [7][8][9], and Mixed-Integer methods [10].Generally all these mathematical techniques are quite fast and are much simple to be implemented but most of them suffer from the problem of numerical convergence and have the following limitations [1]: (i) They do not guarantee the convergence to optimum point globally for non-convex problems such as UC.(ii) The results also are inconsistent due to the various approximations considered while solving the constraints and objective functions which are nonlinear.(iii) The difficulty in reaching the solution due to the consideration of various constraints.
In order to overcome the general difficulties in various approaches, a novel method with the application of cluster algorithms has been proposed in this paper.The method uses Additive and Divisive Cluster Algorithms.The proposed methodology can be unfolded into three stages.In the first stage, four clusters are formed namely base load, intermittent load, semi-peak load and peak load clusters.All the generating units of the plant are segregated into corresponding clusters based on operating costs.In the second stage, UC solution is obtained by developing Additive Cluster (AC) algorithm for increasing load pattern.Finally in the third stage a Divisive Cluster (DC) algorithm is developed for decreasing load pattern.
The remaining paper is organized as follows: Section 2 deals with problem formulation; General purpose additive cluster and divisive cluster algorithms are discussed in the Section 3. Simulation results and discussions are carried out in Section 4 and finally conclusions are drawn in the Section 5.

PROBLEM FORMULATION
Based on the concept of minimization of the costobjective function in the unit commitment problem, certain units are stated to be as 'ON' and remaining as 'OFF'.The following are the various notations considered during the implementation of the problem N : Quantity of generating units in the system T : Time for which the system is running in hours (h);

Objective Functions
The main objective of the problem of UC is to minimize the Total cost (TC) which consists of various components of FC and SC represented by: Where Fuel cost of i th unit: and Start-up cost

System Constraints
The constraints, that need to be taken into view while performing the process of optimization of UC are listed below.

Demand of Load
All the units that have been committed need to generate the power that is the same as load demand given by:

Spinning Reserve
In order to sustain the reliability of the system during sudden changes of load, the system should be able to contain sufficient amount of spinning reserve capacity.

Power Limits of Generators
The output power of each generator should satisfy the criterion given by:

Ramp up/down Rates
The power generation of various units is limited by the following operating limits which are time dependent

Minimum Up/Down Time
Once the commitment of the unit is done, there should be a minimum time before it is to be de-committed given by.

PROPOSED METHODOLOGY
The technique of Cluster Algorithms (CA) can be stated as, to divide a given group of objects into a number of groups or clusters in order that the objects in a particular cluster would be similar among the objects of the other ones.In the first stage of CA, an attempt is made to place an N object in M clusters according to some criterion additive to clusters.Once the criterion is selected, CA searches the space of all classifications and finds the one that satisfies the optimization function.
The proposed methodology for UC problem considers two clustering techniques: Additive Clustering Technique and Divisive Clustering Technique.In the first type of cluster technique, initially individual data points are treated as clusters.Based on necessary criteria (nearest operating costs of units) successively two closest clusters are merged until there is only one cluster remains.

Basic Additive Clustering Algorithm
Step-1: Compute operating cost (proximity) matrix; Step-2: Repeat; Step-3: Merge two closest clusters based on least distance value; Step-4: Update the proximity Matrix to reflect the proximity between the new cluster and the original clusters; Step-5: Until Only one cluster remains.
In the Divisive type clustering technique, successively each cluster is separated from the others until a singleton cluster of individual point(s) remain.A suitable methodology is required to take the decision on which cluster must be removed from the others.The basic algorithm is given below.

Basic Divisive Clustering Algorithm
Step-1: Compute operating cost (Proximity) Matrix; Step-2: Repeat; Step-3: Separate a cluster from other clusters based on maximum distance value; Step-4: Update the proximity Matrix to reflect the proximity between the clusters those remaining; Step-5: Until all the clusters are removed.
The flowchart for the above methodology can be observed from Fig. 1.
The proposed methodology can be unfolded in to three stages.
 The operating cost of each plant is calculated and the plants are clustered based on their objective function values.In this way best clusters are brought out so that they can be employed to take up the load.

Characterization of Various Units
The units that tend to operate for long periods of the day can be seen as generators relating to Base Load (BL) and Intermittent Loads (IL) which generate maximum amount of power.Ideally, it is to be noted that these units need to have minimum value of fuel cost, maximum capabilities of generation but generally they consist of higher start-up costs and start up times since that are ''ON" during most of the working period.Additionally it can be observed that the reliability of the system is dependent on the performance of these units only.The units that have low start up costs and low start up times can be seen as Semi-Peak Load (SPL) and Peak load (PL) units since they can be brought into ''ON" position and ''OFF" position very frequently.As a result it can be observed that these units have low capabilities of generation and high costs as they are connected to the system for loads which are above base load and intermittent load.Based on the generation cost functions, the closet cost function units are segregated into clusters as BL, IL, SPL and PL as given in Table 5. BL: Load upto800MW duration: 0-24 hours IL: Load between 800MW to 1200 MW, duration 0-18 hours SPL : Load 1200MW to 1400 MW, duration 0-6 hours PL : Load 1400MW to 1500MW, duration 0-3 hours The maximum limits for the four loads as: BL-Max: 1000 MW; IL-Max: 1200 MW; SPL-Max: 1400 Mw and PL-Max: 1500 MW.
For carrying out the additive cluster algorithm, objective function values are stored in ascending order and for divisive cluster algorithm the objective function values are stored in the descending order as given in table 4. The closest values are divided into four clusters as BL, PL, Semi PL and IL.

Design of Additive Clustering (AC) Algorithm
Step-1: Read the load value D(t).Spinning Reserve requirement R(t).Threshold values of four clusters.
Step-2: From the load duration curve, identify the load as any: BL, IL, SPL or PL.
Step-3: Commit the units in corresponding cluster by executing subroutine for Economic Dispatch (ED).
Step-4: Check the constraint: D (t) +R (t) < Cluster Threshold value.If condition is satisfied, go to main program.Else, go to next step.
Step-5: Merge next priority list cluster to previous cluster.

Design of Divisive Clustering (DC) Algorithm for UC Problem
This DC algorithm is proposed for UC when the load is decreasing after it stopped from increasing.The DCA starts at the point where some units in various clusters are already under 'on' condition.Now the requirement is to ISSN 1335-8243 (print) © 2016 FEI TUKE ISSN 1338-3957 (online), www.aei.tuke.skput some units under 'off' condition, so as to meet the present D(t).The priority list is prepared based on the startup time/costs.The strategy is, to put off the unit with maximum generation cost.
Step-1: Read the system load.
Step-2: De-Commit the next unit with maximum generation cost according to priority list.
Step-3: Commit the units in corresponding cluster by executing subroutine for Economic Dispatch (ED).
Step-4: Check the constraint: D(t)+R(t) sum of all generations.If condition is satisfied, go to main program.Else, go to step-2.

RESULTS AND DISCUSSIONS
Table 1 shows the daily load pattern on the plant and Table 2 shows the operating characteristics of all the plants.
The average fuel costs (A) and start-up costs (B) for all the units can be calculated as follows: The Euclidian costs of all the units can be calculated as follows: Euclidian costs of the unit = The above calculations of all the units have been tabulated in Table 3 and Table 4 respectively.Table 5 shows the priority order of various units corresponding to their Euclidian costs with respect to additive clustering and divisive clustering and Table 6 shows the segregation of all the 10 units in order to take up the daily load pattern.For the 20-generating unit system, the data of the ten generating unit system was duplicated and the load data was multiplied by 2. Similar procedure has been implemented for evaluating the data for 40 unit, 60 unit, 80 unit and 100 unit system.Table 7 shows the allocation of generation to various units based on the daily load pattern and based on the clusters.It can be observed from the table that the clusters only take up the load allotted to them while the other generators do not take up the load until it falls into the other category.The operating costs of the generators taking the load can be observed from the table.It can be observed that the technique is quite simple and easy to be implemented.Table 8 also shows the allocation of various units under various loads for large scale system extending from 20 units to 100 units along with their operating cost.

CONCLUSIONS
A novel method based on clustering technique has been proposed to mitigate Unit Commitment problem.The proposed method is more realistic and less heuristic.Following load pattern, two individual algorithms based on Additive and Divisive cluster algorithms are proposed for increasing and decreasing load patterns.The Euclidian cost of generation of units is obtained and based on these costs the units are segregated in to clusters.Two separate priorities lists one for increasing and another for decreasing load conditions are prepared based on generation costs.A thermal system in the range of 10-100 units has been considered for simulation study.The strategy employed proved to be quite effective and satisfactory as evident through simulation results.

Fig. 1
Fig. 1 Methodology of additive and divisive cluster algorithm

Fig. 2
Fig. 2 Proposed Algorithm for UC problem

Table 3
Average fuel cost and start-up cost of each unit

Table 4
Euclidian cost of all units

Table 5
Priority list is formed with minimum operation cost

Table 6
Segregation of 10 units into clusters and their priority

Table 7
Generation of 10 units in 24 hour schedule