Solution of Unit Commitment Problem using Stochastic Algorithm

Solution of Unit Commitment Problem using Stochastic Algorithm

Denora S

Electrical and Electronics Engineering St.Josephs College of Engineering Chennai, India

Dr.Ramesh Babu M

Electrical and Electronics Engineering St.Josephs College of Engineering Chennai, India

Abstract In order to solve the unit commitment problem, the new method presented in this paper generates all conceivable logical states for generating units for each hour of the day. Every day's load demand is coded as an integer in the scheduling variables. The Particle Swarm Optimization (PSO) method is used to address and optimize the unit commitment problem by taking into account both production cost and transient cost. The solution to the issue must consider system constraints as well as generator constraints, which include minimum up/down times and maximum/minimum generation for each generating unit (such as reserve capacity). The proposed algorithm is explained in this paper and is used with various thermal and wind units. The outcomes were tallied and contrasted with thermal unit outcomes.

KeywordsPSO, Transient cost, Unit commitment.

INTRODUCTION

The objective of unit commitment (UC), a non-linear complex mixed-integer optimization problem, is to distribute the entire demand of the test system across all generating units at the lowest operating cost, which includes both production cost and transition cost, while also satisfying all period-specific constraints, total load, system losses, and reserve requirements. The UC problem must first ascertain the on/off status of each producing unit at each hour of the planning period in order to calculate how demand and reserve capacity should be distributed among the committed units. UC has the most crucial role in maintaining the power networks. With an increase in producing units, the UC problems become exponentially more complicated, making it more challenging to solve them for power systems.

There have been several methods suggested for resolving the UC issue with the least amount of running expense, which will increase the power system operator's potential savings. However, they vary in terms of computational effectiveness and solution quality. These techniques are divided into deterministic and stochastic search algorithms. Dynamic programming (DP), modified dynamic programming (MDP), improved lagrangian relaxation (ILR), lagrangian relaxation differential evolution (LRDE), branch and bound methods (B&B), and lagrangian relaxation (LR) are examples of deterministic approaches. These techniques work quickly, precisely, and simple to solve power systems of average size. Convergence, solution quality, and complexity are problems for them. The heuristic or stochastic search algorithms, such as Genetic Algorithms, Tabu Search, Effective Hybrid

Particle Swarm Optimization (EHPSO), Discrete PSO (DPSO), Hybrid PSO (HPSO), Fuzzy Adaptive PSO (FAPSO), Evolutionary Programming, Simulated Annealing, Ant Colony Optimization, Mixed integer PSO (MIPSO), Multi-objective PSO. Several hybrid algorithms are also suggested using the two types of algorithms mentioned above. These techniques can manage challenging linear and nonlinear constraints and deliver excellently optimized results. However, the accuracy problem afflicts all these methods. The increased problem size and number of generation units negatively effects the computational time and the quality of the solutions.

In this paper, a novel approach is suggested by generating all possible states of each unit for each particle at each time step. The power system operator can make excellent profits by using PSO to optimize these particle states as opposed to using any of the other methods mentioned above.

PROBLEM FORMULATION

The primary goal of the UC problem is to reduce overall operating costs, which are comprised of the costs of production, startup, and shutdown. This function can be optimized by considering all generator constraints and system constrains.

1. Production cost

   The main component of the objective function of UC problem is to minimize the total production cost subjected to set of generator constraints over the scheduling period. The generator power output, Pi, and the production cost, PCi, for unit i at any given time are quadratic functions.

   (1)

   Where ai, bi and ci are the cost coefficients of unit i.

   Pi is the MW generated of committed unit i.

2. Start-up cost

The second component of the objective function is start-up cost. It depends on the OFF time period TOFF. The start-up cost can be calculated by two methods, they are exponential start-up cost and cold/hot start-up cost. If cold start time is less than OFF time period TOFF, then start- up cost taken as hot-start cost else it taken as cold start cost. The start-up cost SCi at any given time period t, is

given by equ.(2),

Where CTi is cold-start time, INSi is the initial status of unit i, HSCi is the hot start-up cost, CSCi is the cold start-up cost, i is the cooling time constant, and Di(off) is the off time before unit i get committed.

Power balance constraint

The power balance constraints ensure that power load in each time slice is satisfied by the sum of power generation from all types of generation units.

1. Generator Boundary Constraint

   The committed generators must operate between its upper and lower boundary limits as given here,

   (8)

2. Minimum up/down time Constraint

The generators require minimum time to start from the cooling period and to shut down from the running condition as given in Eq. (9)

(9)

where MDTi MUTi is the minimum down/up time limits for the ith unit in hours, and Ton is the time at which the unit has been turned on before the hour. The value of TonToff is expressed as,

H. Ramp rate constraint

(10)

(5)

where PD,t, and PL,t are the total system demand and the losses at hour t in MW.

1. Spinning Reserve Constraint

   The spinning reserve is the amount of unutilized capacity in online energy assets that can make up for power outages or frequency fluctuations during a specific time period. For big synchronous generators, the spinning reserve is a traditional idea

   . (6)

   where Pmax is the upper bound limit of the ith generator, and PSR is the spinning reserve at time t.

2. Prohibited Operating Zone (POZ)

   The generators cannot generate real power in certain operating zones due to mechanical stress or sub synchronous oscillations leading to complete shut-down of the unit. These zones are called as Prohibited Operating Zones, causing discontinuities in the fuel-cost curve. During real-time, generators are restricted in POZ. The realistic operating zones of a generator can be described as follows,

   (7)

   m=2,3,, when, =1

   i= 1,2,..,

   where Pl, Pu are the lower and upper bound limits of the ith generator in the prohibited operating zones, pozi is the number of prohibited operating zones of the ith generator, and npoz is the number of units having prohibited zone.

   The ramp up/down limit of a generator is mathematically given as,

   (11)

   SOLUTION USING PSO

   1. Particle Formation

      This algorithm creates logical states for each particle in order to solve the UC problem. In order to express the on/off status of the generators at each hour of the scheduling period T, Particle uses logical state strings. Maximum 2^n logical states with 1/0 as the numbers are possible for each particle. The unit's ON and OFF state are represented by 1 and 0, respectively. PSO approach to optimize UC,

      Where 0 < k < 1 and

      RESULT AND DISCUSSION

      1. Test case-1

In the test case-1, a 10-unit system that contains 10 generators and different loads at every hour of a day is considered for implementation. The test data and load demand of the 10-unit system is given below in Table 1 and Table 2.

Table 1: Test data of 10-unit system

The Table 1 provides the test data for a 10-unit system that helps to solve the UC problem.

Table 2: Load data of 10-unit system

The Table 2 provides information of hourly load demand that should be matched with the output of the 10-unit system.

Table 3: Comparison of total cost of 10-unit system

TOTAL COST ($)	PSO
BEST	566136
AVERAGE (25 TRIAL CASE)	569687.2
WORST	575760

The Table 3 gives a comparative results of best cost, average cost and worst cost obtained for 25 trial cases using the 10-unit system data.

Fig 1: 10-Unit System GBest vs Iteration of PSO

The Fig 1 shows the graph plotted between Gbest and iterations that provides an information on the best data obtained through several iterations.

Table 4: ON / OFF Time period of 10-unit system

The Table 4 provides an hourly power output from each unit system and the total cost by each unit system.

Fig 2: Load demand of 10-unit system

The Fig 2 shows a comparative graph between the load demand and a constraint applied load demand (spinning reserve).

Fig 3: Total cost obtained in 10-unit system

The Fig 3 gives an information about the total cost obtained for each iteration.

Fig 4: Output power in 10-unit system

Fig 4 shows the statistical data of hourly output power from 10-unit system.

B. Test case-2

In the test case-2, a 26-unit system that contains 26 generators and different loads at every hour of a day is considered for implementation. The test data and load demand of the 26-unit system is given below in Table 5 and Table 6.

Table 5: Test data of 26-unit system

The Table 5 provides the test data for a 26-unit system that helps to solve the UC problem.

Table 6: Load data of 26-unit system

The Table 6 provides information of hourly load demand that should be matched with the output of the 26-unit system.

Table 7: Comparison of total cost of 26-unit system

TOTAL COST ($)	PSO
BEST	312432
AVERAGE (25 TRIAL CASE)	345536
WORST	354733

The Table 7 gives a comparative results of best cost, average cost and worst cost obtained for 25 trial cases using the 26-unit system data.

Fig 5- 26-UNIT SYSTEM GBest vs Iteration of PSO The Fig 5 shows the graph plotted between Gbest

and iterations that provides an information on the best data obtained through several iterations.

TABLE 8: ON/OFF TIME PERIOD OF 26-UNIT SYSTEM

The Table 8 provides an hourly power output from each unit system and the total cost by each unit system.

Fig 6- LOAD DEMAND OF 26-UNIT SYSTEM

The Fig 6 shows a comparative graph between the load demand and a constraint applied load demand (spinning reserve).

Fig 7: TOTAL COST OBTAINED IN 26-UNIT SYSTEM

The Fig 7 gives an information about the total cost obtained for each iteration.

Fig 8: OUTPUT POWER OBTAINED IN 26-UNIT SYSTEM

Fig 8 shows the statistical data of hourly output power from 26-unit system.

CONCLUSION

Algorithms that might efficiently deliver the greatest outcomes in terms of manufacturing cost and start-up cost are needed to solve the hard challenge of UC. When compared to other findings, the suggested methodology's optimal solution properties produce better UC outcomes, which are tabulated. A recently proposed population-based stochastic optimisation approach for distinct state particle generation is called logical state particle swarm optimisation. For some difficult issues, such as UC in actual power systems, PSO has comparable or even better search performance when compared to other stochastic optimisation techniques. Additionally, by employing unique convergence values that can help the particles meet the equality demand restriction and get rid of the extra reserve allocation, the convergence behaviour could be sped up. According to current research, the standard PSO should be modified in order to boost variety and improve convergence, much like our new method does. As a result, the algorithm is able to explore the search area quickly and produce high-quality solutions. By taking into account the wind energy factors, the suggested algorithm can be further adjusted, creating a stochastic unit commitment problem. A proposed approach for the unit commitment problem in the current system makes use of a number of restrictions. We can create a stochastic unit commitment dilemma by including a second variable source (renewable energy – wind (or) solar). The suggested algorithm can be applied to a deregulated power system via the construction of a stochastic unit commitment problem. In a market that has been deregulated, the utility is in charge of managing distribution, maintaining cables and poles, and billing customers for these services. Retail electricity providers, or REPs, are companies that deliver electricity to customers in a deregulated electricity market. (the supply of electricity).

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