Optimal Reactive Power Dispatch Under Load Uncertainty Incorporating Solar Power Using Firefly Algorithm

Abstract

This paper intends a solution towards the optimal reactive power dispatch (ORPD) within power systems incorporating solar power. And also gives an answer to the ORPD problem under load uncertainty for different random loads. The main objectives are to minimize the real power losses and voltage deviations by initiating solar energy source at one of the load bus systems with no change in constraints such as equality and inequality constraints. The reactive power dispatch problem is resolved with the firefly optimization technique. The suggested firefly algorithm is verified on IEEE 14-bus test system.

1 Introduction

An ORPD problem is also a big challenge in power system. ORPD means control the apparatus to optimize reactive power flow to reduces active power, losses and improve voltage quality [1, 2]. There are several challenges faced in power systems in which some of them are voltage instability and optimal reactive power flow. To overcome the operating necessities of a consistent power system is to retain the voltage inside the allowable range [3, 4]. Power balance equations are the equality constraints, and generator voltages, transformer tap settings and shunt capacitors are the inequality constraints, respectively. Minimize the active power loss and voltage deviation through the optimal modifications in control variables of the power system, at the same time fulfilling different constraints detected by the electrical network. This problem requires the finest use of the obtainable voltage magnitude of the generator, tapings of the transformer and the output of capacitor banks to reduce the losses and voltage deviation of the system [5]. The uncertainty of loads and solar power in [6] is simplified by a scenario reduction approach [7, 8].

In power system, economical operation is to run the network at low cost by maintaining reliability and security of the power system, and it also studied the estimation of active power loss. Real power regulation as well as imaginary power dispatch is the goals of the economic operation in power system. The way to regulate real power and distribute the reactive power is to introduce the renewable source. The sustainable sources show a vital role in the electrical power systems. By incorporation of renewable source into power system, the cost of generation is reduced, and system performance is also enhanced [9]. The uncertainty at different solar power outputs and different loading conditions is observed by considering the renewable energy source.

Here, the firefly algorithm is engaged for the solution of the ORPD problem, and it effectively tests on IEEE 14-bus test system. The remaining of this paper listed as follows: The problem formulation is depicting in Sect. 2; the procedure of the firefly algorithm is illustrated in Sect. 3. The simulation outcomes that initiate the firefly algorithm designed for ORPD are exposed in Sect. 4. To end with, the paper conclusions are programmed in Sect. 5.

2 Problem Formulation

2.1 Objective Functions

2.1.1 Real Power Loss

The aim of these function remains minimizing losses of real power (P_loss) in the network is calculated as

$$F_{1} = {\text{Min}}\left\{ {P_{{{\text{Loss}}}} (x,y)} \right\} = \sum\limits_{m = 1}^{{N_{{\text{l}}} }} {P_{{{\text{loss}}}} }$$

(2.1)

where N_l stands transmission line number, and $P_{{{\text{loss}}}}$ represents real power loss usually expressed in MW.

2.1.2 Voltage Deviation

For improvement of the profile of voltage, the additional objective be situated to diminish voltage deviation in any way load buses, and it is calculated as

$$F_{2} = {\text{Min}}\left\{ {V_{{{\text{deviation}}}} (x,y)} \right\} = \sum\limits_{n = 1}^{{N_{{\text{b}}} }} {\left| {V_{n} - V_{{{\text{spec}}}} } \right|}$$

(2.2)

where N_b is quantity of load bus, and V_spec is a pre-specified voltage amount at load bus.

For multi-objective function, above-stated two functions in Eqs. (2.1) and (2.2) are given equal weight such that

$$F_{3} = 0.5*F_{1} + 0.5*F_{2}$$

(2.3)

2.2 Constraints

To minimize the objective functions of the ORPD problem is subjected to the following constraints, one is equality, and another is inequality constraints.

2.2.1 Equality Constraints

The active power and reactive power balance equations are given by

$$P_{Ga} - P_{Da} - V_{a} \sum\limits_{b = 1}^{{{\text{NB}}}} {V_{b} \left[ {G_{ab} \cos \left( {\delta_{ab} } \right) + B_{ab} \sin \left( {\delta_{ab} } \right)} \right] = 0}$$

(2.4)

$$Q_{Ga} - Q_{Da} - V_{a} \sum\limits_{a = 1}^{{{\text{NB}}}} {V_{b} \left[ {G_{ab} \sin \left( {\delta_{ab} } \right) - B_{ab} \cos \left( {\delta_{ab} } \right)} \right] = 0}$$

(2.5)

2.2.2 Inequality Constraints

$$V_{{{\text{Gm}},\min }} \le V_{{{\text{Gm}}}} \le V_{{{\text{Gm}},\max }} ;\,\,m \in N_{{\text{g}}}$$

(2.6)

$$Q_{{{\text{Gm}},\min }} \le Q_{{{\text{Gm}}}} \le Q_{{{\text{Gm}},\max }} ;\,\,m \in N_{{\text{g}}}$$

(2.7)

$$T_{m,\min } \le T_{m} \le T_{m,\max } ;\,\,m \in N_{{\text{t}}}$$

(2.8)

$$Q_{{{\text{cm}},\min }} \le Q_{{{\text{cm}}}} \le Q_{{{\text{cm}},\max }} ;\,\,m \in N_{{\text{c}}}$$

(2.9)

where

N_g:

number of generators,

N_t:

number of transformers,

N_c:

number of capacitors.

2.3 PV Array Output

The calculation of the PV power output requires the some significant electrical characteristics which are open-circuit voltage, short-circuit current and fill factor. The open-circuit voltage is obtained by number of cell connected in series, and short-circuit current is calculated by number of cells connected in parallel. The fill factor is formulated by using the following expression.

$$P_{\max } = {\text{FF}} \times V_{{{\text{oc}}}} \times I_{{{\text{sc}}}}$$

(2.10)

The power output of array can be calculated using the product of number of PV modules arranged in series as well as in parallel, and module output power can be given below

$$P_{{\text{A}}} = N_{{\text{p}}} \times N_{{\text{s}}} \times P_{{\text{M}}}$$

(2.11)

The PV array power output is calculated by using different parameters are given in [10, 11]. In given reference paper, total 20 units are considered to generate 6.967 MW, and every unit is having 10 modules both in series and parallel. The output power of one PV module is 348.35 kW. The total power generated from PV system is given to the load bus 14.

2.4 Uncertainty of Load and Solar

PDF means that probability density function and lognormal probability density are the methods to model the load uncertainty and solar irradiance, respectively. Different loading conditions are considered randomly as 25, 50, 75, 100 and 105%. Solar irradiance is available only 12 h in a day. The generation from solar PV system is zero during night-time. Zero irradiance is set through 50% possibility, and the leftover 50% possibility protects the conditions of solar irradiance generate by lognormal distribution function.

3 Firefly Algorithm

The firefly algorithm (FA) is an advanced optimization technique implemented by Yang in 2008 [12], and FA is motivated by the flashing performance of fireflies. FA is simple, flexible and easy to implement. The three main rules of firefly algorithm are given in [13,14,15]. The light intensity varies according to the inverse square law for a particular distance.

$$I_{0} \propto \frac{1}{{r^{2} }}$$

(3.1)

The brightness $I$ of a firefly at an exacting position x preserve stay selected as $I(x) \propto f(x)$. The distance between the fireflies is directly proportional to the light intensity which is given as

$$I(r) = I_{0} \exp ( - \gamma r^{2} )$$

(3.2)

The attractiveness denoted with β of a firefly is defined through following expression

$$\beta = \beta_{0} \exp ( - \gamma r^{2} )$$

(3.3)

where $\beta_{0}$ is the attractiveness at r = 0.

In general, the values of $\alpha ,\beta ,\gamma$ are considered in between 0 and 1.

The distance between the two fireflies located at m and n is given as

$$r_{mn} = \sqrt {\sum\limits_{k = 1}^{d} {(x_{m,k} - x_{n,k} )^{2} } }$$

(3.4)

The faction of a firefly m is involved to one more attractive firefly, and n is resolute by

$$x_{m} = x_{m} + [\beta_{n} (r)](x_{n} - x_{m} ) + \alpha ({\text{rand}})$$

(3.5)

3.1 Implementation of FA for ORPD

• Step 1: Initialize

• Number of fireflies

• Amount of iterations

• Set the standards of α, β, and γ

• In this, the standards of α, β and γ are considered as 0.2, 0.1 and 1, respectively.

• Step 2: Set the iteration count i = 0 and increase the iteration by i = i + 1.

• Step 3: Determine the fitness result of each firefly by substituting in the objective function stated in Eqs. (2.1), (2.2) and (2.3).

• Step 4: Sort the fireflies depending on their light intensities and for every iteration find the best firefly. Light intensity is varied based on top of the space involving them.

• Step 5: Progress fireflies (control variables) derived from their light intensity.

• Step 6: Continue the process until stopping criteria is reached.

4 Simulation Results

IEEE 14-bus system contains five generator buses in which bus number 1 represented as reference bus, and remaining buses stay generator buses. In between buses, 20 branches are connected and also have three tap-changing transformers. The real power output of solar PV system is additional to load bus 14, without changing limits of the control variables. Total generation from PV system is 6.967 MW, and it is given to the load bus, i.e. at the 14th bus. In the 14-bus system, the control variables are 11, after including PV array at the 14th bus and their restrictions are scheduled in Table 1.

Table 1 Limits of control variables

Uncertainty of Load and Solar Power Output

Initially, the load data is calculated as some particular value in percentages, and similarly, the availability of solar PV power is also divided in percentages as 0%, 25%, 50%, 75% and 100%, respectively. For each percentage of solar power output, different loading conditions like 25, 50, 75, 100 and 105% are given a total of 25 scenarios. For each time, the firefly optimization is applied and tabulated the optimal values of objective functions which are loss minimization and minimization of deviancy of voltage. The minimization of real power losses for random loads and solar power (PV) output is shown in Table 2, minimization of voltage deviation for random loads and solar power output is presented in Table 3, and minimization of multi-objective function for random loads and solar power output is given in Table 4. From the tabular columns 2, 3 and 4, it is observed that for different cases of objectives, the values are low after incorporating full solar power into the system compared to other cases. After increasing load, also the increase in losses is less after incorporating solar power into the system.

Table 2 Minimization of losses for random loads and solar power output

Table 3 Minimization of voltage deviation for random loads and solar power output

Table 4 Minimization of multi-objective function for random loads and solar power output

Case 1—Minimizing Real Power Loss

The first and foremost objective is to minimize the total transmission line active power losses, and the main goal of dispatch problem is reducing loss in transmission line. In this case, we only consider the real power loss minimization as objective. The firefly algorithm is run for both with solar and without solar and tabulated the parameters represented in Table 5. Optimal values of control variables corresponding to different objectives and its meeting characteristics are exposed in Fig. 1.

Table 5 Parameters of control variables and power loss

Fig. 1

Convergence characteristics of losses

The average value along with standard deviation value is calculated for the given IEEE 14-bus system is exposed in Table 6.

Table 6 Mean and standard deviation

Case 2—Minimizing Voltage Deviation

The control variables here are tuning such so as to the whole voltage deviation is minimized. The voltage deviations are listed in Table 7 with its convergence characteristics are shown in Fig. 2.

Table 7 Optimal values of bus voltage deviation

Fig. 2

Convergence characteristics of voltage deviation

Case 3—Multi-objective Function

Nothing like the two earlier cases, this case reflects equally both objective functions given which are real power loss then voltage deviation all at once. This technique is utmost appropriate for optimizing the all parameters of reactive power involved. FA performs excellently in optimizing equally the both objective functions.

From Table 8, it is clear that real power loss is 14.381 MW, and the voltage deviation is 0.06533 p.u. Since equal weight is given to both the objectives, the optimal value is shown in Fig. 3. Comparison of results for three different cases without and with solar is presented in Table 9.

Table 8 Optimal values of multi-objective function

Fig. 3

Convergence characteristics of multi-objective function

Table 9 Comparison of results

Optimal value = 0.5 * 0.14381 + 0.5 * 0.06533 = 0.10457 p.u.

5 Conclusion

In this paper, the objective functions aimed at the minimizing real power losses; also voltage deviation is performed on IEEE 14-bus system using FA. The algorithm gives the best results for both with solar and without solar in three cases. When comparing the results, losses are less when solar power is added at load bus. Uncertainty at different loading conditions and different solar power outputs also gives better results when integrated with solar power. Therefore, the access of further sustainable power in network produces less loss in ORPD problem.