Keywords

1 Introduction

The complicated structure of the demand–supply, rapidly increasing power consumption, and contemporary electrical gadgets make it difficult to maintain economical and reliable distribution systems. In the competitive power market, the sustainability of the distributed generation (DG) resources is playing key role for the better operation and management. The main objective of a modern power distribution system is to provide quality and uninterrupted power supply to each consumer. Therefore, an effective distribution control system should increase system efficiency overall through loss reduction and management of power quality. A passive distribution method was used in the past, however present distribution n system is having bidirectional power flows due to the integration of renewable energy sources and distributed generation meeting the increasing load demand of various types. However, there are numerous challenges in the competitive regime of distribution systems in terms of the losses, voltage and reactive power requirements and compensation; DGs cost, distributed energy resources and their location, and energy storage facilities, which require fundamental change in the operation and management of the network. Distributed energy resources location and sizing, dispatching of the units, practical load consideration, load growth are key issues that require attention for the distribution system planner.

There are a number of cost-effective solutions available to enhance the operation and performance of distribution networks. These methods include strengthening feeders, rearranging networks, installing dispersed generation, placing reactive power sources, and integration of distributed generation with renewable sources. The planning and operation of distribution system require its efficient operation and management with reduced losses and higher efficiency. With the introduction of the competitive power system structure, it has added the new sources of power generation along with conventional sources in the system for sustainable green energy. The distribution network thus has grown with bidirectional power flow acting as active distribution system. The additional components, increasing load growth, the customers, and electricity providers are affected by distributed generation in terms of voltage profile, power flow, continuity, stability, power supply quality, and short circuit level [1]. To overcome these effects there is a need to address the issues of operation and management with distributed generation having proper location and sizes. Various optimization techniques are being used for optimal placement and sizing of the distributed generation. In certain ways, the distribution network operator has almost no effect on the placement and sizing of DG because the choice to locate it depends on stockholders, the availability of the fuel source, land rights of way, and climatic circumstances [2]. However, these DGs require the proper integration in the network which has impact on the loss’s reduction and voltage profile improvement with optimal sizes. The load pattern and its nature are essential for better management as they have impact on the losses and voltage profile.

Small sources with modular power technologies, ranging from 1 kW to 50 MW, are known as distributed generation. They produce electricity closer to the points of consumption. Conventional and non-conventional both types of distributed generation are available. Solar power, wind power, small hydro, natural gas generators, biogas, biomass, and geothermal power are the non-conventional types, and in conventional types of fuel cells, diesel generators, sterling engines, gas turbines, and internal combustion reciprocating engines are there [3].

Various techniques and approaches have been proposed by different authors to address the issues of optimal sizing and placement of distributed generation considering different load models and different types of DGs. The number, size, and position of multi-distributed generation (multi-DG) units in distribution networks with different load models have been chosen by authors in [4] using a multi-objective index-based Particle Swarm Optimization technique. In [5] voltage-dependent load model of the distribution system has been taken into consideration for the appropriate location and size of distributed generators. A cuckoo search algorithm-based multi-objective index-based technique was implemented in [6] to optimize the DG's size and site under various load scenarios. A soft computing technique has been proposed in [7] for the Optimal sizing and sitting of DG with considering different distribution load models. In [8] Various load models’ effects on distributed generation planning are proposed by considering single and multiple DG units. The optimal positioning and size of DG units considering different load models have been addressed in [9] using a unique multi-objective quasi-oppositional grey wolf optimizer approach. The impacts of load models and load demand in the distribution system when distributed generators are present have been taken into consideration in [10]. Performance of Voltage Step Constraints and Load Models in the Optimal Location and Size of Distributed Generation, using Incremental Power Flow and the Exhaustive Search Approach, [11].

All loads that are taken into account are of constant P, Q loads during the conventional power flow analysis. This presumption is unworkable in the actual operation of a dynamic, complicated power system. Such load modeling may produce contradictory findings and erroneous conclusions, which might result in inaccurate assessments of power loss, cost, deferral values, and other system indices [12]. Loads are often modeled as voltage or frequency dependent in actual power system static or dynamic studies. Loads come in a variety of forms and classifications. Voltage-dependent loads are classified based on the ZIP model as polynomial load model and exponential load model. In the previous proposed works in literature, load models are formulated based on their types and categories as constant, residential, commercial, and industrial loads.

In this paper, the exponential load model (ZIP model) is considered and problem is formulated for the optimal allocation and sizing of DG units. Different nature-based optimization techniques have been utilized to find the optimal DG placement and sizing considering multi-objectives in the objective function. The multi-objective function contains minimization of cost of line losses, real power generated by DG, and voltage deviation. The main contribution of the papers is:

  • Improving the technical, economic, and environmental benefits by integrating different types of DGs into the distribution network.

  • Three different nature-based evolutionary algorithms have been implemented and their result has been compared to get the best optimization among them.

  • The lowering of DG unit generation costs, improvement of bus voltage variation, and reduction of power loss.

  • To research various load models used in a real-world distribution system operation situation.

  • The IEEE 85-BUS test radial distribution system was used to compare the solutions of various load models and types of DGs.

This paper is organized as follows: Sect. 2 introduces the formulation of distribution system load models. The multi-objective function formulation is done in Sect. 3. In a discussion in Sect. 4, the simulation findings for the test system are provided. Finally, the conclusions and references of the suggested work are provided.

2 Formulation of Distribution System Load Model

2.1 Exponential Load Model

The power-voltage relationship at the load bus is represented by exponential equations in the exponential model. These equations are essentially described in the ZIP model, except they contain fewer coefficients. To explain the algebraic connection between active & reactive power with applied voltage V, the exponential load model comprises two coefficients (exponents) termed \({n}_{p}\) and \({n}_{q}.\)

$$\mathrm{\rm P}={\mathrm{\rm P}}_{0}{\left(\frac{\upsilon }{{\upsilon }_{0}}\right)}^{{n}_{p}}$$
(1)
$$\mathrm{Q}={\mathrm{Q}}_{0}{\left(\frac{\upsilon }{{\upsilon }_{0}}\right)}^{{n}_{q}}$$
(2)

where P and Q are the real and reactive power of the actual load, V is the applied voltage of the load bus, \({V}_{0}\) is nominal voltage, \({\mathrm{P}}_{0}\), \({\mathrm{Q}}_{0}\) are the nominal active and nominal reactive power of the load, and \({n}_{p},{n}_{q}\) are exponential model coefficients.

The load behavior is expressed by the exponential models using the exponent’s \({n}_{p}\) and \({n}_{q}\) as follows:

  • Constant Impedance load (CI) when \({n}_{p}\)=\({n}_{q}\)=2

  • Constant Current load (CC) when \({n}_{p}\)=\({n}_{q}\)=1

  • Constant Power load (CP) when \({n}_{p}\)=\({n}_{q}\)=0

2.2 Polynomial Load Model (ZIP)

In practical distribution systems, the actual load is the combination of different nature of loads like constant power, constant current, and constant impedance. Such types of loads can be represented by the ZIP model, also known as the polynomial load model. In order to calculate the actual power (active power), the ZIP model adds Constant Impedance (CI), Constant Current (CC), and Constant Power (CP) to create a polynomial equation that describes the connection between the load power's characteristics and the applied voltage. Equations (3)–(4) mentioned here reflect the ZIP model algebraically:

$$\mathrm{P}={\mathrm{P}}_{0}\left[{\mathrm{a}}_{\mathrm{p}}{\left(\frac{\mathrm{V}}{{\mathrm{V}}_{0}}\right)}^{2}+{\mathrm{b}}_{\mathrm{p}}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{0}}\right)+{\mathrm{c}}_{\mathrm{p}}\right]$$
(3)
$$\mathrm{Q}={\mathrm{Q}}_{0}\left[{\mathrm{a}}_{\mathrm{q}}{\left(\frac{\mathrm{V}}{{\mathrm{V}}_{0}}\right)}^{2}+{\mathrm{b}}_{\mathrm{q}}\left(\frac{\mathrm{V}}{{\mathrm{V}}_{0}}\right)+{\mathrm{c}}_{\mathrm{q}}\right]$$
(4)

where P denotes the load’s real active power demand, Q denotes its actual reactive power demand, V denotes the load's actual voltage at the load bus, \({V}_{0}\) denotes the load's nominal voltage, \({P}_{0}\) denotes the load's nominal active power, and \({Q}_{0}\) denotes the load's nominal reactive power. The ZIP load model parameters for active power are \({\mathrm{a}}_{\mathrm{p}}, {\mathrm{b}}_{\mathrm{p}},\) and \({\mathrm{c}}_{\mathrm{p}}\), and \({\mathrm{a}}_{\mathrm{q}}, {\mathrm{b}}_{\mathrm{q}},\) and \({\mathrm{c}}_{\mathrm{q}}\) are the ZIP load model coefficients for reactive power; where the values of \({\mathrm{a}}_{\mathrm{p}}\) = 0.1, \({\mathrm{b}}_{\mathrm{p}}\) = 0.1, \({\mathrm{c}}_{\mathrm{p}}\) = 0.8, \({\mathrm{a}}_{\mathrm{q}}\) = 0.1, \({\mathrm{b}}_{\mathrm{q}}\) = 0.1, and \({\mathrm{c}}_{\mathrm{q}}\) = 0.8.

3 Mathematical Formulation

Finding the optimal location and size of DG units considering the impact of different distribution load models to reduce the overall cost of the system while taking equality and inequality limitations into consideration is the main goal of the multi-objective problem formulated here.

3.1 Energy Loss Costs (CL)

The annual cost of power loss is given below:

$$\mathrm{CL}=\left(\mathrm{TPRL}\right)*\left(\mathrm{Kp}+\mathrm{Ke}*\mathrm{Lsf}*8760\right)\mathrm{ \$}$$
(5)

The loss factor is presented below as a function of the load factor (Lf).

$$\mathrm{Lsf}=\mathrm{k}*\mathrm{Lf}+\left(1-\mathrm{k}\right)*{\mathrm{Lf}}^{2}$$
(6)

where CL = annual cost of energy losses, TPRL = total real power loss of the system, LSF = loss factor, k = 0.2, Lf = 0.47, Kp = 57.6923 $/kW, and Ke = 0.00961538 $/kWh.

3.2 Cost of the DG for Reactive and Actual Power

$$\mathrm{C}\left({\mathrm{P}}_{\mathrm{DG}}\right)=\mathrm{\alpha }*{\mathrm{P}}_{\mathrm{DG}}^{2}+\mathrm{b}*{\mathrm{P}}_{\mathrm{DG}}+\mathrm{c \$}/\mathrm{MWh}$$
(7)

The following cost coefficients are used: \(\mathrm{\alpha }=0,\mathrm{b}=20,\mathrm{c}=0.25\)

Based on the most complex power that DG can deliver, the cost of reactive power is determined as follows:

$$\mathrm{C}\left({\mathrm{Q}}_{\mathrm{DG}}\right)=\left[\mathrm{C}\left({\mathrm{S}}_{\mathrm{Dg}}\right)-\mathrm{C}\left(\sqrt{{\mathrm{S}}_{\mathrm{DgMAX}}^{2}-{\mathrm{Q}}_{\mathrm{Dg}}^{2}}\right)\right]*\mathrm{k}$$
(8)
$${\mathrm{S}}_{\mathrm{DgMAX}}=\frac{{\mathrm{P}}_{\mathrm{DgMAX}}}{\mathrm{cos\varnothing }}$$
(9)

To conduct the analysis, the power factor has been taken as 0.9 lagging and unity with \({\mathrm{P}}_{\mathrm{DgMAX}}\) = 1.1* \({\mathrm{P}}_{\mathrm{DG}}\). k is between 0.05 and 0.1, 0.1 is used in this study.

3.3 Reflection of Voltage Deviation on Cost

The distribution system voltage may change as a result of the penetration of DG units. The voltage violation should thus be kept to a minimum. The voltage deviation is described as follows.

$${V}_{D}={\sum }_{i=1}^{{N}_{l}}\left|{V}_{i}-{V}_{M}\right|$$
(10)

The following is the formulation of how the voltage variation impacts the cost:

$${C}_{VD}={V}_{D}{W}_{VD}$$
(11)

where \({V}_{D}\) is the voltage deviation at load buses, \({V}_{M}=1.0 p.u\) is the maximum allowable voltage, \({C}_{VD}\) is the cost due to voltage deviation, \({W}_{VD}\) is the economic operator of voltage deviation.

3.4 Multi-objective Function

$$\mathrm{min}F={\alpha }_{1}{CL+\alpha }_{2}\mathrm{C}\left({\mathrm{P}}_{\mathrm{DG}}\right)+{\alpha }_{3}{C}_{VD}$$
(12)

where F is the multi-objective function, CL is the annual cost of line loss, \(\mathrm{C}\left({\mathrm{P}}_{\mathrm{DG}}\right)\) is the total annual DG generation cost, \({C}_{VD}\) is the annual voltage deviation cost, \({\alpha }_{1}\), \({\alpha }_{2}\), and \({\alpha }_{3}\) are the weight factor of the objective function and the values are 0.786, 0.0265, and 0.1875 respectively.

The constraints are:

Voltage limits:

$$ 0.95 \le V_{i} \le 1.06 $$
(13)

Power balance limitation:

without considering DG units:

$${P}_{Grid}={\sum }_{j=1}^{{N}_{l}}{P}_{d}\left(j\right)+{P}_{loss}$$
(14)
$${Q}_{Grid}={\sum }_{j=1}^{{N}_{l}}{Q}_{d}(j)+{Q}_{loss}$$
(15)

with considering DG units

$${P}_{Grid}+{\sum }_{i=1}^{{N}_{DG}}{P}_{DGi}={\sum }_{j=1}^{{N}_{l}}{P}_{d}(j)+{P}_{loss}$$
(16)
$${Q}_{Grid}+{\sum }_{i=1}^{{N}_{DG}}{Q}_{DGi}={\sum }_{j=1}^{{N}_{l}}{Q}_{d}(j)+{Q}_{loss}$$
(17)

Upper and lower limits of DG

$${60 \;\mathrm{KW} \le P}_{DG}\le 3500\;\mathrm{ KW}$$
(18)

4 Simulation Results and Discussion

Figure 1 depicts the IEEE-85 Bus standard 12.66 kV radial distribution system, which contains 85 buses and 84 branches. The one-line diagram of the IEEE test system is shown in Fig. 1. The analysis is carried out for a multi-objective function using metaheuristic technique and the results are obtained and compared with three algorithms. The total load on the system is 2.3788 MVAR and 2.661 MW, respectively are the total load on this test system bus and line data were extracted from [13]. This simulation takes into account the backward-forward sweep approach to carry out the distribution system load flow analyses. The DG’s size and location are obtained for a test system at the given load scenario. The maximum iteration size taken is 100 and the population size is taken as 50. The simulation study of the test system has been carried out using MATLAB 2021a software and an Intel Core I3 10th Gen CPU with 8.0 GB of RAM. Three different optimization techniques, whale optimization from [14], grey wolf optimization from [15], and firefly algorithm from [16], are implemented for the analysis of optimal DG allocation and the results are compared to validate the above multi-objective problem formulation.

Fig. 1
A diagram of the radial distribution system. The numbers from 1 through 85 present the buses and the lines joined through dots present the branches connecting the bus.

One-line diagram of IEEE-85 BUS radial distribution system

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The losses, DG size, minimum voltage, maximum voltage bus, cost of emery loss are obtained for different types of loads at unity power factor is given in Table 1. Comparing the results with different techniques, the firefly algorithm has performed well in three cases and losses obtained are minimal. The voltage profile curve with different load models and different techniques at unity power factor is shown in Fig. 2. The blue line in the graph represents the base case without DG placement and the other three colors represent results with the three DG placement.

Table 1 Comparison of different parameters with different algorithms at unity power factor
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Fig. 2
2 point-to-point line graphs of voltage profile versus bus number for I E E E bus system 85. These plot fluctuating trends for the base case without D G, D G with W O, D G with G W O, and D G with F F. The highest peaks are for D G with W O.figure 2

Voltage profiles for various load models in an 85-BUS system with three DG operating at unity power factor a Constant Power (CP) b Constant Current (CC) c Constant Impedance (CI) d ZIP load

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The optimal DG placement and sizing problem may have a solution with higher sizes of DG units, for power loss reduction, but the cost of DG allocation would rise. Therefore, in order to conduct a realistic feasibility analysis of DG installation at a site, it is required to examine various aspects of cost, power loss, and voltage enhancement at the same time taking a multi-objective case. Maximum three DGs are taken into consideration in this study to obtain the best location and size for DGs. It is observed from Table 1, that grey wolf optimization algorithm gives the lowest DG power generation cost.

In Table 2, the results are obtained with three DG at 0.9 lagging power factor case and are tabulated. The lowest annual line loss cost is obtained with firefly algorithm and the lowest DG generation cost is obtained with Grey Wolf Optimization algorithm. The improved voltage profile curve with 0.9 lagging power factor for different distribution load models taking different heuristic techniques is presented in Fig. 3. The voltage profile is better with DGs for all the cases of loads.

Table 2 Comparison of different parameters with different algorithms at 0.9 lag power factor
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Fig. 3
2 point-to-point line graphs of voltage profile versus bus number for I E E E bus system 85. These plot fluctuating trends for the base case without D G, D G with W O, D G with G W O, and D G with F F. The highest peaks are for D G with W O.figure 3

Voltage profiles of 85-BUS system with three DG at 0.9 lagging power factor for different types of load models a Constant Power (CP) b Constant Current (CC) c Constant Impedance (CI) d ZIP load

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5 Conclusions

This paper presented three different metaheuristic optimization algorithms for optimal DG allocation and sizing considering the effect of different distribution load models. Exponential and polynomial load model of distribution system has been considered in this study. The results of various approaches presented in the paper have been compared for different types of load models. The formulation of a MOF with several different objectives, DG annual generation cost, annual cost of power loss, and voltage deviation cost are determined. Investigations of the results with different load models and their effect on voltage profile are presented. The results have been compared with different algorithms at varying power factor. Whale Optimization, Grey Wolf Optimization, and Firefly Algorithms are compared to loss reduction and DG cost. IEEE-85 BUS radial distribution system has been taken as a case study to validate the above scenarios. Future research can take into account the shifting demand profile taking demand response program and its impact on the operation and management considering the EVs load and storage devices.