Crisscross Optimization Algorithm and Monte Carlo Simulation for Solving Optimal Distributed Generation Allocation Problem

: Distributed generation (DG) systems are integral parts in future distribution networks. In this paper, a novel approach integrating crisscross optimization algorithm and Monte Carlo simulation (CSO-MCS) is implemented to solve the optimal DG allocation (ODGA) problem. The feature of applying CSO to address the ODGA problem lies in three interacting operators, namely horizontal crossover, vertical crossover and competitive operator. The horizontal crossover can search new solutions in a hypercube space with a larger probability while in the periphery of each hypercube with a decreasing probability. The vertical crossover can effectively facilitate those stagnant dimensions of a population to escape from premature convergence. The competitive operator allows the crisscross search to always maintain in a historical best position to quicken the converge rate. It is the combination of the double search strategies and competitive mechanism that enables CSO signiﬁcant advantage in convergence speed and accuracy. Moreover, to deal with system uncertainties such as the output power of wind turbine and photovoltaic generators, an MCS-based method is adopted to solve the probabilistic power ﬂow. The effectiveness of the CSO-MCS method is validated on the typical 33-bus and 69-bus test system, and results substantiate the suitability of CSO-MCS for multi-objective ODGA problem.


Introduction
In recent years, environmental concerns, fossil fuel resource depletion and advances in technology have resulted in the increase of distributed generation (DG) in distribution networks [1,2].Reasonable application of DGs can bring many advantages, such as voltage profile improvement and pollutant emission reduction [3][4][5].However, inappropriate allocation of DGs may also lead to voltage fluctuations and system instability due to the uncertain nature of renewable resources [6,7].It is crucial to develop proper models and methodologies to identify the optimal allocation of DGs, the aim of which is to determine the best types, locations and sizes of DGs taking into account system uncertainties.
The optimal DG allocation (ODGA) problem in distribution networks has been investigated in the literature from different perspectives.In [8], an optimization model was proposed for optimizing DG allocation problem, but it mainly focused on total planning cost minimization.In [9], four indices including real and reactive power loss, voltage profile and feeder capacity were considered for the ODGA problem in distribution systems with different load models.The optimization model Energies 2015, 8, 13641-13659

Modeling of Wind Turbines (WT) Generation
Many experiments have demonstrated that the stochastic behavior of wind speed in most regions approximately follows the Weibull distribution [17]; the probability density function (PDF) of wind speed v can be expressed as: where k and c are the shape index and the scale index of Weibull distribution.
The output power of wind turbines varies with the wind speed, and its models can be formulated as: where P WT_n is rated power; v is the wind speed; v ci , v co and v n are the cut-in wind speed, cut-out wind speed and rated wind speed.

Modeling of Photovoltaic (PV) Generation
The solar illumination intensity is considered as the dominant factor affecting the PV output power, and it approximately follows the Beta distribution with a PDF as Equation (3) [20]: where r max represents the maximum intensity of sunlight in a certain period of time; and α and β are the shape parameters of the Beta distribution, respectively.The output power of PV is related to the solar illumination intensity, their relationship can be described as: P s ptq " P stc rptq r stc r1 `kpTptq ´Tstc qs (4) where P s (t) is the output power of photovoltaic under illumination intensity t; T(t) is the surface temperature of the photovoltaic array; r stc , T stc and P stc are the intensity of sunlight, the surface temperature of the photovoltaic array and the maximum output power in the standard testing environment (1 kW/m 2 ), 25 ˝C; k is the temperature coefficient.

Modeling of Load Data
The load P L is assumed to be a random variable that follows the normal distribution [17].The probability density function of P L is given by the following expression: where µ P is the mean value; and σ P is the standard deviation.

Formulation of the ODGA Problem
A multi-objective ODGA model considering environmental benefit, total DG cost and power loss is established under the chance constrained programming (CCP) framework.In this section, the objective functions and constraints are explained.

Pollutant Emissions
DG allocation problem considering environmental benefits can achieve the best balance between economic and environmental benefits.The pollutant emissions of the distribution system mainly consist of two parts, namely, the pollutant emissions from thermal power units and the pollutant emissions from of DG units.An environmental cost index is given as follows: S DG,n ˆp f DG,n ˆTDG,n `Nsub ÿ k"1 P sub,n ˆTsub,n q ˆK ÿ i"1 Q n,i pV n,i `PN n,i qs (6) where 1/(1+η) b represents the present value of the annual cost considering discount rate η in the bth year; T is the total number of DG planning years; S DG,n and pf DG,n are the installed capacity and power factor of the nth DG; P sub,n is the power of nth thermal power unit; T DG,n and T sub,n are the equivalent generation hours of DG and thermal power generation in one year; N DG and N sub are the total number of DGs and substations in the distribution network; Q n,i is the amount of the ith pollutant emissions from the nth generator per active power (kg/kWh); and V n,i and PN n,i are the environmental treatment cost and penalty fee of the ith pollutant ($/kg), respectively [21]; K is the types of pollutant.

Total DG Cost
The total DG cost factor including the investment cost C I DG , operating and maintenance (O&M) cost C OM DG along the planning period is formulated as the expression given in Equation (7).
where C I DGk and C OM DGk are the per-unit investment cost and operation and maintenance cost of the kth type of DG, respectively; P DGik is the installed capacity of the kth type of DG in candidate location i; T DGik is the equivalent generation hours of the kth type of DG in candidate location i in one year; N DG is the set of candidate buses for installing DG; N type is the number of types of DGs.
Due to the uncertain output power of renewable DGs, it is not an easy task to accurately evaluate the O&M costs.In this paper, the O&M costs are equivalent to the cost for per unit of electricity generated (i.e., $/kWh) as [22] and [23] did.

System Power Losses
This objective is to minimize the system power losses of distribution network.The power losses cost can be expressed as follows: where, p r is the electricity price; T max is the equivalent time of load consumption (i.e., annual maximum load utilization hours), which expression can refer to [24]; I n is the current corresponding to the annual maximum load utilization hours of the nth branch; R n is the resistance of the nth branch; N branch is the number of branches.

Objective Function
The cost of ODGA problem is considered in one unique objective function that formulated as below: min F " µC ems `λC DG `ξC loss (9) Energies 2015, 8, 13641-13659 where C ems is the environmental cost; C DG is the total DG cost; C loss is the power losses cost; and µ, λ and ξ are weighting coefficients for each sub objective.Due to the multi-objective feature of Equation ( 9), a method named analytic hierarchy process (AHP) is employed to determine the weighting coefficients as [17] did.The AHP method was first proposed by T. Saaty in 1970, and now it has gradually become an algorithm with extensive applications in multi-objective comprehensive evaluations [17].When determining the weight coefficients, experts from related fields should be invited to score for sub objectives, and then the weight coefficients can be by the AHP method.
In some developing countries, DG technologies are still in the primary stage of development.Because of the immature technology, high costs or specific power market system, either the social investors or private investors have high barriers to participate in the field of DG at present or even some time in the future.Therefore, in this paper, proceeding from the realities in some developing countries, we studied the optimization problem from the view point of power grid companies.However, the established ODGA model can also be used for other stockholders through adjusting the weight coefficients of the three sub objectives in Equation ( 9) using the AHP method.

Deterministic Constraints
The deterministic constraints are the load-flow equations and the DG capacity constraints that showed in Equation (10).
where P Gi and Q Gi are the active and reactive output power of the generators at node i; P Li and Q Li are the active and reactive load power at node i; V i and V k are the voltage amplitudes at node i and node k; G ik and B ik are the conductance and susceptance between node i and node k; δ ik is the voltage angle between node i and node k.P DGimax and P DGimin are the upper and lower limits of active output power of DGs; Q DGimax and Q DGimin are the upper and lower limits of reactive output power of DGs.P Li is the active load power at node i, PEN max is the maximum allowable penetration of DG; and N DG and N bus are the number of installed DGs and buses of network, respectively.

Chance Constraints
Chance constraint programming (CCP) is a stochastic programming method that suitable for solving optimization problem with uncertainties.More details about CCP can refer to [17,25].In the CCP optimization framework, some constraints are allowed to be satisfied within a certain confidence level when uncertain nature of renewable DGs are considered.The chance constraints of node voltage and the permissible power flow in this work are given as: where, α and β are the confidence level of permissible voltage and power flow under the CCP framework.V min and V max are the the upper and lower limits of voltage amplitude; S ij and S ijmax Energies 2015, 8, 13641-13659 are the power flow and the permitted maximal power flow limit in the feeder between node i and node j, respectively.

Proposed CSO-MCS Solution Approach
In this paper, an approach that embedded crisscross optimization algorithm with Monte Carlo simulation (CSO-MCS) is developed to find the best solution for the ODGA problem under uncertainties.In the solution approach, the CSO algorithm is dedicated to search the best solution among a number of feasible solutions, and the MCS-based probabilistic power flow (PPF) method is used to deal with the system uncertainties and solve the power flow.

Determination of Feasible Solutions Using CSO
The crisscross optimization algorithm (CSO) is a recent evolutionary algorithm inspired by Confucian doctrine of gold mean and the crossover operation of genetic algorithm [26].Compared to other heuristic algorithms, the CSO algorithm has a huge advantage in solution accuracy and convergence speed when addressing complex optimization problems [26,27].The CSO algorithm is mainly made of three components: the horizontal crossover, the vertical crossover and the competitive operator.The horizontal crossover and vertical crossover execute within each iteration and reproduce their offspring solutions called moderation solutions by performing different crossover operations.Meanwhile, after each crossover operation, the competitive operator should be carried out to choose the better particle.Only those new solutions that outperform their parent particles can survive while the others will be eliminated in the competition.It is the combination of the double search strategies and competitive mechanism that enables CSO significant advantages in convergence speed and solution accuracy.The detailed introduction of the three components is described in the following subsections.

Horizontal Crossover
The horizontal crossover is used to generate the moderation solutions by executing a special arithmetic crossover operation on all the dimensions between two different particles so as to enlarge searching scopes.Suppose the ith parent particle X(i) and the jth parent particle X(j) are chosen to perform the horizontal crossover operation at the dth dimension, their offspring, namely the moderation solution, can be reproduced through the equations as shown in Equation (12).

#
MS hc pi, dq " r 1 ˆXpi, dq `p1 ´r1 q ˆXpj, dq `c1 ˆpXpi, dq ´Xpj, dqq MS hc pj, dq " r 2 ˆXpj, dq `p1 ´r2 q ˆXpi, dq `c2 ˆpXpj, dq ´Xpi, dqq where X(i,d) and X(j,d) are the dth dimension of two parent particle X(i) and X(j); r 1 and r 2 are random values uniformly distributed between 0 and 1; c 1 and c 2 are expansion coefficients uniformly distributed between ´1 and 1; MS hc (i,d) and MS hc (j,d) are the offspring of X(i,d) and X(j,d), respectively.
In the process of the horizontal crossover, all particles of CSO are randomly divided into M/2 pairs.Each pair of particles reproduces two moderation solutions according to Equation (12).Through the horizontal crossover operator, new solutions can be searched in a hypercube space with a larger probability while in the periphery of each hypercube with a decreasing probability, which had been proved in [26].This search mechanism gifts CSO with powerful global search ability.After horizontal crossover search, the moderation solutions should compete with their corresponding parent particles.The winners will save in the updated population and serve as the parent population of vertical crossover.

Vertical Crossover
The vertical crossover search is an arithmetic crossover operated on all the particles between two different dimensions.Suppose the d 1 th and d 2 th dimensions of the particle i (i.e., X(i,d 1 ) and X(i,d 2 )) are used to carry out the vertical crossover operation, the d 1 th dimension offspring of particle i (i.e., MS vc (i,d 1 )) can be reproduced by Equation (13).
where r is a uniformly distributed random value between 0 and 1; M is the population size; D is the number of dimensions.
In the process of vertical crossover, all particles are randomly divided into D/2 pairs.The role of vertical crossover search is to facilitate some stagnant dimensions of the population and escape from dimension premature convergence.Unlike horizontal crossover, only a few paired dimensions are chosen to perform the vertical crossover operation because that about 10%-40% of the dimensions in the swarm may be simultaneously trapped into stagnancy when optimizing the multimodal functions [26].Therefore, the vertical crossover probability P vc is suggested to be set in the range [0.2, 0.8] in [26].Similar to the horizontal crossover search, the new solutions should compete with the parent particle and those solutions that outperform their competitor can survive.

Competitive Operator
In CSO, a greedy selection strategy that is similar to the differential evolutionary algorithm is used to update the population, as expressed in Equation ( 14).
Every time when the crossover operation is finished, the competitive operator need to be executed to choose the particle that has better fitness value.Only those moderation solutions that outperform their parent particles can survive, while all the rest are eliminated in the competition.Such competitive method allows the crisscross search always maintain in a historical best position and quicken the converge rate to the global optima.

Basic Procedure of CSO Algorithm
The basic procedure of CSO algorithm is given as follows: Step 1. Population initialization and parameters set.
Step 2. Execute horizontal crossover with the competitive operator.
Step 3. Execute vertical crossover with the competitive operator.
Step 4. Repeat Steps 2 and 3 until the number of iterations is larger than the specified maximum value.

MCS-Based Probabilistic Power Flow Calculation
As mentioned before, ODGA problem becomes more complex when system uncertainties are considered.In order to obtain more feasible planning results, probabilistic power flow (PPF) calculation methods must be employed.Compared to the deterministic power flow, the PPF method characterizes the uncertainty in system information by describing the variation in terms of a suitable probability distribution.In this paper, the Monte Carlo simulations (MCS)-based PPF method proposed in [28] is adopted.The MCS is used to produce a large number of random variables according to the statistical models that established in Section 2. In case of uncertainties in the input variables of the power system, it is desirable to assess the system output variables (bus voltages and line flows) for many load and generation conditions.It is necessary to run the deterministic power flow routine many times in order to evaluate possible system states.More detailed principles about Energies 2015, 8, 13641-13659 MCS-based probabilistic power flow method can refer to [28] and [29].The flowchart of MCS-based PPF method is shown in Figure 1.
Energies 2015, 8, page-page 8 MCS-based probabilistic power flow method can refer to [28] and [29].The flowchart of MCS-based PPF method is shown in Figure 1.

Solving Steps of the CSO-MCS Method to ODGA Problem
The flowchart of the CSO-MCS algorithm is illustrated in Figure 2 and its procedure for solving ODGA problem can be described as follows: Step 1: Define the input data, including the network data and the specified parameters of CSO-MCS algorithm; Step 2: Randomly generate the initial population as showed in Equation ( 15).

Solving Steps of the CSO-MCS Method to ODGA Problem
The flowchart of the CSO-MCS algorithm is illustrated in Figure 2 and its procedure for solving ODGA problem can be described as follows: Step 1: Define the input data, including the network data and the specified parameters of CSO-MCS algorithm; Step 2: Randomly generate the initial population as showed in Equation (15).Step 3: Execute the MCS-based PPF calculation and check the feasibilities of CSO particles.
Step 4: Calculate the fitness function of CSO particle according to Equations ( 6)-( 9), using the results of the PPF.Step 3: Execute the MCS-based PPF calculation and check the feasibilities of CSO particles.
Step 4: Calculate the fitness function of CSO particle according to Equations ( 6)-( 9), using the results of the PPF.
Step 5: Check the constraints of ODGA model with the results of the PPF.If the constraints are satisfied, go to the next step; otherwise, add a large value M p to the fitness value as a penalty.Repeat Steps 3 to 5 for all particles of the initial population; Step 6: Sort the particles in descending order of fitness value and save current best solutions in the repository G best ; Step 7: Update the population using horizontal crossover and vertical crossover.
Step 9: Select the best solution found in the above solving procedure as the best DG allocation scheme.

Numerical Results
To comprehensively demonstrate the validity of the proposed CSO-MCS algorithm, simulation studies are conducted on two test systems which are widely used as benchmarks in the power system planning field for solving the ODGA problems.The two test systems are the 33-bus and 69-bus distribution systems, respectively.The first 33-bus system is a radial system with total load of 3.715 MW.The second 69-bus system is a widely used distribution system in the literature.The detailed data of the systems appear in [30] and [31].
The parameters of the proposed model are specified as follows: (1) The planning period of DG is 15 years; the discount rate is 12%; (2) The voltage magnitude cannot exceed ˘5% of the nominal voltage and the power flow on the lines should not exceed 4 MVA with the confidence levels of 90% (i.e., α = β = 0.9); (3) Weighting coefficients of Equation ( 9): µ = 0.2, λ = 0.38, ξ = 0.42.The electricity price is 0.089 $/kWh.The investment cost, operation, and maintenance cost and other technical parameters of WT/PV/MT generators can be seen in Table 1.In a practical situation, it cannot be expected that the O&M cost and electricity price always keep constant along the time.However, in order to simplify the problem, it is assumed that the O&M costs and the electricity price are as constant in the simulations.The pollutant emission rates (kg/MWh) of different kinds of DGs are given in Table 2.The environmental value standard and penalty for pollutant emissions are shown in Table 3.When planning DG, a series of candidate places that suitable for installing specific types of DGs will be given, from the aspects of climate, geography and technology conditions.Then, the CSO-MCS will be used to determine the best schemes (i.e., the best types, locations and sizes of DGs) among the given candidate locations taking into account the voltage amplitude, power flow and total costs of the distribution system.It is assumed that the candidate types and locations of DGs in the 33-bus and 69-bus systems are as shown in Tables 7 and 8 respectively; the size of the each DG unit is within the limit between 10 and 500 kVA with discrete interval capacity of 10 kVA.To find the effectiveness and improvement of the developed CSO-MCS algorithm, the test results are compared with those obtained by another algorithm named PSO-MCS in [32].All the experiments are implemented using Matlab2009b at a Core2 2.40-GHz machine with 2-GB RAM.The parameters of the algorithms are set as follows: In CSO-MCS, the horizontal crossover probability and vertical crossover probability are set as P hc = 1 and P vc = 0.8 according to the suggestion in [26].In PSO-MCS, the acceleration coefficients are set to c 1 = c 2 = 0.8, the inertia weight is set to w = 0.4 [32].In the two algorithms, the population sizes are set to N pop = 50, the maximum number of iterations is set to Iter max = 500, the sample sizes of Monte Carlo are set to N s = 500, other parameters are set to the same as those used in the optimization model.To reduce statistical errors, the test is repeated 30 times independently.

Case 1: IEEE 33-Bus System
In this case, the typical 33-bus system is used to test the proposed method.The convergence characteristics of the developed CSO-MCS algorithm and PSO-MCS are illustrated in Figure 3. From this figure, it can be seen that the PSO-MCS algorithm is fast in convergence.The iteration number required by PSO-MCS is only 57.However, the convergence accuracy is not entirely satisfactory.This is due to the shortcoming of PSO that is easily trapped in the local optimum and appeared premature convergence.As to CSO-MCS, the experimental results show that its iteration number is greater than PSO-MCS, while the optimal result of CSO-MCS is 1.88% better than that of PSO-MCS.
Energies 2015, 8, page-page 11 To find the effectiveness and improvement of the developed CSO-MCS algorithm, the test results are compared with those obtained by another algorithm named PSO-MCS in [32].All the experiments are implemented using Matlab2009b at a Core2 2.40-GHz machine with 2-GB RAM.The parameters of the algorithms are set as follows: In CSO-MCS, the horizontal crossover probability and vertical crossover probability are set as Phc = 1 and Pvc = 0.8 according to the suggestion in [26].In PSO-MCS, the acceleration coefficients are set to c1 = c2 = 0.8, the inertia weight is set to w = 0.4 [32].In the two algorithms, the population sizes are set to Npop = 50, the maximum number of iterations is set to Itermax = 500, the sample sizes of Monte Carlo are set to Ns = 500, other parameters are set to the same as those used in the optimization model.To reduce statistical errors, the test is repeated 30 times independently.

Case 1: IEEE 33-Bus System
In this case, the typical 33-bus system is used to test the proposed method.The convergence characteristics of the developed CSO-MCS algorithm and PSO-MCS are illustrated in Figure 3. From this figure, it can be seen that the PSO-MCS algorithm is fast in convergence.The iteration number required by PSO-MCS is only 57.However, the convergence accuracy is not entirely satisfactory.This is due to the shortcoming of PSO that is easily trapped in the local optimum and appeared premature convergence.As to CSO-MCS, the experimental results show that its iteration number is greater than PSO-MCS, while the optimal result of CSO-MCS is 1.88% better than that of PSO-MCS.The simulation results including the pollutant emissions, the power losses cost, the total DG cost and the CPU time obtained by CSO-MCS and the compared algorithm are shown in Table 4.The comparisons of statistics of the above indices, such as mean value, standard deviation, maximum and minimum value are presented as well.In terms of solution quality, we see that the mean pollutant emissions and power losses cost achieved by CSO-MCS is 25.08% and 14.40% less than those by PSO-MCS.The best fitness value CSO-MCS achieved is better than PSO-MCS as analyzed earlier.
These results demonstrate the effectiveness of the developed CSO-MCS algorithm in addressing DG allocation problems.With respect to the computing time, the mean CPU time of CSO-MCS is 227.02 s, which is 24.65% faster than PSO-MCS.In stability, we can see that the standard deviation of CPU time of CSO-MCS is 6.42 s, which is 47.38% less than PSO-MCS.Similar results can be found in other indices.For example, the standard deviation value of emissions index obtained by CSO-MCS is 68.89% less than that of PSO-MCS.These results show that CSO-MCS has good performance in robustness.The simulation results including the pollutant emissions, the power losses cost, the total DG cost and the CPU time obtained by CSO-MCS and the compared algorithm are shown in Table 4.The comparisons of statistics of the above indices, such as mean value, standard deviation, maximum and minimum value are presented as well.In terms of solution quality, we see that the mean pollutant emissions and power losses cost achieved by CSO-MCS is 25.08% and 14.40% less than those by PSO-MCS.The best fitness value CSO-MCS achieved is better than PSO-MCS as analyzed earlier.
These results demonstrate the effectiveness of the developed CSO-MCS algorithm in addressing DG allocation problems.With respect to the computing time, the mean CPU time of CSO-MCS is 227.02 s, which is 24.65% faster than PSO-MCS.In stability, we can see that the standard deviation of CPU time of CSO-MCS is 6.42 s, which is 47.38% less than PSO-MCS.Similar results can be found in other indices.For example, the standard deviation value of emissions index obtained by CSO-MCS is 68.89% less than that of PSO-MCS.These results show that CSO-MCS has good performance in robustness.Figure 4 compares the voltage profiles of the 33-buses system before and after DG installation.The voltage profile is one of the main criterions for power quality improvement.As seen in Figure 4, the voltage quality of the origin system is dissatisfactory.The voltage of many nodes, especially the end nodes like node-31 and node-32, is seriously below the permitted range.After installing DGs according to the optimal schemes, the voltage profile is obviously improved.It can be observed that both of the methods (i.e., CSO-MCS and PSO-MCS) provide better voltage profile in the simulation.According to the statistical results, the average voltage deviation (AVD) of the scheme obtained by CSO-MCS and that by PSO-MCS is 0.00615 p.u and 0.01472 p.u, which present a reduction of 89.15% and 74.02% compare to the origin system.Meanwhile, we can see that the voltage profile obtained by CSO-MCS method is better than that by PSO-MCS in terms of average voltage deviation and voltage stability.Figure 4 compares the voltage profiles of the 33-buses system before and after DG installation.The voltage profile is one of the main criterions for power quality improvement.As seen in Figure 4, the voltage quality of the origin system is dissatisfactory.The voltage of many nodes, especially the end nodes like node-31 and node-32, is seriously below the permitted range.After installing DGs according to the optimal schemes, the voltage profile is obviously improved.It can be observed that both of the methods (i.e., CSO-MCS and PSO-MCS) provide better voltage profile in the simulation.According to the statistical results, the average voltage deviation (AVD) of the scheme obtained by CSO-MCS and that by PSO-MCS is 0.00615 p.u and 0.01472 p.u, which present a reduction of 89.15% and 74.02% compare to the origin system.Meanwhile, we can see that the voltage profile obtained by CSO-MCS method is better than that by PSO-MCS in terms of average voltage deviation and voltage stability.From the above analyses, we can draw a conclusion that CSO-MCS has good performance in terms of convergence accuracy and robustness, and it is suitable for solving DG allocation problems.
From the above simulation results, we can also evaluate the benefits brought about by optimal DG allocation.Before placing DGs in the distribution system, the emission of air pollutants is 1.3582 × 10 5 tons.After allocating DGs using CSO-MCS algorithm, a maximal reduction of 0.298 × 10 5 tons or approximately 21.96% of pollutant emissions can be achieved, as shown in Table 4.The significant reduction in pollutant emission reveals the great contribution of DGs to environmental protection.In terms of power losses, after placing DGs, the system loss costs reduce From the above analyses, we can draw a conclusion that CSO-MCS has good performance in terms of convergence accuracy and robustness, and it is suitable for solving DG allocation problems.
From the above simulation results, we can also evaluate the benefits brought about by optimal DG allocation.Before placing DGs in the distribution system, the emission of air pollutants is 1.3582 ˆ10 5 tons.After allocating DGs using CSO-MCS algorithm, a maximal reduction of 0.298 ˆ10 5 tons or approximately 21.96% of pollutant emissions can be achieved, as shown in Table 4.The significant reduction in pollutant emission reveals the great contribution of DGs to environmental protection.In terms of power losses, after placing DGs, the system loss costs reduce from 7.8783 ˆ10 5 $ to 5.1482 ˆ10 5 $, which proves that reasonable application of DGs can effectively reduce the system network losses.

Case 2: PG&E 69-Bus System
A larger test system proposed in [31] is considered in this case.The base power of this system is 10 MVA and the base voltage is 12.66 kV.
Figure 5 shows the convergence characteristics obtained by CSO-MCS and PSO-MCS.As shown in this figure, it is obvious that CSO-MCS outperforms PSO-MCS in terms of searching for better converged solutions.The optimal fitness value of CSO-MCS is 0.704% better than that of PSO-MCS.
The simulation results obtained by CSO-MCS are shown in Table 5 and compared with the results of PSO-MCS.According to the results reported in Table 5, it can be observed that CSO-MCS is capable of finding better solutions.The minimum of pollutant emission applying CSO-MCS is 1.0772 ˆ10 5 t, which is 0.706% less than the best results of PSO-MCS.Similar results can be found in the power losses cost index.As seen in Table 5, the power losses cost achieved by CSO-MCS is 22.39% less than that by PSO-MCS.

Case 2: PG&E 69-Bus System
A larger test system proposed in [31] is considered in this case.The base power of this system is 10 MVA and the base voltage is 12.66 kV.
Figure 5 shows the convergence characteristics obtained by CSO-MCS and PSO-MCS.As shown in this figure, it is obvious that CSO-MCS outperforms PSO-MCS in terms of searching for better converged solutions.The optimal fitness value of CSO-MCS is 0.704% better than that of PSO-MCS.
The simulation results obtained by CSO-MCS are shown in Table 5 and compared with the results of PSO-MCS.According to the results reported in Table 5, it can be observed that CSO-MCS is capable of finding better solutions.The minimum of pollutant emission applying CSO-MCS is 1.0772 × 10 5 t, which is 0.706% less than the best results of PSO-MCS.Similar results can be found in the power losses cost index.As seen in Table 5, the power losses cost achieved by CSO-MCS is 22.39% less than that by PSO-MCS.With respect to computing speed, the mean CPU time of CSO-MCS is 583.92 s, which is 5.46% less than that of PSO-MCS.Besides, it is worthwhile to note that CSO-MCS also shows good performance on robustness.It can be observed from Table 5 that the standard deviation of emissions index achieved by CSO-MCS is 0.0027 × 10 5 t, whereas the standard deviation value obtained by PSO-MCS is 0.0097 × 10 5 t.In terms of other indices, the power losses cost index, total  With respect to computing speed, the mean CPU time of CSO-MCS is 583.92 s, which is 5.46% less than that of PSO-MCS.Besides, it is worthwhile to note that CSO-MCS also shows good performance on robustness.It can be observed from Table 5 that the standard deviation of emissions index achieved by CSO-MCS is 0.0027 ˆ10 5 t, whereas the standard deviation value obtained by PSO-MCS is 0.0097 ˆ10 5 t.In terms of other indices, the power losses cost index, total DG cost index and CPU time index achieved by the proposed algorithm are smaller than those achieved by PSO-MCS.The above analysis results of 69-bus test system further confirm the effectiveness and robustness of CSO-MCS in solving DG allocation problems.
Energies 2015, 8, 13641-13659 Figure 6 shows and compares the mean voltage magnitude of each node for the 69-bus system using different methods.From this figure, it can be observed that the voltage amplitude of many nodes of the origin system is seriously below the permitted range.After placing DGs using the optimization methods, we can see that a marked improvement in voltage profiles can be achieved.As seen in Figure 6, before placing DGs, the original voltage magnitude of node 54 is 0.909 p.u.After optimizing DG allocation in the distribution system, the voltage magnitude of node 54 increases to 0.986 p.u. Similar results have been seen in other nodes.In addition, according to the statistical results, the average voltage deviation (AVD) obtained by CSO-MCS is 0.005 p.u, which has a reduction of 81.62% compare to the origin system.When comparing the voltage profiles obtained by the two algorithms (i.e., CSO-MCS and PSO-MCS), we can see that the AVD achieved by CSO-MCS is also less than the results that of PSO-MCS (0.007 p.u), which also indicates the validity of CSO-MCS in solving ODGA problem.
Energies 2015, 8, page-page 14 DG cost index and CPU time index achieved by the proposed algorithm are smaller than those achieved by PSO-MCS.The above analysis results of 69-bus test system further confirm the effectiveness and robustness of CSO-MCS in solving DG allocation problems.
Figure 6 shows and compares the mean voltage magnitude of each node for the 69-bus system using different methods.From this figure, it can be observed that the voltage amplitude of many nodes of the origin system is seriously below the permitted range.After placing DGs using the optimization methods, we can see that a marked improvement in voltage profiles can be achieved.As seen in Figure 6, before placing DGs, the original voltage magnitude of node 54 is 0.909 p.u.After optimizing DG allocation in the distribution system, the voltage magnitude of node 54 increases to 0.986 p.u. Similar results have been seen in other nodes.In addition, according to the statistical results, the average voltage deviation (AVD) obtained by CSO-MCS is 0.005 p.u, which has a reduction of 81.62% compare to the origin system.When comparing the voltage profiles obtained by the two algorithms (i.e., CSO-MCS and PSO-MCS), we can see that the AVD achieved by CSO-MCS is also less than the results that of PSO-MCS (0.007 p.u), which also indicates the validity of CSO-MCS in solving ODGA problem.

Influence of Uncertainties of Renewable DGs
In order to investigate the effect of uncertainties of renewable energy on the planning results, two scenarios have been designed in Table 6.The stochastic characteristics of renewable energy can be described by their probability density functions (PDF).In this simulation, we design two sets of PDF parameters of renewable energy (i.e., shape and scale parameters of the Weibull distribution of wind speed; shape parameters of the Beta distribution of solar radiation).Scenario B shows higher level of wind speed and lower level of solar illumination in comparison to Scenario A. The results associated with the schemes for the 33-bus and 69-bus systems are shown in Tables 7 and 8.With respect to the chance constraints of voltage amplitude and power flow, statistical results indicate that, for the 33-bus system, the probabilities of voltage chance constraint Pr{Vmin ≤ V ≤ Vmax} are 0.954 and 0.962; for the 69-bus system, the probabilities of voltage chance constraint Pr{Vmin ≤ V ≤ Vmax} are 0.937 and 0.942.In terms of the chance constraint of power flow, the probabilities Pr{Sij ≤ Smax} are 0.922 and 0.937 for the 33-bus system; 0.920 and 0.929 for the 69-bus system.Both of the probabilities can satisfy the given value (i.e., α = β = 0.9), so the optimization results are acceptable.Besides, from Table 7, we can observe that the obtained results vary with the uncertain characteristics of renewable energy.The penetration of WT-based DG is increased from 300 kW (Scenario A) to 500 kW (Scenario B), whereas the installation of PV-based DG is decreased from 610 kW (Scenario A) to 370 kW (Scenario B).

Influence of Uncertainties of Renewable DGs
In order to investigate the effect of uncertainties of renewable energy on the planning results, two scenarios have been designed in Table 6.The stochastic characteristics of renewable energy can be described by their probability density functions (PDF).In this simulation, we design two sets of PDF parameters of renewable energy (i.e., shape and scale parameters of the Weibull distribution of wind speed; shape parameters of the Beta distribution of solar radiation).Scenario B shows higher level of wind speed and lower level of solar illumination in comparison to Scenario A. The results associated with the schemes for the 33-bus and 69-bus systems are shown in Tables 7 and 8.With respect to the chance constraints of voltage amplitude and power flow, statistical results indicate that, for the 33-bus system, the probabilities of voltage chance constraint Pr{V min ď V ď V max } are 0.954 and 0.962; for the 69-bus system, the probabilities of voltage chance constraint Pr{V min ď V ď V max } are 0.937 and 0.942.In terms of the chance constraint of power flow, the probabilities Pr{S ij ď S max } are 0.922 and 0.937 for the 33-bus system; 0.920 and 0.929 for the 69-bus system.Both of the probabilities can satisfy the given value (i.e., α = β = 0.9), so the optimization results are acceptable.Besides, from Table 7, we can observe that the obtained results vary with the uncertain characteristics of renewable energy.The penetration of WT-based DG is increased from 300 kW (Scenario A) to 500 kW (Scenario B), whereas the installation of PV-based DG is decreased from 610 kW (Scenario A) to 370 kW (Scenario B).Similar change of optimization results appeared in the 69-bus system.The main reason for this change is due to the higher level of wind speed and the lower level of solar illumination intensity in Scenario B.  The voltage probability density curves of node 4 and node 32 indicate that the voltage profile is obviously improved after placing DGs properly.From Figure 7 to Figure 10, we can see that in most cases of the 33-bus system, the voltage of node 32 distributes between 0.98 p.u and 1.00 p.u, and the voltage of node 4 distributes between 0.99 p.u and 1.03 p.u.As for the 69-bus system, we can observe that the voltage of node 50 distributes between 0.985 p.u and 0.995 p.u, and the voltage of node 14 distributes between 0.98 p.u and 1.02 p.u.Both of them are within the allowable range for power systems to ensure the operation safety.However, it is worthwhile to note that the voltage amplitudes of the two nodes may sometimes exceed the limitation, due to the intermittent of DG power supply.Thus, for those important customers whom are sensitive to voltage quality and locate close to DG connected nodes, it is necessary to make measures to keep stable voltage quality.The voltage probability density curves of node 4 and node 32 indicate that the voltage profile is obviously improved after placing DGs properly.From Figure 7 to Figure 10, we can see that in most cases of the 33-bus system, the voltage of node 32 distributes between 0.98 p.u and 1.00 p.u, and the voltage of node 4 distributes between 0.99 p.u and 1.03 p.u.As for the 69-bus system, we can observe that the voltage of node 50 distributes between 0.985 p.u and 0.995 p.u, and the voltage of node 14 distributes between 0.98 p.u and 1.02 p.u.Both of them are within the allowable range for power systems to ensure the operation safety.However, it is worthwhile to note that the voltage amplitudes of the two nodes may sometimes exceed the limitation, due to the intermittent of DG power supply.Thus, for those important customers whom are sensitive to voltage quality and locate close to DG connected nodes, it is necessary to make measures to keep stable voltage quality.

Conclusions
Integration of distributed generations into the distribution system may change various operating characteristics of the system, especially at increased penetration of renewable DGs like wind power and photovoltaic generation.This paper has proposed a novel approach integrating crisscross optimization algorithm and Monte Carlo simulation called CSO-MCS is proposed to effectively obtain the best sizes, locations and types of DGs in the distribution system.The proposed CSO-MCS method is compared with PSO-MCS in terms of convergence and computation time to demonstrate its effectiveness and feasibility.Furthermore, an optimization framework for optimal DG allocation problem using chance constrained programming (CCP) is presented in this paper.The objective functions encompass minimization of total pollutant emission, DG cost and active power losses over the planning period.In order to assure the feasibility of planning results, the system uncertainties covering wind speed, solar radiation and load consumption are taken into account using their probabilistic models.
This proposed work can help the system operators in evaluating DG allocation schemes to decide the best feasible solution for practical implementation and enable them to evaluate the impact of system uncertainties on DG planning work.As future work, we are preparing to apply CSO-MCS to address more practical ODGA problems considering reliability impacts and other factors, then conduct a research on the system operation and energy management of a smart distribution grid connecting large numbers of DGs and PEVs.

Conclusions
Integration of distributed generations into the distribution system may change various operating characteristics of the system, especially at increased penetration of renewable DGs like wind power and photovoltaic generation.This paper has proposed a novel approach integrating crisscross optimization algorithm and Monte Carlo simulation called CSO-MCS is proposed to effectively obtain the best sizes, locations and types of DGs in the distribution system.The proposed CSO-MCS method is compared with PSO-MCS in terms of convergence and computation time to demonstrate its effectiveness and feasibility.Furthermore, an optimization framework for optimal DG allocation problem using chance constrained programming (CCP) is presented in this paper.The objective functions encompass minimization of total pollutant emission, DG cost and active power losses over the planning period.In order to assure the feasibility of planning results, the system uncertainties covering wind speed, solar radiation and load consumption are taken into account using their probabilistic models.
This proposed work can help the system operators in evaluating DG allocation schemes to decide the best feasible solution for practical implementation and enable them to evaluate the impact of system uncertainties on DG planning work.As future work, we are preparing to apply CSO-MCS to address more practical ODGA problems considering reliability impacts and other factors, then conduct a research on the system operation and energy management of a smart distribution grid connecting large numbers of DGs and PEVs.

Conclusions
Integration of distributed generations into the distribution system may change various operating characteristics of the system, especially at increased penetration of renewable DGs like wind power and photovoltaic generation.This paper has proposed a novel approach integrating crisscross optimization algorithm and Monte Carlo simulation called CSO-MCS is proposed to effectively obtain the best sizes, locations and types of DGs in the distribution system.The proposed CSO-MCS method is compared with PSO-MCS in terms of convergence and computation time to demonstrate its effectiveness and feasibility.Furthermore, an optimization framework for optimal DG allocation problem using chance constrained programming (CCP) is presented in this paper.The objective functions encompass minimization of total pollutant emission, DG cost and active power losses over the planning period.In order to assure the feasibility of planning results, the system uncertainties covering wind speed, solar radiation and load consumption are taken into account using their probabilistic models.
This proposed work can help the system operators in evaluating DG allocation schemes to decide the best feasible solution for practical implementation and enable them to evaluate the impact of system uncertainties on DG planning work.As future work, we are preparing to apply CSO-MCS to address more practical ODGA problems considering reliability impacts and other factors, then conduct a research on the system operation and energy management of a smart distribution grid connecting large numbers of DGs and PEVs.
Energies 2015, 8, 13641-13659 Energies 2015, 8, page-page 9 candidate locations for WT, PV and MT.The initial population is filled with M randomly generated solutions.

Figure 2 .
Figure 2. Flowchart of CSO-MCS for solving the ODGA problem.

Step 5 :
Check the constraints of ODGA model with the results of the PPF.If the constraints are satisfied, go to the next step; otherwise, add a large value Mp to the fitness value as a penalty.Repeat Steps 3 to 5 for all particles of the initial population;Step 6: Sort the particles in descending order of fitness value and save current best solutions in the repository Gbest; Step 7: Update the population using horizontal crossover and vertical crossover.

Figure 2 .
Figure 2. Flowchart of CSO-MCS for solving the ODGA problem.

Figure 7 .
Figure 7. PDF curve of voltage amplitude in Scenario A of 33-bus system: (a) Node 32 and (b) Node 4.

Figure 10 .
Figure 10.PDF curve of voltage amplitude in Scenario B of 69-bus system: (a) Node 50 and (b) Node 14.

Figure 10 .
Figure 10.PDF curve of voltage amplitude in Scenario B of 69-bus system: (a) Node 50 and (b) Node 14.

Figure 10 .
Figure 10.PDF curve of voltage amplitude in Scenario B of 69-bus system: (a) Node 50 and (b) Node 14.
capacity of the ith WT unit, jth PV unit and kth MT unit, respectively.N w , N p and N m are the number of candidate locations for WT, PV and MT.The initial population is filled with M randomly generated solutions.

Table 1 .
Technical specification parameters of different types of distributed generation (DGs).

Table 2 .
Pollutant emission rate of different types of DGs.

Table 3 .
Environmental value standard and penalty for pollutant emissions.

Table 4 .
Results obtained by crisscross optimization algorithm and Monte Carlo simulation (CSO-MCS) and particle swarm optimization and Monte Carlo simulation (PSO-MCS) (30 runs for Case 1).

Table 4 .
Results obtained by crisscross optimization algorithm and Monte Carlo simulation (CSO-MCS) and particle swarm optimization and Monte Carlo simulation (PSO-MCS) (30 runs for Case 1).