Maximum Hosting Capacity Assessment of Distribution Systems With Multitype DERs Using Analytical OPF Method

Renewable energy is being increasingly integrated into distribution systems worldwide in response to technological, economic, and environmental challenges. The assessment of hosting capacity allows us to determine the maximum installation capacity of distributed energy resources (DERs) in a distribution system within its operational limits to obtain more benefits. In this study, a new multistage algorithm is developed based on an analytical approach and optimal power flow (OPF) for the assessment of DERs’ hosting capacity (DERHC) with single and multiple multi-type DERs. In the first stage, the optimal locations of DERs are determined analytically, and the second stage involves the calculation of optimal DERs sizes for the assessment of the maximum locational and total DERHC. This method provides mathematical and global optimum certainty considering the constraints to maintain the reliability and protection of the system. Moreover, the proposed method is tested using a standard IEEE 33-bus distribution system, and different scenarios are created based on the number and type of DERs to achieve the best-case results of DERHC. The obtained results are compared with those of the conventional OPF iterative method that are encouraging and validate the accuracy and robustness of the proposed methodology.

stage is based on the novel concept of DERHC to obtain 99 the optimal locations for the placement of multi-type single 100 and multiple DERs using an analytical approach. The second 101 stage of the algorithm evaluates the maximum DERs capac-102 ity injection into the distribution system using the OPF to 103 determine the locational and total maximum DERHC. The 104 numerical and simulation results were obtained using IEEE 105 33-bus distribution test systems. 106 The main contribution of this study can be summarized as 107 follows: 108 • Novel direction toward concept of DERHC assessment: 109 In this article, a comprehensive analysis of the assessment of 110 the DERHC of a distribution system is presented based on a 111 novel direction toward assessment of DERHC.

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• Analytical OPF method: A multistage analytical OPF 113 method is proposed for the assessment of the DERHC of dis-114 tribution system with single and multiple multi-type DERs. 115 This method provides accuracy and robustness with the high 116 probability of finding a global optimum solution, and its 117 mathematical certainty is verified through the results obtained 118 on the IEEE 33-bus test system. In addition, the robustness of 119 the proposed method is also verified for an increased number 120 of buses, i.e., the IEEE 69-bus distribution system.

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• Assessment of multi-type DERHC: The DERHC of a 122 distribution system with single and multiple multi-type 123 DERs are evaluated using the proposed method. The 124 simulation results obtained for the IEEE 33-bus distribu-125 tion system are compared with those of the iterative OPF 126 method that are encouraging. Furthermore, the DERHC 127 of the distribution system is also determined by combin-128 ing all types of DERs.

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The remainder of this paper is organized as follows. A new 130 direction for the DERHC concept is presented in Section II. 131 Section III focuses on the proposed DERHC methodology, 132 including the mathematical formulation and classification of 133 DERs. Section IV presents a detailed description of the test 134 system, the simulation and numerical results, and a discussion 135 of the obtained results. Section V presents the conclusion of 136 this study.

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The study of DERHC has been recently started to research 139 the impacts of increasing the penetration of DERs into the 140 distribution network of power systems. The literature review 141 defines the DERHC as, ''the maximum integration of DERs 142 into the distribution system, subjected to the operational 143 constraints, at which the power system operates satisfacto-144 rily'' [25], [26], [27]. A locational DERHC is usually referred 145 to as the maximum capacity of the DER that can be integrated 146 at each location (bus) of a distribution system without disturb-147 ing the operation of the power system.

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The concept of the DERHC of a distribution system is illus-149 trated in Fig. 1. The distribution grids, substations, and trans-150 mission system are modeled based on the specified limits 151 of operational indices for protection and better performance. 152   The distribution systems are radial with a high R/X ratio 167 and unbalanced loads. Therefore, conventional power flow 168 approaches, such as the Newton-Raphson method and fast 169 decoupled power flow analysis, are not preferred when 170 dealing with distribution system challenges. However, it is 171 essential to use the distribution version of the power flow 172 method [29].

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The iterative based OPF is a conventional method used 174 for the assessment of DERHC by the Pacific Gas and Elec-175 tric (PG&E) distribution system operator [30]. This method 176 involves creating two matrices of all possible combinations of 177 the number of buses with respect to the number of DERs and 178 DER capacities ranging from minimum to maximum. It fol-179 lows a systematic procedure by iteratively selecting combina-180 tions. Power flow calculations are performed, and the limits 181 of the operational constraints are checked at every iteration. 182 The DER capacities that are within the boundaries of the oper-183 ational constraints are saved in an array, and the same pro-184 cedure is performed until all possible combinations of both 185 matrices are investigated. Thus, the combination with the 186 maximum sum of DER capacities is selected as the DERHC 187 of distribution system. The flow diagram of this method is 188 presented in Fig. 2.  The objective function for the assessment of DERHC is for-219 mulated based on the definition, ''the summation of total 220 DERHC capacity injection into a distribution system sub-221 jected to operational constraints,'' as follows: where S DERs i is the total apparent power from the DERs at 227 bus number i of a distribution system with n number of buses. 228 Equations (2) and (3) represent the maximum and minimum 229 limits of the voltage and current constraints, respectively. The 230 minimum and maximum limits of voltage are selected as 231 0.95 and 1.05 p.u., respectively, and the branch current limits 232 are obtained from [31]. 233

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The equations for the active and reactive power injections of 235 a distribution system with n number of buses is as follows: where P inj i and Q inj i are the active and reactive power injec-239 tions, respectively, at bus i of the distribution system. k is 240 the number of buses starting from the first to the last bus; 241 and V i and V k are the voltage magnitudes at buses i and k, 242 respectively. G ik + jB ik is the line admittance. To determine the power injection for multiple DERs, a matrix 245 of all possible combinations, N C , is constructed using (6) 246 with the total number of eligible buses N B and the number 247 of integrated DERs N DERs [23].
The active power injection for multiple DERs into bus (i) 250 P inj i,multi is then calculated by summing the power injection of 251 N DERs number of DERs on all combinations using (7), and 252 the combination of buses with the maximum power injection 253 is obtained using (8).
The reactive power injection for multiple DERs Q inj i,multi is 257 determined by rewriting (7) and (8) as follows: powers from the DERs are as follows: The most common example of a type 1 DER is a photovoltaic 276 array with a static power converter. distribution system subjected to operational constraints. The 302 methodology is described in two stages. Step 1) Read the line and load data of the distribution sys-305 tem, the number of DERs to be integrated, and their 306 type.

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Step 2) Run single-time power flow analysis to obtain the 308 bus voltages.

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Step 5) Extract the number of locations where the power 314 injection is maximum according to the number of 315 DERs to be integrated.

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Step 6) Print them as an optimal location.

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The assessment of the maximum DERHC of the distribution 318 system is dependent on the placement of the DERs; therefore, 319 the optimal locations for the integration of DERs need to be 320 determined. The first stage provides an accurate and robust 321 analytical optimal location for the placement of DERs with 322 mathematical certainty. The flow diagram of this stage is 323 shown in Fig. 3. Step 1) Import the optimal location from the first stage of 326 algorithm.

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Step 2) Create a matrix of all possible combinations of 328 DER capacities, starting from the minimum to the 329 maximum capacity of a DER with a step increase 330 in capacity using (6).

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Step 3) Start iterations from 1 to the length of the matrix of 332 DER capacities. 333 VOLUME 10, 2022 iteration and create an array of those iterations 335 whose DER capacities are within the operational 336 constraints.

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Step 5) Select the combination of DER capacities whose 338 sum is the maximum.

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Step 6) Print the optimal locations, maximum locational 340 and total DERHC of the distribution system with 341 simulation time.

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The second stage of the OPF was used to calculate the max-

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This section presents a verification of the proposed algorithm.

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Intensive simulation tests were conducted for the integration 356 of multi-type single and multiple DERs into the distribution 357 system for the assessment of DERHC compared with the OPF 358 iterative method. The locational and total DERHC were deter-359 mined for different scenarios created for single and multiple 360 DERs and combinations of multi-type DERs. The proposed 361 algorithm and tests were programmed in MATLAB with a 362 3.20 GHz PC with 16.00 GB of RAM.

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The IEEE 33-bus radial distribution system is used as a 364 test system to investigate the performance of the proposed 365 algorithm and to determine the best-case scenario of the max-366 imum locational and total DERHC of the distribution system. 367 The topology of the standard IEEE 33-bus distribution system 368 is illustrated in Fig. 5. The test system is a 12.66 KV system 369 with one feeder substation as bus 1 (slack bus) and 32 PQ 370 buses that are considered candidate buses for DER integra-371 tion. The total active and reactive powers provided by the load 372 buses are 3715 KW and 2300 KVAR, respectively. The locational DERHC was evaluated for the IEEE 33-bus 374 system using the OPF iterative method. It is plotted against 375 the calculated power injection without considering the oper-376 ational constraints using the first step of the proposed algo-377 rithm to validate the accuracy of the proposed algorithm. The 378 results are presented in Fig. 6. In the figure, the DERHC of 379 each bus and the calculated power injection are represented 380 by bars. These bars on each bus have the same nature of 381 increasing or decreasing the KVA capacity with respect to 382 other buses but with different capacities. This indicates that 383 if any bus has a high DERHC, the calculated power injection 384 value is also high, and vice versa. This validates the accuracy 385 of finding the optimal location for the placement of DERs 386 for the assessment of DERHC using the first stage of the 387 proposed algorithm.

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The assessment of the maximum DERHC with multi-type 389 single and multiple DERs is presented in the following 390 subsections.

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The proposed algorithm was first applied to single and mul-393 tiple type 1 DERs that generate only active power. The sim-394 ulation results of the maximum locational and total DERHC 395 of the 33-bus distribution system with the integration of one, 396 two, and three type 1 DERs are obtained through the proposed 397  Table 2. The simulation results of the total DERHC of the distribu-433 tion system with the integration of multi-type single and mul-434 tiple DERs are presented in Table 2. The locational DERHC 435 on the optimal number of buses are also mentioned. With 436 the integration of the multi-type single and multiple DERs, 437 the obtained results show that the total DERHC is maximum 438 with the integration of three DERs all of type 3, as compared 439 with the DERHC with three DERs of type 1 and type 2. The 440 locational DERHC is also maximum with the integration of 441 single and multiple DERs of type 3 at the optimal locations of 442 the distribution system. Furthermore, it can be observed that 443 three is the optimal number of integrated DERs in the 33-bus 444 distribution system. A detailed comparison of the DERHC 445 and simulation time of the proposed analytical OPF algo-446 rithm and the OPF iterative method is presented in Table 2. 447 The simulation time of the proposed methodology shows a 448 significant improvement, and the DERHC results are almost 449 similar to those of the OPF iterative method. The computa-450 tional time of the OPF iterative method is much longer, but in 451 VOLUME 10, 2022 this study, the step increase in the capacity of multiple DERs The optimal locations for these scenarios were obtained 487 using the first stage of the proposed algorithm as buses 2, 11, 488 and 26. Different types of DERs were sequentially placed at 489 the optimal locations, and the second stage of the proposed 490 algorithm was used to calculate the DERHC.  results are also presented in the bar graph in Fig. 11. Here, considered to maintain the reliability and protection of the 530 distribution system. Furthermore, the simulation time for this 531 solution decreased, which improved the robustness of the 532 grid system. The investigation was supported by a cohesive 533 and critical analysis of the simulation results of the DERHC 534 obtained for a 33-bus radial distribution system that verified 535 the accuracy of the proposed algorithm. The effectiveness of 536 proposed method in terms of computational robustness is also 537 verified on higher bus system i.e., IEEE 69-bus system. Based 538 on the simulation results, the following conclusions can be 539 drawn:

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• The DERHC is a location-dependent concept, and a 541 higher capacity of DERs can be integrated by deter-542 mining the optimal locations without any enhancement 543 techniques.

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• The DERHC of the distribution system with the same 545 type of DERs is maximum at 9503 KVA for three DERs 546 of type 3 in comparison with the integration of three type 547 1 and type 2 DERs.

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• For a combination of different DER types, Scenario 549 2 had the maximum DERHC of 11953 KVA compared 550 with the other scenarios.

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This study makes a significant contribution to the research 552 on DERHC analysis assessment. In the future, it would be 553 more interesting to study the enhancement of hosting capacity 554 by using energy storage devices, reconfiguration or reinforce-555 ment, and smart inverter techniques considering the DER 556 uncertainties and load variability.