Research Article | Open Access

Volume 2019 |Article ID 7576093 | https://doi.org/10.1155/2019/7576093

Boyu Zhu, Jun Zhou, Guangchuan Liang, Xuan Zhou, Liulin Zhou, "A General MINLP Model for the Multiway Valve Channel-Limited Location-Allocation Problem", Mathematical Problems in Engineering, vol. 2019, Article ID 7576093, 20 pages, 2019. https://doi.org/10.1155/2019/7576093

# A General MINLP Model for the Multiway Valve Channel-Limited Location-Allocation Problem

Accepted31 Jul 2019
Published10 Sep 2019

#### Abstract

As an important aspect of oil and gas field construction, oil and gas field surface engineering often affects the efficiency and safety of oil and gas production, and it requires a large investment. In the past, much work has been done on layout optimization for gathering pipeline networks. However, few studies have considered multiway valves in the optimization of pipeline networks. Compared to traditional metering processes, a process using multiway valves can reduce construction and operation costs and enable automation of the well-selection operation. In this paper, an MINLP model is established in which the number of multiway valves and their numbers of channels are considered special constraints and the number of multiway valves and the associations between wells and multiway valves are treated as optimization variables. A specific heuristic algorithm for solving this problem is also proposed in this paper. We consider the coordinates of real-world wells and of randomly generated well locations as different examples to analyze the performance of this algorithm. These examples demonstrate that the algorithm can initially adjust the network through single-step iteration, while double-step iteration is efficient when all channels of a multiway valve have been associated with pipelines, and multistep iteration can help the objective function escape from local optima. Finally, numerical analysis results prove that the proposed algorithm can be used to efficiently solve the problem of interest and exhibits stable convergence.

#### 1. Introduction

In recent years, as the production scale of oil and gas fields has expanded, the construction and labor costs incurred for the measurement of oil and gas production per well have also increased. In addition, the practical implementation of conventional metering methods is subject to certain limitations, especially in areas such as deserts and deep-sea regions. There are three main metering methods for oil and gas fields: single-well metering, metering by manual rotation, and automatic rotation metering. The single-well metering process requires the installation of metering equipment for each well, as shown in Figure 1(a). The initial investment is relatively expensive, especially when the well production rate is below the initial estimate. In the manual rotation measurement process, as shown in Figure 1(b), each valve that is to be opened or closed needs to be operated manually. Although it is not necessary to install metering devices at each well site, metering a well will require the manual operation of more than a dozen valves, and after metering, these valves will need to be operated again to discharge the liquid in the metering separator, thus requiring more operation time and incurring a higher cost. In addition, the metering device occupies a large area, and such devices are not suitable for offshore platforms. Regarding the third method, retrofitting an actuator onto the original manual valve of an existing manifold will greatly increase the construction cost. Therefore, it is necessary to find a device that can enable automatic well-selection metering while incurring lower construction and operation costs.

At present, many oil and gas field production enterprises use multiway valves (MVs) to achieve automatic rotation metering. The MV-based metering process can reduce the amount of manual operation required in remote areas, which is greatly beneficial for reducing operation costs. The metering process with an MV is shown in Figure 1(c). The blue dashed arrow indicates the flow that needs to be metered at present, and the black arrows indicate the flows from production wells. When switching to metering another well, the control system requires only remote operation, thus leading to considerable savings in operation cost. Compared to the traditional methods, the MV-based metering process reduces the numbers of valves, gauges, and actuators that need to be installed on the pipelines. Each device also occupies a smaller space, making these devices more suitable for a wider variety of oil and gas fields, especially for coalbed methane mining, shale gas, and other unconventional oil and gas resources. From here on, when the term “valve” is used in this paper, it refers to an MV.

The number of channels of an MV is a characteristic that is defined by the manufacturer and determines how many pipelines can access the MV. If an MV has 16 channels, this means that at most 16 wells can be connected to the MV. Manufacturers make MVs with various channel counts, such as 12, 14, and 16. Depending on the conditions in the field or other requirements such as consistency principle, the designer needs to select MVs with an appropriate number of channels to meet the requirements of the project. Therefore, during the design process, the number of wells connected to an MV is constrained by the number of channels of the MV, which depends on the designer’s selection. Depending on the constraint relaxation degree, there are three possible ways to specify MV-related constraints:(1)The number of MVs is unrestricted(2)Allowable ranges are given for the number of MVs and their channel counts(3)The channel counts of the MVs and the number of MVs are fixed in advance

For a given target oil and gas reservoir, the traditional solution is to invite experts to design feasible solutions based on real scenarios, but this approach does not necessarily guarantee the most economical cost. By contrast, optimization methods can help a designer to obtain the best solution. However, in the past, researchers have rarely considered constraints related to the channel counts of the MVs and the number of MVs, although such constraints are important to ensure that the optimal solution satisfies the required service conditions. In this paper, the problem of interest is formulated as a special P-median problem, and the constraints mentioned above are integrated into the mathematical model. By treating the associations between wells and MVs and the number of MVs as variables and the minimum cost as the objective function, a general mixed integer nonlinear programming (MINLP) model is established. Based on the connection and foresight matrices of the production wells and MVs, a foresight matrix search algorithm (FMSA) is proposed in this paper to solve the MINLP problem. In this algorithm, the nonlinear part of the problem is treated as a subproblem, and the optimization results are obtained through iteration. By examples, the convergence of the algorithm can be ensured, as can the quality and efficiency of the solution. In addition, it is proven that, with this algorithm, the solution process can be completed in a reasonable time even if the problem is large in scale.

This paper is divided into six sections: In Section 1, we have introduced the problem motivating this paper and what we have done to address it. In Section 2, we review the related research contributions that have been reported in the past. In Section 3, the detailed mathematical model is presented, and the special constraints are formulated. Subsequently, the details of solving the mathematical model using the proposed algorithm are introduced in Section 4. The results are analyzed in Section 5. Finally, conclusions and discussions are presented in Section 6. The data used in this paper can be found in Tables 15.

 x (m) y (m) Number x (m) y (m) Number x (m) y (m) Number 7430 2770 W1 4176 5039 W65 5570 3473 W129 7177 2698 W2 3975 4957 W66 5801 3634 W130 7193 3084 W3 4068 5358 W67 4985 3832 W131 7112 3341 W4 4429 5300 W68 7389 5655 W132 7572 2515 W5 4722 5370 W69 8522 5800 W133 8827 4759 W6 4904 5150 W70 8103 5715 W134 9009 5366 W7 4273 5543 W71 7542 5441 W135 9587 5795 W8 4558 5660 W72 7186 5593 W136 9458 5509 W9 5138 5270 W73 6566 5406 W137 9135 5732 W10 4969 5575 W74 6475 4116 W138 8935 5562 W11 5231 5060 W75 6024 4415 W139 8746 5209 W12 6030 5549 W76 6221 4302 W140 8774 5516 W13 5468 4806 W77 6465 4443 W141 8565 5454 W14 6472 5940 W78 6295 4625 W142 8261 4764 W15 7846 5656 W79 6120 4893 W143 8253 4440 W16 6208 5351 W80 6374 5033 W144 8396 4255 W17 5788 5334 W81 6650 5172 W145 8479 4649 W18 5557 5251 W82 6523 4829 W146 8713 4407 W19 5749 4979 W83 6711 4567 W147 9132 4226 W20 5994 5116 W84 5997 3457 W148 8857 4207 W21 6756 4798 W85 5990 3836 W149 8479 4998 W22 5723 4398 W86 6292 3902 W150 722 1355 W23 5573 4520 W87 6206 3526 W151 1779 1152 W24 5965 2209 W88 6492 3690 W152 1497 1374 W25 6276 2157 W89 6760 3709 W153 1971 1398 W26 6395 2580 W90 6593 3934 W154 1952 952 W27 6095 2413 W91 6250 2881 W155 1617 989 W28 5827 2630 W92 5532 2840 W156 1671 763 W29 5791 2352 W93 5819 2923 W157 1352 1125 W30 5384 2478 W94 5774 3200 W158 890 1183 W31 5587 2195 W95 6151 3119 W159 535 738 W32 5421 1986 W96 6405 3284 W160 264 29 W33 5718 2003 W97 6638 3422 W161 140 460 W34 6953 1171 W98 6812 3137 W162 459 285 W35 6758 1383 W99 6962 2906 W163 284 2160 W36 6489 1199 W100 6693 2722 W164 1805 2128 W37 6641 978 W101 6525 2993 W165 1466 1868 W38 6971 904 W102 7089 4415 W166 4185 4644 W39 7321 1026 W103 7254 4146 W167 4266 4411 W40 7397 1444 W104 7067 4733 W168 3996 4274 W41 7178 1236 W105 7036 5093 W169 4167 4079 W42 7074 1509 W106 7340 5187 W170 4047 3589 W43 6615 1674 W107 7832 5149 W171 4213 3355 W44 6959 2201 W108 8062 5110 W172 4280 3785 W45 6836 2432 W109 7609 4981 W173 4439 4188 W46 6529 2291 W110 7273 4862 W174 4642 3905 W47 6646 2086 W111 7835 4858 W175 4506 3500 W48 6450 1842 W112 7626 4730 W176 4745 3665 W49 6228 1732 W113 7819 4479 W177 4902 3436 W50 7117 1848 W114 7883 4211 W178 4624 3289 W51 7220 2425 W115 8024 4616 W179 4849 4060 W52 6837 1771 W116 7521 4356 W180 4417 4898 W53 6076 1312 W117 7333 4573 W181 4553 4667 W54 6304 1463 W118 7031 3908 W182 4614 4356 W55 5601 1025 W119 7409 3886 W183 4767 4755 W56 5263 808 W120 7728 4028 W184 4910 4514 W57 5429 1272 W121 7150 3733 W185 4671 5018 W58 5809 1246 W122 7370 3515 W186 5016 4291 W59 5721 1462 W123 7599 3580 W187 5021 4926 W60 5238 1591 W124 7878 3778 W188 5243 4666 W61 5533 1698 W125 7735 3369 W189 5220 4048 W62 5887 1725 W126 7497 3253 W190 5352 4367 W63 5219 3276 W127 7883 3225 W191 5465 4197 W64 5441 3674 W128 7696 2928 W192
 x (m) y (m) Number x (m) y (m) Number x (m) y (m) Number 264 29 W1 4213 3355 W13 4767 4755 W25 140 460 W2 4280 3785 W14 4910 4514 W26 459 285 W3 4439 4188 W15 4671 5018 W27 284 2160 W4 4642 3905 W16 5016 4291 W28 1805 2128 W5 4506 3500 W17 5021 4926 W29 1466 1868 W6 4745 3665 W18 5243 4666 W30 1793 2404 W7 4902 3436 W19 5220 4048 W31 4185 4644 W8 4624 3289 W20 5352 4367 W32 4266 4411 W9 4849 4060 W21 5465 4197 W33 3996 4274 W10 4417 4898 W22 4176 5039 W34 4167 4079 W11 4553 4667 W23 3975 4957 W35 4047 3589 W12 4614 4356 W24
 x (m) y (m) Number x (m) y (m) Number x (m) y (m) Number 855.158 2624.822 W1 1365.531 7212.275 W15 9786.806 7126.945 W29 8010.146 292.2028 W2 1067.619 6537.573 W16 5004.716 4710.884 W30 9288.541 7303.309 W3 4941.739 7790.517 W17 596.1887 6819.719 W31 4886.09 5785.251 W4 7150.371 9037.206 W18 424.3114 714.4546 W32 2372.836 4588.488 W5 8909.225 3341.631 W19 5216.498 967.3003 W33 9630.885 5468.057 W6 6987.458 1978.098 W20 8181.486 8175.471 W34 5211.358 2315.944 W7 305.4095 7440.743 W21 7224.396 1498.654 W35 4888.977 6240.601 W8 5000.224 4799.221 W22 6596.053 5185.949 W36 6791.355 3955.152 W9 9047.222 6098.666 W23 9729.746 6489.915 W37 3674.366 9879.82 W10 6176.664 8594.423 W24 8003.306 4537.977 W38 377.3887 8851.68 W11 8054.894 5767.215 W25 4323.915 8253.138 W39 9132.868 7961.839 W12 1829.225 2399.32 W26 834.6981 1331.71 W40 987.1228 2618.712 W13 8865.119 286.7415 W27 3353.568 6797.28 W14 4899.014 1679.271 W28
 x (m) y (m) Number x (m) y (m) Number x (m) y (m) Number 1269.868 9133.759 W1 6554.779 1711.867 W28 5383.424 9961.347 W55 1576.131 9705.928 W2 7060.461 318.328 W29 8173.032 8686.947 W56 4217.613 9157.355 W3 6550.980 1626.117 W30 8692.922 5797.046 W57 971.318 8234.578 W4 5852.678 2238.119 W31 8530.311 6220.551 W58 3516.595 8308.286 W5 5472.155 1386.244 W32 9027.161 9447.872 W59 1621.823 7942.845 W6 4505.416 838.214 W33 2784.982 5468.815 W60 2289.770 9133.374 W7 5498.602 1449.548 W34 1868.726 4897.644 W61 1523.780 8258.170 W8 4387.444 3815.585 W35 2760.251 6797.027 W62 1066.528 9618.981 W9 4455.862 6463.130 W36 1189.977 4983.641 W63 46.342 7749.105 W10 5059.571 6990.767 W37 119.021 3371.226 W64 2598.704 8000.685 W11 6160.447 4732.888 W38 3112.150 5285.331 W65 4314.138 9106.476 W12 5852.641 5497.236 W39 1656.487 6019.819 W66 3377.194 9000.538 W13 3804.458 5678.216 W40 2629.713 6540.791 W67 9571.669 4853.756 W14 5688.237 4693.906 W41 781.755 4426.783 W68 8002.805 1418.863 W15 3509.524 5132.495 W42 844.358 3997.826 W69 7431.325 3922.270 W16 4908.641 4892.526 W43 2769.230 461.714 W70 6948.286 3170.995 W17 8147.237 9057.919 W44 1492.940 2575.083 W71 9502.220 344.461 W18 9575.068 9648.885 W45 1965.953 2510.839 W72 9597.440 3403.857 W19 7922.073 9594.924 W46 758.543 539.501 W73 7512.671 2550.951 W20 8491.293 9339.932 W47 1818.470 2638.029 W74 8407.173 2542.822 W21 6787.352 7577.401 W48 1455.390 1360.686 W75 8142.848 2435.250 W22 7655.168 7951.999 W49 4018.080 759.667 W76 9292.636 3499.838 W23 7093.648 7546.867 W50 2399.162 1233.189 W77 9171.937 2858.390 W24 8909.033 9592.914 W51 1839.078 2399.525 W78 9340.107 1299.062 W25 7572.002 7537.291 W52 4172.671 496.544 W79 6323.592 975.404 W26 5307.976 7791.672 W53 3692.468 1112.028 W80 6557.407 357.117 W27 6892.145 7481.516 W54
 x (m) y (m) Number x (m) y (m) Number x (m) y (m) Number 5438.859 7210.466 W1 6951.63 4991.16 W65 2547.902 2240.4 W129 3658.162 7635.046 W2 377.3887 8851.68 W66 1886.62 2874.982 W130 3934.564 6714.311 W3 1365.531 7212.275 W67 2702.704 1970.538 W131 5522.913 6298.834 W4 1067.619 6537.573 W68 2702.943 2084.614 W132 4670.682 6481.984 W5 305.4095 7440.743 W69 2955.338 3329.363 W133 4241.668 5078.583 W6 596.1887 6819.719 W70 8443.088 1947.643 W134 4886.09 5785.251 W7 154.8713 9840.637 W71 8010.146 292.2028 W135 5000.224 4799.221 W8 526.77 7378.581 W72 6987.458 1978.098 W136 5004.716 4710.884 W9 326.0082 5611.998 W73 8865.119 286.7415 W137 5268.758 4167.995 W10 1564.05 8555.228 W74 7224.396 1498.654 W138 4896.876 3394.934 W11 911.1346 5762.094 W75 9990.804 1711.211 W139 5985.237 4709.243 W12 444.5409 7549.333 W76 8177.606 2607.28 W140 4400.851 5271.427 W13 5.223754 8654.386 W77 6951.405 679.9277 W141 4624.492 4243.49 W14 834.8281 6259.598 W78 9159.912 11.51057 W142 5822.492 5407.393 W15 319.9102 6147.134 W79 8699.41 2647.79 W143 4045.8 4483.729 W16 426.5241 6351.979 W80 6962.663 938.2003 W144 1671.684 1062.163 W17 252.2818 8422.066 W81 7688.543 1672.535 W145 1781.325 1280.144 W18 3507.271 9390.016 W82 8013.476 2278.429 W146 1614.847 1787.662 W19 3111.023 9233.796 W83 8448.557 2094.051 W147 2362.306 1193.962 W20 3674.366 9879.82 W84 9479.331 820.7121 W148 2186.766 1057.983 W21 4323.915 8253.138 W85 7378.417 634.045 W149 1096.975 635.9137 W22 5478.709 9427.37 W86 4709.233 2304.882 W150 1920.283 1388.742 W23 4177.441 9830.525 W87 4302.074 1848.163 W151 1117.057 1362.925 W24 4607.259 9816.38 W88 4388.7 1111.192 W152 1476.082 549.7415 W25 4574.244 8753.716 W89 5211.358 2315.944 W153 169.8294 1208.596 W26 5180.521 9436.226 W90 4899.014 1679.271 W154 1057.094 1420.411 W27 5224.953 9937.046 W91 3724.097 1981.184 W155 1339.313 308.8955 W28 4386.45 8335.006 W92 4252.593 3127.189 W156 2276.643 4356.987 W29 5144.235 8842.81 W93 4228.857 942.2934 W157 2580.647 4087.198 W30 5827.91 8153.972 W94 4734.86 1527.212 W158 2372.836 4588.488 W31 6125.665 9899.502 W95 3180.741 1192.145 W159 1733.886 3909.378 W32 4980.943 9008.525 W96 3477.127 1499.973 W160 2919.841 4316.512 W33 5746.612 8451.782 W97 3773.955 2160.189 W161 2691.194 4228.356 W34 5216.498 967.3003 W98 3624.115 495.3258 W162 1904.333 3689.165 W35 4820.221 1206.116 W99 4895.7 1925.104 W163 1909.237 4282.53 W36 5895.075 2261.877 W100 4170.29 2059.755 W164 2427.854 4424.023 W37 5840.693 1077.69 W101 4888.977 6240.601 W165 3308.579 4243.095 W38 5943.563 225.1259 W102 4941.739 7790.517 W166 1998.628 4069.548 W39 6385.308 336.0384 W103 3846.191 5829.864 W167 1897.104 4950.058 W40 6753.321 67.15314 W104 5308.643 6544.457 W168 2818.669 5385.967 W41 4713.572 357.6273 W105 4076.192 8199.812 W169 1239.323 4903.573 W42 4423.054 195.7762 W106 4609.164 7701.597 W170 1776.025 3985.895 W43 5880.261 1547.523 W107 3411.246 6073.892 W171 855.158 2624.822 W44 5340.641 899.5068 W108 4257.288 6444.428 W172 987.1228 2618.712 W45 5737.098 520.7789 W109 4587.255 6619.448 W173 604.7118 3992.578 W46 6224.751 5870.447 W110 4794.632 6393.17 W174 688.061 3195.997 W47 6028.431 7112.158 W111 5447.161 6473.115 W175 908.2329 2664.715 W48 6596.053 5185.949 W112 9132.868 7961.839 W176 1536.567 2810.053 W49 6568.599 6279.734 W113 7150.371 9037.206 W177 1230.837 2054.942 W50 6663.389 5391.265 W114 6176.664 8594.423 W178 1465.149 1890.722 W51 6981.055 6665.279 W115 8181.486 8175.471 W179 6443.181 3786.094 W52 6959.493 6998.878 W116 8313.797 8033.644 W180 5948.961 2622.117 W53 6833.632 5465.931 W117 7183.589 9686.493 W181 6791.355 3955.152 W54 6476.176 6790.168 W118 6377.091 9576.939 W182 6447.645 3762.722 W55 6278.964 7719.804 W119 6678.327 8443.922 W183 6170.909 2652.809 W56 6834.159 7040.474 W120 6357.867 9451.741 W184 7302.488 3438.77 W57 7386.403 5859.87 W121 8419.292 8329.168 W185 5313.339 3251.457 W58 6609.446 7297.519 W122 8085.141 7550.771 W186 6021.705 3867.712 W59 7690.291 5814.465 W123 7487.057 8255.838 W187 6073.039 4501.377 W60 5649.796 6403.118 W124 2259.218 1707.08 W188 6620.096 4161.586 W61 2077.423 3012.463 W125 2217.467 1174.177 W189 5860.921 2621.453 W62 2966.759 3187.783 W126 424.3114 714.4546 W190 6877.961 3592.282 W63 1829.225 2399.32 W127 834.6981 1331.71 W191 6786.523 4951.77 W64 2518.061 2904.407 W128

#### 2. Literature Review

The complexity and large scale of the corresponding mathematical models make MINLP problems a popular and difficult topic to address for scholars around the world. The methods for solving MINLP problems can be generally divided into deterministic methods and heuristic methods. Methods of the former type can yield globally optimal solutions for small-scale problems but are often inefficient for large-scale problems, whereas the quality of the solutions obtained with the heuristic methods that have been proposed to date usually cannot be guaranteed, and these methods cannot be used to solve large-scale problems within a limited time.

The problem of interest here is different from the well group partitioning (WGP) problem, which usually considers factors such as the gathering radius and pipeline diameter. The purpose of the conventional WGP model is to determine the optimal mapping between wells and stations. In other words, subject to well constraints and other constraints, the wells are divided among different gathering stations to save construction costs. This problem is a kind of location-allocation problem. In a discrete location-allocation problem, P facilities are selected from a set of available candidate sites such that a given objective function is minimized. This problem is specifically referred to as the P-median problem. However, there are some differences between the model established in this paper and the general P-median model. In the problem considered here, the alternative facility (MV) locations are the same as the well locations, meaning that , whereas in the conventional P-median problem, is often less than n. Moreover, the optimization target of the conventional P-median problem is usually to minimize the total distance from facilities to customers, while in this paper, the target is to minimize the cost of all equipment, including pipelines and devices, and the model is constrained by the special constraints of MVs. In addition, special constraints on the numbers of channels and facilities (MVs) are imposed in the model considered here. All of these factors increase the scale of the optimization problem, making the problem more complicated and difficult to solve. In [1, 2], the authors have proven that the P-median problem is an NP-hard problem when the number of facilities is uncertain. That is, the MINLP problem solved in this paper is a special case of the location-allocation problem. In this problem, the number of MVs m and the associations between wells and valves are treated as optimization variables. The associations between pipelines and various facilities are precisely the elements of interest in oil and gas field surface engineering, and reasonable optimization of the pipeline network structures between facilities can effectively reduce the engineering and construction costs. There are many possible pipeline network structures, including star, tree, and ring structures. With the development of methods for exploiting unconventional oil and gas resources, tree-tree network structures [3, 4] have been used in marginal gas reservoirs for resources such as coalbed gas and shale gas. Such a structure can considerably reduce costs by virtue of the connection configuration between the wells and co-construction with existing wells and stations. Similarly, research on star-tree  and star-star  pipeline networks has become quite mature.

A pipeline network also includes facilities such as manifolds, platforms, and compressors. Many studies have investigated oil and gas network models that contain various types of facilities. Rodrigues et al.  proposed a 0-1 programming model to determine the connections between FPSO units, manifolds, and facilities as well as the sizes and locations of platforms. In [8, 9], the authors proposed a mixed integer linear programming (MILP) model to optimize the connections between wells and platforms while considering not only the installation time but also the linear drop of the reservoir pressure. Furthermore, Ramos Rosa et al.  considered the reservoir dynamics and secondary development to establish an MILP model seeking the maximum net present value (NPV) while simultaneously optimizing the manifold layout, routing, and pipeline diameter. Most of the oil and gas field layout optimization problems mentioned above can be transformed into mixed integer programming (MIP) problems, i.e., programming problems in which some of the independent variables take integer values while others take continuous values. MIP first became an independent branch of mathematical problem-solving when Gomory  proposed the cutting-plane method. Generally, MINLP problems can be solved using two types of methods: deterministic algorithms and heuristic algorithms. A deterministic method of solving an MINLP problem is to simplify it to an MILP problem, which can be effectively solved by means of dual simplex or sequential quadratic programming. However, the main difficulty is that appropriate linear approximations must be adopted for nonlinear functions. To this end, Möller , Martin et al. , Tomasgard , and Nørstebø  et al. used piecewise approximation to deal with the nonlinear functions in the model. Mikolajková et al.  proposed a linearization method for solving MINLP problems. Deterministic methods can guarantee a globally optimal solution when solving an MINLP problem, but they usually require considerable resources to solve large-scale problems.

Because of the complexity of MINLP problems, metaheuristic algorithms, including genetic algorithms (GAs)  and particle swarm optimization (PSO) , are also used to solve problems related to oil and gas pipeline networks. However, the computational burden of heuristic algorithms is usually heavy, making them an inefficient means of solving large-scale problems. In addition, either heuristic methods converge slowly or the quality of their results cannot be guaranteed. Because of the special characteristics of MVs, no related research on optimization problems involving MVs has been reported in recent years; therefore, to address this lack, a corresponding MINLP model and an algorithm for solving it are proposed in this paper.

#### 3. Problem Definition and Formulation

##### 3.1. Connection Matrix and Assumptions

In general, optimal solutions to various problems can be found through various optimization methods, but the solution found may not necessarily be suitable for the actual conditions and engineering design requirements of a particular situation. For a given oil and gas field, especially an unconventional reservoir with scattered wells, it is necessary to reserve some MV channels for future connections to additional production wells; therefore, either the channel counts of the MVs and the number of MVs need to be fixed in advance or ranges should be defined for these parameters to satisfy the need to reserve valves or channels for future use. If experts define several possible schemes in accordance with the given requirements, the most appropriate solution can be obtained by comparing the results of these schemes, but the quality of a solution chosen in this way cannot be guaranteed; thus, it is necessary to formulate and solve the corresponding optimization problem with constraints on the number of MVs and their channel counts.

Suppose that there are n wells that need to be divided into m sets and connected to the corresponding MVs. The MVs will be placed at well sites because no additional land acquisition is allowed; therefore, all well sites are possible alternative locations for MVs, and the number of MVs m is unknown. Let be the connection matrix, which reflects the connection relationships between the wells and the MVs. If , then well i is in well set j, which means that well i is connected to MV j. Thus, the remaining elements in the ith column of the matrix are all equal to 0. Here, a matrix , with dimensions of is introduced to indicate whether an MV is placed at the ith well site. If , this means that an MV is placed at the ith well site. If , this indicates that there is no MV placed at the ith well site. By incorporating into the matrix , we obtain the matrix , as shown in Figure 2. If in the matrix , then the corresponding MV can be identified by finding the row j that contains the element in the ith column of (below the first row) that has a value of 1; that is, MV j is placed at the ith well site. Thus, and the other wells connected to valve j can be found at the same time. Now, considering the characteristics of MVs, the following assumptions are adopted before solving the mathematical model in this paper:(1)Generally, the number of channels of an MV ranges from 6 to 16. According to the designer’s requirements, the corresponding MV specification will be given as either a constant or a range before optimization. Thus, the special constraints on the MVs will be specified beforehand.(2)All wells need to connect directly to an MV because this is the only way in which wells can be selected for metering; there are no pipeline connections between wells.(3)The possible locations of the MVs are preselected by the engineering company. One of the strategies that is usually adopted is to place the MVs at well sites to reduce land acquisition, which means that all well sites are potential alternative locations for MVs.(4)In this paper, we use the Euclidean distance to quantify the length of the pipeline between a well and an MV. Additionally, terrain and obstacles are not considered.(5)The costs of valves and instruments other than the MVs are not considered. These costs are neglected in the calculation since the use of MVs simplifies the metering process.(6)We assume that the same material and diameter are used for all pipelines. Therefore, factors such as pressure and flow rate are not considered.

##### 3.2. MINLP Model
###### 3.2.1. Objective Function

Given the above assumptions, the objective function can be simply expressed as follows:where represents the cost of all pipelines, which is equal to the sum of the costs of the pipelines connected to each MV:where represents the total cost of all pipelines connected to MV j, which can be expressed as follows:where “” denotes the matrix dot product operation.

The total cost of the pipelines connected to valve j is the dot product of , , and , where is the pipeline cost coefficient matrix, is the jth row of the connection matrix, and is a matrix consisting of the pipeline lengths from the wells to the valve. represents the coordinates of the ith well, . Specifically, , means that the ith well is connected to the jth MV. Then, represents the pipeline length from well i to the corresponding MV under the assumption that this well is connected to valve j; this pipeline length can be expressed as

represents the coordinates of the jth MV, which are unknown and are determined by solving a subproblem. The detailed solution method will be described in Section 4. Then, the cost of all pipelines can be written as follows:

Considering the costs of the MVs themselves, the total cost of all MVs can be expressed aswhere is a matrix with dimensions of that is used as a compact representation of the connection matrix, in which is a binary variable. If all elements in the jth row of the connection matrix are 0, then ; otherwise, . is the matrix of the MV cost coefficients. is the dot product of and . By summing the costs of the pipelines and valves, the objective function is obtained as follows:

###### 3.2.2. Constraints

The objective function has been established, as shown in equation (7). Now, we will describe the constraints of the model. In this paper, the variable m is constrained by and , as follows:where is the upper bound on the number of MVs, while is the minimum value that m can take; both and are integers. As the range of m shrinks, the number of MVs will finally be specified; that is, . Only one nonzero element is allowed to exist in each column of the connection matrix because each well can be connected to only one MV. This constraint can be expressed as follows:

Given that row in the connection matrix represents one MV, the sum of the elements in each row of this matrix cannot be greater than the channel count of the corresponding MV. Since the channel counts of the MVs selected by the designer might differ, is used to represent the maximum MV channel count provided by the manufacturer, and is used to represent the specified channel count of the jth MV:

The following constraints should be satisfied:

If is the channel count of MV j as chosen or required by the designer and is a binary variable that represents whether a specified channel count for the jth valve is actually used to constrain the mathematical model, according to the constraint relaxation degree, then the maximum number of wells that can actually be connected to MV j is defined as

By combining this equation with equation (11), we obtain

Usually, it is not permitted to connect zero wells to a given MV. Therefore, taking 1 as the default lower bound for each row of the matrix, we have

Sometimes, it is unreasonable to connect only a few wells to an MV. In this case, we use to indicate a specified lower bound on the number of wells to which an MV is allowed to be connected. The corresponding constraint can be expressed as

The minimum number of wells that can be connected to MV j is determined by . is a binary variable that controls the value of . If , a specific lower bound is not required and the default is used, whereas if the value of this variable is 0, a specific lower bound is applied. Thus, using the same formulation used for the upper bound, we obtain the following constraint:

By combining equations (16)∼(18), we can write the constraint related to the lower bound as follows:

The following relationships should be satisfied:

Equations (20) and (21) are used to ensure that the lower bound will not exceed the upper bound. Equations (22) and (23) ensure that both the required lower bound and the bound that is actually applied are at least as large as the default lower bound. Finally, equation (24) stipulates that the real lower bound should not exceed the required lower bound. Since the matrix represents the MVs, the sum of the elements of the matrix should satisfy

Because only one MV is allowed in each well group, when we subtract from any row of the connection matrix L and then add the absolute value of the results together, we will have for each row of . Then, the constraint that only one MV is allowed in each well group can be written as

According to the above equations, the mathematical model can be expressed as

The decision variables used in the mathematical model represent the associations between the wells and the MVs, and the number of valves and their channel counts are treated as constraints. The binary variables and determine the range of the channel counts of the MVs, while and determine the range of the number of MVs.

#### 4. The Proposed Algorithm for Solving the MINLP Problem

The underlying principle of methods such as steepest gradient descent  is to find the direction in which the objective function decreases the fastest at the present point, take a step in that direction, and then repeat this process, thus iteratively reducing the value of the objective function. We use the same idea to search for the optimal solution in a discrete space. In this paper, the mathematical model is described in the matrix form; thus, the step size and the number of steps must be expressed in terms of a measure of the distance between two matrices. If we treat the whole matrix as a point, then we can move one nonzero element in the matrix to obtain a new matrix, and this new matrix can be treated as a point adjacent to the original point; that is, the new matrix is one step away from the original matrix. All the points obtained by changing one individual element are the points closest to the current point. As an example, let us consider a connection matrix that is equivalent to the 5 × 5 unit matrix; then, we can obtain a foresight matrix corresponding to this connection matrix, as shown in Figure 3. The square columns represent the objective function values, where each row of the matrix represents a well and each column represents an MV. Then, the values in the jth column of the matrix represent the values of the objective function after the corresponding nonzero elements are moved to the jth valve; that is, the square column in the ith row and jth column represents the objective function value after well i is connected to MV j.

If the initial connection matrix is a unit matrix, then initially, the values on the diagonal of the foresight matrix are equal to the objective function value of the original matrix because the values on the diagonal of the foresight matrix, represented by square columns with stripes, are obtained only when no element is moved. For a case in which the constraints are not satisfied after a single nonzero element is moved to another position, the corresponding objective function value is represented by a dark gray square column; all such objective function values are replaced by a sufficiently large penalty value M to ensure that the algorithm will not select such a point. White columns are used to represent the feasible domain, meaning that these positions in the matrix represent points that satisfy the constraints of the mathematical model. The minimum value in the feasible domain, denoted by , is found as the result of the current iteration, as shown in Figure 3. As shown in the figure, the minimum objective function value is found in the 4th row and 1st column; therefore, we set the element in the 4th row and 1st column to 1 and set the remaining elements in the 4th row to 0. In this way, we can find the path that causes the objective function value to drop the fastest from the current point, thus completing one single-step iteration. When all values in the feasible domain are greater than or equal to the current value, the current point is considered to be a locally optimal solution.

We can always construct a foresight matrix such as that shown in Figure 3 after every step and then find the point that minimizes the objective function for the next step until no value less than the current value of the objective function is found. To obtain a better solution once the current point has fallen into a local optimum, we can consider increasing the step size, thus enabling the algorithm to escape from the local optimum and find a better feasible solution after further iterations.

##### 4.1. Solving the Subproblem

All wells connected to MV j are assigned to the same well set. The MV will be placed at one of the wells in this set such that the total weighted length from the wells to the MV is as short as possible, as expressed in equation (23). Thus, a subproblem must be solved to find the placement of valve j that results in the shortest pipeline length in well group j:

This subproblem can be solved using a simple algorithm. The coordinates of the wells connected to valve j are represented by . Because the location of MV j is unknown, is assumed to represent the valve coordinates that minimize the sum of the pipeline lengths connected to valve j. The valve will be placed at each well site in turn; then, we can compare the results to find the optimal placement position for the valve.

##### 4.2. Details of the Algorithm

The algorithm for solving the MINLP problem obtains the solution through iteration. The iterative process mainly comprises 4 steps; this process is depicted in Figure 4 and described in detail below.

###### 4.2.1. Step 1: Initialization

If the constraint on the number of MVs m is relaxed, m can range from 1 to n. In the extreme case, a unit matrix is taken as the initial feasible solution, meaning that each MV is initially assumed to be connected to only one well. Generally, if , we can initially set , and the wells will be sequentially assigned to the n MVs in accordance with the constraint conditions.

###### 4.2.2. Step 2: Iteration

The initial costs can be obtained by substituting the initial distribution matrix into the objective function. Then, a blank matrix with dimensions of is initialized as the foresight matrix.

Starting from the first column, each nonzero element 1 in the connection matrix is moved up or down by one row; each movement will change the distribution, and the objective function value will change as the connection matrix changes. The corresponding objective function value is calculated for every movement, and the new objective function value is filled into the corresponding position of the foresight matrix.

In the example shown in Figure 5, element 1 in the first row and first column is moved to the second row and first column to obtain a new connection relationship. After substituting this new connection relationship into the objective function, the value corresponding to the new connection relationship is obtained. is then filled into the second row and first column of the foresight matrix, corresponding to the new position of the nonzero element in the connection matrix. If the constraints are not satisfied by this new connection relationship, a sufficiently large penalty value M is placed in the corresponding position instead. Once the foresight matrix has been completely filled, the path of fastest descent can be determined. This process is illustrated in Figure 5.

According to the constraints, the nonzero elements in the connection matrix can move only up and down; they cannot move left or right. Since each column of the matrix represents a well, when a nonzero element is moved from the jth row to the kth row in column i, this represents that this well is disconnected from MV j and connected to MV k, and the value of the element in the ith row and jth column returns to 0.

In this way, each 1 value in the connection matrix is moved from the top row to the bottom. Each time a 1 value is moved to a new position, a new objective function value is generated, filling in the corresponding position in the foresight matrix. When the new matrix represents a connection relationship that does not satisfy the constraints, a sufficiently large penalty value M is filled into the foresight matrix instead. Once the foresight matrix has been completed, we then search for the minimum value in this matrix that is smaller than the current value. For example, if is the smallest value in the foresight matrix, then for the next iteration, 1 in the ith column of the connection matrix is moved to the jth row, and the value at the original position is set to 0; thus, one single-step iteration is completed. By repeating this process, the current value of the objective function can be iteratively reduced, and new connection relationships can be formed. Figure 5 illustrates this iterative process for a 5 × 5 foresight matrix.

If all nonzero elements in a given row of the connection matrix leave that row by moving up or down, this means that the corresponding MV does not exist; in this case, the empty row should be removed from the connection matrix, decreasing the feasible domain. In this way, the number of MVs can be found. Specifically, if the elements of the jth row are all 0, then the jth row should be removed from the connection matrix, as shown in Figure 6.

###### 4.2.3. Step 3: Increasing the Step Size

If a lower cost cannot be obtained by moving a single element in the connection matrix up or down, this means that the solution has become stuck in a local optimum. Therefore, the step size must be gradually increased to allow the algorithm to escape from this local optimum. For example, suppose that the matrix L is initially set as the unit matrix and that the initial step size is 1. When the objective function value cannot continue to decrease after a certain number of iterations, then increasing the step size can help to find a new connection relationship that will allow the objective function to continue to decrease. More specifically, suppose that the ith row of the matrix L contains two nonzero elements, meaning that MV i is connected to two wells, while the jth row contains three nonzero elements. In this case, both nonzero elements in the ith row are added to the jth row, setting all elements in the ith row to zero, and the new total cost under the new connection relationship is obtained. This cost value is filled into the ith row and jth column of the foresight matrix. Then, the matrix is restored to its previous state, and the algorithm proceeds to fill in the other elements in the foresight matrix in the same way. Each element in the foresight matrix is an objective function value, and these values are used as the basis for iteration.

In the foresight matrix for multistep iteration, the values on the diagonal should be equivalent to the objective function value when no wells are moved because for each element on the diagonal, the row and column correspond to the same well; therefore, the penalty value M is filled into the positions on the diagonal of the matrix. Moreover, the matrix elements below the diagonal to the left are symmetric to the matrix elements above the diagonal to the right; therefore, all matrix elements in either the lower left or the upper right can be directly set equal to M to avoid redundant calculations. As before, if the constraints are not satisfied after the elements of the connection matrix are moved, the penalty value M is also directly filled into the corresponding position. After the foresight matrix has been fully filled in, the smallest value in this matrix can be found. If this minimum value is smaller than the cost obtained in the previous iteration, then we can find the row and column corresponding to this minimum value, add the elements of the ith row to the jth row, and remove the ith row, whose elements are now all 0. At this time, the number of MVs m is set to m minus 1.

If the remaining channels of an MV do not allow connection to any further wells, the wells will be sequentially assigned to other MVs. This iterative process will repeat until the objective function value can no longer be reduced by such multistep iterations. Then, the algorithm will either return to Step 1 or proceed to Step 4.

If there is no MV whose channels are fully occupied or for which the number of wells connected to that valve has reached the lower bound , then the optimal solution can be obtained through single-step or multistep iteration. However, according to the constraints, if an MV is fully occupied, then it cannot be connected to any new wells, whereas if a valve has reached its lower bound , then no more wells can be moved out of the corresponding row. In this case, the iterative process can be continued as follows:(1)The step size can be increased to 2 to test whether the objective function value will continue to decrease. First, we look for the fully occupied MV. Suppose that wells are connected to this MV; then, the remaining wells will each be moved to this MV in turn, replacing one of the existing wells, which is moved to another MV that meets the constraints. By repeating this process, the new values of the objective function are obtained and written into the foresight matrix based on the corresponding well numbers i and j. Finally, the path of steepest descent can be determined in the same way as before.(2)For an MV j that has been fully connected, we look for an MV t with more channels than MV j, where denotes the number of channels of MV j. Then, , and the total number of wells connected to each valve satisfies . Since , must be satisfied to allow the jth and tth valves to be exchanged; that is, the following conditions should be satisfied simultaneously:

If these conditions are satisfied, then the channel counts of the jth and tth MVs are exchanged, and we continue iterating steps 2 and 3 mentioned above. However, if i and t do not satisfy , , and , then the penalty value M is written into the corresponding position of the foresight matrix. The process above is repeated; if, after the Nth iteration, the objective function value cannot continue to decrease, meaning that , then the current objective function value corresponds to the optimal solution.

#### 5. Analysis of Examples

##### 5.1. Verification of the Validity of the Algorithm

First, a simple example is presented to demonstrate that the proposed algorithm can effectively solve the problem of interest. The associations between the wells and the MVs and the number of MVs are taken as variables, and their values are found by the algorithm via iteration; in particular, during the iterative process, multistep iteration can help the objective function value to escape from a local optimum. Let us consider a simple example for which we can make a reasonable prediction of the expected solution. Then, we can apply the proposed algorithm to solve the example and compare its solution with our expected result to prove that the algorithm can effectively solve the presented MINLP model.

For all examples in this paper, we consider steel pipes with an inner diameter of 53 mm and a wall thickness of 3.5 mm as the pipes linking the wells and valves; thus, the cost coefficient is a constant. Moreover, the cost of an MV is a known constant once the vender is decided. In this paper, the cost of an MV is treated as an average value that does not vary with the number of channels. Here, we consider an example of a field with 26 wells (Table 1; W10∼W35), for which the channel counts of the MVs and the number of MVs are not limited; therefore, the number of channels of each MV is uniformly set to 16, which is the maximum number of channels per MV that can be provided by general manufacturers.

We first perform single-step iteration, during which the number of MVs will usually stay at a relatively large value, as shown in Figure 7. When the identity matrix is used as the initial matrix, the number of MVs cannot be decreased below 8 via this process. However, increasing the step size can lead to further reduction in the value of the objective function, demonstrating that the reason that the objective function cannot be further reduced via single-step iteration is that the step size of each advance is too small. Thus, the objective function becomes stuck in a local optimum, and it is necessary to increase the step size to escape from that local optimum to allow the objective function to continue to decrease. The objective function decreases almost linearly during the initial 18 steps because the main factor affecting the decrease in the objective function value during the initial iterations is the number of MVs rather than the pipeline cost. However, the value of the objective function no longer decreases once it drops into a local optimum. After the 18th iteration, the step size must be increased to continue to reduce the objective function value. The single-step iteration process converges prematurely after the 18th iteration; thus, the findings demonstrate that the strategy of increasing the step size can efficiently help the objective function to escape from a local optimum.

The result of optimization with an increase in step size is shown in Figure 8. The hollow square symbols represent the positions of the MVs, the triangular symbols represent the wells connected to MV 1, and the circular symbols represent the wells connected to MV 2. For the 26 wells considered in this calculation, the algorithm completely divides them into two groups, and the solution satisfies all the constraints, which is obviously consistent with our expectations. This example proves that the proposed algorithm can be applied to solve the problem of interest and obtain reasonable optimization results.

##### 5.2. Convergence of the Algorithm

Usually, we can judge whether an algorithm is valid according to whether the value of the objective function converges during the iterative process. If the algorithm is divergent, no matter how many times it is iterated, the value of the objective function will always fluctuate and will not approach a constant. In contrast, convergence of the algorithm means that the objective function value will tend toward stability after some numbers of iterations. In summary, convergence is a necessary condition for the algorithm to be valid.

Next, various examples are presented to analyze the convergence of the algorithm. First, we consider a field with 35 wells (Table 2; W1∼W35) as an example. MVs with four different channel counts are specified: . Thus, the number of channels of each MV is fixed in advance, the limitations on the numbers of pipelines that are allowed to be connected to different valves are different, and the number of MVs is also given, i.e., m = 4.

Figure 9 shows the iterative process for these 35 wells, which includes single-step iteration, double-step iteration, and redistribution of different kinds of valves. The iterative process still converges, and the optimal solution is obtained after the 21st iteration. After 14 iterations, the redistribution of different valve types is necessary to allow the objective function value to continue to decrease. Finally, a difference of 0 between two consecutive steps indicates that the final solution has been obtained.

As seen from Figure 9, the objective function value gradually tends toward stability and converges as iteration continues. The result is shown in Figure 10. Red circular symbols, blue triangular symbols, purple diamond symbols, and yellow triangular symbols are used to represent the individual wells connected to MVs 1, 2, 3, and 4, respectively, and square symbols are used to represent the MVs. The initial channel counts of the MVs corresponding to these different well groups are 14, 12, 10, and 8, respectively, while the final channel counts are 12, 10, 14, and 8, in sequence. In the case of tight constraints, the minimum value of the objective function can be obtained by using the proposed optimization algorithm.

In addition to the example above, an example with 40 wells (Table 1; W69∼108) and various channel counts is considered to prove the convergence of the algorithm; the results are shown in Figures 11(a) and 11(b). It can be seen from the figures that, during the initial iterations, the objective function value decreases at a relatively fast rate. Then, the rate of decrease slowly declines as iteration proceeds. The algorithm eventually converges when .

Again, the value of the objective function gradually converges and tends toward stability. In Figure 11(a), the iterative processes with a total of 46 and 48 channels are identical, indicating that once the number of channels exceeds a certain threshold, a further increase in the number of channels has no effect on the iterative process or the result. Additionally, although the iterative process for an initial feasible solution with 40 channels is different from the other 4 cases, the final result is not significantly different from the others, indicating that the initial solution does not have a great influence on the result.

The iterative processes for total channel counts of 46 and 48 are also fundamentally similar. Once the total number of channels exceeds 46, the iterative processes are basically the same. The results show that the algorithm presented above can stably converge. Therefore, these examples demonstrate that even under different constraints and in different scenarios, the algorithm can drive the objective function value toward stable convergence.

##### 5.3. Parameter Analysis

In general, the designer of a pipeline system should consider the possibility of reserving channels for new wells in the future along with other factors when specifying the channel counts of the MVs and the number of MVs. However, different combinations of MVs with different numbers of channels might affect the results; therefore, we will analyze the effects of different channel counts and different numbers of MVs on the results of our algorithm. We consider a field with 40 wells (Table 1; W69∼108) as our example here, varying the constraints on the numbers of channels and valves to obtain different results. The results are shown in Table 6, from which we can see the influence of these different constraint parameters on the cost of the pipeline system.

 Total channels Cost (¥) Number of MVs Individual channel counts 40 1.424 × 106 3 16, 16, 8 42 1.396 × 106 3 16, 16, 10 44 1.396 × 106 3 16, 14, 14 46 1.396 × 106 3 16, 16, 14 48 1.396 × 106 3 16, 16, 16 50 1.304 × 106 3 16, 16, 18 40 1.306 × 106 4 10, 10, 10, 10 42 1.287 × 106 4 12, 10, 10, 10 44 1.287 × 106 4 12, 12, 10, 10 46 1.287 × 106 4 12, 12, 12, 10 48 1.287 × 106 4 12, 12, 12, 12 50 1.287 × 106 4 14, 12, 12, 12 40 1.647 × 106 5 8, 8, 8, 8, 8 42 1.520 × 106 5 8, 8, 8, 8, 10 44 1.426 × 106 5 8, 8, 8, 10, 10 46 1.426 × 106 5 8, 8, 10, 10, 10 48 1.426 × 106 5 8, 8, 10, 10, 12 50 1.426 × 106 5 8, 8, 10, 12, 12 40 1.641 × 106 6 7, 6, 7, 7, 7, 6 42 1.619 × 106 6 8, 8, 6, 6, 6, 8 44 1.615 × 106 6 8, 8, 8, 8, 6, 6 46 1.592 × 106 6 8, 8, 8, 10, 6, 6 48 1.592 × 106 6 10, 8, 8, 10, 6, 6 50 1.592 × 106 6 10, 8, 8, 8, 8, 8 40 1.893 × 106 7 6, 6, 6, 6, 6, 4, 6 42 1.870 × 106 7 6, 6, 6, 6, 6, 6, 6 44 1.775 × 106 7 6, 8, 6, 6, 6, 6, 6 46 1.766 × 106 7 8, 8, 6, 6, 6, 6, 6 48 1.766 × 106 7 6, 8, 6, 8, 8, 6, 6 50 1.766 × 106 7 8, 8, 6, 8, 6, 6, 6 40 1.957 × 106 8 5, 5, 5, 5, 5, 5, 5, 5 42 1.950 × 106 8 6, 6, 5, 5, 5, 5, 5, 5 44 1.941 × 106 8 6, 6, 5, 5, 6, 6, 5, 5 46 1.941 × 106 8 6, 6, 5, 5, 6, 6, 6, 6 48 1.941 × 106 8 6, 6, 5, 5, 6, 6, 8, 6 50 1.919 × 106 8 8, 6, 6, 6, 6, 6, 6, 6

Six different numbers of MVs are specified: 3, 4, 5, 6, 7, and 8. The table shows that the requirement of the use of too few MVs will lead to additional costs when the total number of channels is the same. The cost also rises if more than 4 MVs are used, and the smaller the total number of channels is, the faster the increase of cost is. When the total number of channels is fewer than or equal to 42, the cost rises at a significantly higher rate than in the other cases. Therefore, the total number of channels should be set to greater than 42 to make it easier to achieve lower costs. However, increasing the number of channels does not significantly reduce the overall cost once the total number of channels exceeds 48. The number of MVs has a greater impact on the cost. These findings suggest that the optimization of the number of MVs is the dominant factor affecting the value of the objective function, while the number of channels is the secondary influencing factor. These observations can provide a reasonable basis for practical design.

Next, a set of well distribution data from the field and a set of data randomly generated by MATLAB are used as examples to analyze how the well distribution density affects the optimization result. In the first of these examples, the coordinates of 40 wells (Table 1; W69∼108) that are densely distributed in an actual production field are considered for optimization. The number of MVs m is treated as an optimization variable, and each MV is specified to have 16 channels; in the optimization result obtained, four MVs are used. It can be seen from Figure 12(a) that the optimization result is reasonable.

In the second example, the positions of the 40 wells are randomly generated using the rand function (Table 3; W1∼W40), and the number of MVs is constrained to a range. The upper limit on the number of channels is uniformly set to 16 for each MV, and the problem is solved using the proposed algorithm. The well connections of the six MVs are shown in Figure 12(b). These optimization results are also reasonable.

##### 5.4. Stability of the Algorithm

In this section, scatter plots are presented in Figure 13 to show the well and MV distribution results calculated using the proposed algorithm under the condition that the maximum number of channels per MV is constrained to 16 and the number of MVs is constrained to a certain range. The results prove that the algorithm can stably solve the MINLP problem and obtain reasonable results. Figures 13(a) and 13(b) show the result generated using the data based on W140∼W180 (Table 1), Figures 13(c) and 13(d) show the result generated using the data based on W67∼W128 (Table 1), Figure 13(e) shows the result generated using the data based on W1∼W80 (Table 1), and Figure 13(f) shows the result generated using the data based on W1∼W80 (Table 4).

##### 5.5. Algorithm Performance

The scale of the computations performed in the proposed algorithm mainly depends on the size of the foresight matrix and the number of iterations. Under the assumption that all solutions are feasible, the maximum time complexity of the algorithm is , where m and n denote the numbers of rows and columns, respectively, of the matrix and N is the number of iterations. If the coordinate data are given in order, the time consumption is smaller, and if they are random, the time consumption is larger, as seen by comparing the examples with 191 wells in Table 7. The random data for 191 wells are shown in Table 5. For the examples with 26 and 41 wells, for the first example of each type listed in Table 7, the identity matrix is considered the initial feasible solution, and the computational scale is then reduced through multistep iteration. By comparison with the second example in each case, it is confirmed that the multistep iteration strategy can efficiently reduce the scale of the computations performed, thus saving time. However, regardless of the type of iteration used, the same optimal solutions can be found. In the second listed examples with 35, 60, and 80 wells, there is a specified upper bound on the number of channels for each MV, and the number of MVs is constant. The computation times are increased in these cases mainly because the single-step iteration process requires more iterations to approach the optimal solution. The solution times shown in Table 7 essentially meet our expectations with regard to the algorithm complexity. Thus, it is proven that the proposed algorithm can be used to solve various problem instances within a short time.

 Wells Type Channels per MV m (range) f(x) (¥) m (outcome) Time 26 Field All 16 2∼6 0.866 × 106 2 0.255 s 26 Field 16, 16 2 0.866 × 106 2 0.335 s 35 Field All 16 3∼6 1.118 × 106 3 0.836 s 35 Field 8, 10, 8, 10 4 1.394 × 106 4 0.854 s 41 Field All 16 3∼8 1.316 × 106 4 0.877 s 41 Field All 16 4 1.316 × 106 4 1.277 s 60 Field All 16 6∼10 1.91 × 106 6 2.554 s 60 Field 16, 8, 12, 12, 8, 10 6 1.945 × 106 6 3.418 s 80 Random All 16 6∼10 4.330 × 106 7 5.553 s 80 Random 10, 10, 16, 16, 8, 8, 12, 12 8 4.381 × 106 8 8.336 s 191 Field All 16 12 5.350 × 106 12 42.817 s 191 Random All 16 12 7.382 × 106 12 78.645 s

Finally, a large-scale problem with 192 wells (Table 1; W1∼W192) is considered, and the result is shown in Figure 14. Here, the channel count for each valve is constrained to 16, and the number of valves is allowed to vary from 15 to 39. It can be seen that the algorithm can still successfully separate different wells into different blocks in the face of a large number of wells with multiple valves. The results show that the algorithm can still effectively solve such a large-scale problem and yield reliable results. All of the above solution processes were implemented in MATLAB on an ASUS GX501VSK laptop (Intel Core i7-7700HQ Processor, 16 GB of DDR4 RAM).

#### 6. Conclusion

This paper proposes the replacement of traditional metering processes with a process based on MVs, thus simplifying and automating the necessary operations and reducing the investment in redundant valve instrumentation and other equipment, the amount of space occupied by this equipment, and the cost of the station. A generalized MINLP model is presented, and a corresponding heuristic algorithm for MV optimization design is proposed for solving the presented MINLP problem. The numbers of channels of the MVs and the number of MVs are treated as special constraints in the MINLP problem.

The coordinates of actual production well sites and randomly generated data are used as examples for calculation and analysis. From the analysis of these examples, it can be seen that the behavior of the objective function value in the proposed algorithm is strictly convergent. A large collection of optimization results demonstrates that the algorithm shows stable convergence and good reliability. The analysis results show that the proposed heuristic algorithm can quickly and effectively solve the presented MINLP model for the design of a pipeline system with MVs. In this paper, we do not consider the optimization of the MV process itself in an integrated gathering system; this problem will be addressed in the future.

#### Nomenclature

 : Binary variable. If , then a specific number of channels are given. By default, if , the number of channels of the jth MV is given by the manufacturer : Binary variable. If , then a specific lower bound on the number of wells to be connected to the jth MV is given, whereas a value of 0 indicates that the default lower bound is used : Number of channels of a fully connected MV : Number of channels of MV j : Matrix used as a compact representation of the connection matrix : Binary variable used to indicate MVs without any connected channels. If all elements in the jth row of are 0, then ; otherwise, : Total cost : Total cost of pipelines : Total cost of MVs : Total cost of the pipelines connected to the jth MV : Total length of pipelines : Length of the pipelines connected to MV j : Set of the total pipeline lengths obtained by placing MV j at all possible different well sites : Matrix representing the connection relationship between the wells and the valves : Element in the ith row and jth column of . indicates that the ith well is connected to the jth valve, and indicates that there is no connection between the ith well and the jth valve : Matrix with dimensions of used to express where the valves are placed. indicates that an MV is placed at the ith well site, and a value of 0 indicates no MV is placed at the corresponding well site : Connection matrix incorporating the information about the placement of the MVs : Penalty value m: Number of MVs n: Number of wells : Number of channels ultimately selected for MV j : Lower bound actually applied on the number of wells connected to MV j : Specified number of channels for MV j : Specified lower bound on the number of wells to be connected to valve j : Default lower bound on the number of wells to be connected to valve j : Maximum number of channels per MV that a manufacturer can provide and also the maximum value that can take : Set of well coordinates : Indicator that the ith well is connected to the jth MV : Set of MV coordinates : Objective function value obtained by moving the nonzero element in the jth column to the ith row : Foresight matrix : Minimum objective function value after N iterations : Lower bound on the number of MVs : Upper bound on the number of MVs : Cost coefficient matrix for pipelines : Cost coefficient matrix for MVs.

#### Data Availability

The coordinate data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (51704253).

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