{"id":3390,"date":"2026-05-19T10:03:33","date_gmt":"2026-05-19T10:03:33","guid":{"rendered":"https:\/\/gwits.pl\/?p=3390"},"modified":"2026-09-07T11:32:41","modified_gmt":"2026-09-07T11:32:41","slug":"1065545gwits20260504","status":"publish","type":"post","link":"https:\/\/gwits.pl\/en\/1065545gwits20260504\/","title":{"rendered":"Wykrywanie i kontrolowanie wyciek\u00f3w gazu \/ Gas leakage detection and control"},"content":{"rendered":"\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"wp-block-paragraph\"><em><a href=\"https:\/\/orcid.org\/0000-0001-8543-2336\" target=\"_blank\" rel=\"noreferrer noopener\">Andrzej J. Osiadacz<\/a><a href=\"https:\/\/orcid.org\/0009-0003-7458-0888\" target=\"_blank\" rel=\"noreferrer noopener\"> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" class=\"wp-image-2724\" style=\"width:16px\" src=\"http:\/\/gwits.pl\/wp-content\/uploads\/2025\/03\/ORCID.png\" alt=\"ORCID ID\"><\/a> <sup>1*<\/sup><\/em><\/p>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>Afiliacja<\/summary>\n<p class=\"wp-block-paragraph\"><sup>1<\/sup> Wydzia\u0142 In\u017cynierii \u015arodowiska, Politechnika Warszawska<br>* Kontakt \/ Correspondence: <a href=\"mailto:andrzej.osiadacz@pw.edu.pl\">andrzej.osiadacz@pw.edu.pl<\/a><\/p>\n<\/details>\n\n\n\n<p class=\"wp-block-paragraph\"><br>DOI: <strong>10.65545\/GWITS.2026.05.04<\/strong>, GWiTS 5\/2026, maj 2026<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/gwits.pl\/wp-content\/uploads\/2026\/05\/1065545GWITS20260504.pdf\" data-type=\"page\" data-id=\"273\" target=\"_blank\" rel=\"noreferrer noopener\">Pobierz PDF<\/a> (Open Access)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Abstract:<br><\/strong>Gas leakage in low-pressure distribution networks remains a significant technical and economic challenge, amplified by pressure fluctuations and aging infrastructure. This study introduces a practical framework for estimating and reducing leakage using pressure-based network analysis. A simplified leakage model relates losses to nodal pressures and pipe characteristics and is integrated with steady-state hydraulic simulations. Leakage reduction is formulated as a constra\u00adined nonlinear optimization problem targeting optimal regulator settings and pressure smoothing. Tests on networks of varying size confirm consistent leakage reductions while preserving minimum supply pressure, demonstrating the method\u2019s value as an industry-ready decision support tool.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>Keywords: <\/strong>gas leakage, leakage modeling, gas network simulation, pressure optimization<\/em><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Streszczenie:<\/strong><br>Wycieki gazu w niskoci\u015bnieniowych sieciach dystrybucyjnych s\u0105 powa\u017cnym wyzwaniem technicznym i ekonomicznym, pot\u0119gowanym przez wahania ci\u015bnienia i starzej\u0105c\u0105 si\u0119 infrastruktur\u0119. Przedstawiono model praktycznego szacowania i ograniczania wyciek\u00f3w gazu z wykorzystaniem analizy sieci opartej na ci\u015bnieniu gazu. Uproszczony model wycieku wi\u0105\u017ce straty z ci\u015bnieniami w w\u0119z\u0142ach i charakterystyk\u0105 ruroci\u0105g\u00f3w. Redukcj\u0119 wyciek\u00f3w sformu\u0142owano jako problem optymalizacji nieliniowej z ograniczeniami, ukierunkowany na optymalne ustawienia regulatora i stabilizacj\u0119 ci\u015bnienia. Testy przeprowadzone na sieciach o r\u00f3\u017cnej wielko\u015bci potwierdzaj\u0105 sta\u0142\u0105 redukcj\u0119 wyciek\u00f3w przy jednoczesnym zachowaniu minimalnego ci\u015bnienia zasilania, co dowodzi warto\u015bci tej metody jako narz\u0119dzia wspomagaj\u0105cego decyzje w przemy\u015ble.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em><strong>S\u0142owa kluczowe:<\/strong> wyciek gazu, modelowanie wyciek\u00f3w, symulacja sieci gazowej, optymalizacja ci\u015bnienia<\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h3 class=\"wp-block-heading\">1. Introduction<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Approximately 2\u20133% of the total gas sent out by Gas Companies is classed as \u201eunaccounted for\u201d. This figure is made up of many components, but it is believed that a significant proportion of this is due to small undetected leaks at pipe joints in connection with the conversion of networks to supply natural gas. The re\u00adason for this is the potential change in the volume of leakage due to three factors:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>it is necessary to supply natural gas at a higher pressure than manufactured gas for satisfactory combustion,<\/li>\n\n\n\n<li>natural gas is drier than manufactured gas and could cause leakage as a result of old joints drying out,<\/li>\n\n\n\n<li>the thermal loss for a given volume of gas in the case of natural gas is more than twice that of manufactured gas.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">These factors potentially could result in greater leakage losses, and the industry is taking substantial steps to ensure that leaka\u00adge is minimised. This leakage loss represents a considerable loss of revenue to each Gaz Company and any reduction in leakage would be extremely beneficial economically [9]. To assist in con\u00adtrolling leakage rates, it is necessary to be able to estimate pre\u00adsent leakage rates and quantify the improvements which can be made by a variety of leakage control methods. Since leakage is believed to be directly related to the distribution of pressure in low pressure networks, network analysis programs offer the ca\u00adpability of quantifying leakage in distribution networks [1, 2, 9].<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. Leakage estimation in distribution networks<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Medium and low-pressure systems are usually either contro\u00adlled remotely from a regional centre where set points on parti\u00adcular regulators can be altered to satisfy the current load on the network or are individually set to satisfy loading conditions for a specified period of time. In this latter case the experience of distribution engineering is relied upon to ensure that demands will be met, and network restrictions are not violated. In general, this means that the set points are determined so that peak de\u00admands are always met. This has the consequence, unfortunately, that networks are working at higher pressures than necessary during off-peak periods of demand, thus of course, inducing hi\u00adgher leakage rates. The main exception to this is where clock or automatic controls are used to reduce pressures at times of low demand.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Network analysis programs are viewed as tools by distribution engineering to compute steady state pressures and flows in ne\u00adtworks given the demand loads together with source (regulator outlet) and regulator settings [12]. These settings are usual\u00adly chosen by experience and a knowledge of a given network&#8217;s characteristic behaviour [8] and to a certain extent they may be arbitrary but must ensure that the pressures in the network do not fall below a specified minimum at any time. By considering a model to estimate leakage, algorithms can be developed not only to achieve the above but also to reduce leakage rates and to recommend new source settings.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. Leakage model<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In order to estimate leakage rates for various loading levels and governor settings it is necessary to formulate some simple leaka\u00adge model. The leakage rate at a particular pipe joint will be de\u00adpendent on many factors associated with that pipe, e.g. pressure of gas in pipe at joint, pipe diameter, age of joint, type of joint and pipe material [3, 5].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Davenport in [4] has formulated a mathematical model which assigns the pipe joint leakages in a network to each node in the network. For large distribution networks the leakage at joints along a pipe is assumed to be equally distributed between the two end nodes of the pipes. The leakage in that pipe is assumed to be:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>proportional to the pressure at that node,<\/li>\n\n\n\n<li>proportional to the length of pipe (no. of joints).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">For a complete network the total leakage:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>q<\/mi><mrow><mi>l<\/mi><mi>e<\/mi><mi>a<\/mi><mi>k<\/mi><\/mrow><\/msub><mo>=<\/mo><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><\/mrow><msub><mi>q<\/mi><mi>i<\/mi><\/msub><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">q_{leak}=\u2211^n_{i=1}q_i \\quad(1)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">q<sub>i<\/sub> \u2013 leakage at node i.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This leakage model assumes a uniform leakage rate throughout the network, which will only be true if the network is made up of pipes of the same material, same age, sane diameter, and of the same joint type. In most practical cases this will not be true, however, the overall leakage rate determined using this model should be of the correct order of magnitude.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The model outlined above has been used to estimate le\u00adakage losses in several low-pressure distribution networks operating under maximum load conditions and conventio\u00adnal governor settings. For an estimated leakage constant of e = 3.5\u00d710\u20135 m<sup>3<\/sup>\/h\/m\/mbar the results shown that for the ne\u00adtworks chosen the estimated leakage rate is of the order of 2\u20133% of the maximum demand.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For a specimen single feed network the estimated leakage rate was calculated as a function of demand loading for a constant so\u00adurce pressure setting. The leakage rate increases with decreasing demand loading. The increased pressure in the network when it is operating below its maximum demand level gives rise to the observed increase in leakage rate. A similar effect can be demon\u00adstrated for multi-feed networks. The exact form of the curve will depend on the topology of the network, the total pressure drop across the network and the individual governor settings.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The effect of seasonal and diurnal gas demand variation has a considerable effect on gas leakage rates. The total annual leaka\u00adge from a gas distribution network could be up to 3% of the total gas send-out, even though the leakage rate estimated at peak de\u00admand may only be 2% of the maximum demand loading. Hence by using typical daily and yearly profiles it is possible to estimate total leakage from a particular network, rather than leakage at a given demand level. It is the overall leakage that we wish to minimise, and only by estimation of leakage over a period, can we estimate the effect of strategies for network leakage control.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. Methods of leakage reduction<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The problem of gas leakage at pipe joints can be approached at various levels:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>replacement of mains,<\/li>\n\n\n\n<li>repair of joints,<\/li>\n\n\n\n<li>mains reinforcement,<\/li>\n\n\n\n<li>governor pressure control [11].<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Control of leakage is being achieved gradually by mains repla\u00adcement and system treatment, but the more immediate remedy is the minimisation of system pressures [9].<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.1. Mains reinforcement<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">To reduce leakage in existing networks to a minimum it is ne\u00adcessary to minimise the pressure in pipes where leakage rates are significant, since the leakage rate is dependent on pressure. If it is known that particular areas or pipes in a network are likely to be susceptible to leakage, then the network must be examined carefully to reduce pressure in this area. If however leakage is believed to be evenly distributed throughout the network, or no specific leakage information is known, then it is necessary to re\u00adduce the overall pressure in the network while maintaining the statutory minimum supply pressure in the network.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From network analysis results it may be possible to pinpoint pipes where large pressure drops are occurring or areas of low pressure being fed from a distant source. In these cases, there will be in other parts of the network over-pressurised pipes or areas where leakage rates are greater than they need be. It should be possible to improve on present leakage rates by redistributing the pressure contours for the network, either by strategic upsizing\/reinforcement of particular pipes or by the addition of extra so\u00adurces. The effect of an even pressure distribution throughout the network is to smooth the pressure profile and so reduce leakage. In theory, for minimum pressurisation of a multi-feed network it is necessary to have the highest pressure at the centre of distri\u00adbution with the pressure minima at the outskirts of the network.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.2. Governor pressure control<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">In a typical multi-feed network, the minimum supply pressure (MSP) in the network, at maximum demand loading, is depen\u00addent on the source pressure settings. The MSP can be maintained in the network for different combinations of governor settings and for each combination we can calculate an estimate of the le\u00adakage rate for the network. For maximum gas savings it is desira\u00adble to operate the distribution network at the optimum governor settings for which the estimated leakage is a minimum while still maintaining the MSP in the network [5, 6, 10].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For a single feed network, it is trivial to determine the optimum governor pressure setting since the minimum pressure in the network, Pmin is linearly dependent on the governor pressure. For a two-source network the optimum pressure settings can be determined graphically, but for more complex networks the solution must be determined mathematically. Davenport has de\u00adveloped a mathematical optimisation method suitable for multi\u00ad-feed networks. Singer et al [6] have outlined a similar computer method for the optimal control of networks. Both methods rely on a Newton-Raphson type iterative method, and the complete solution may require more than 10 analyses of the network.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">An alternative method for solving leakage minimisation pro\u00adblem has been developed at Warsaw University of Technology (WUT). A governor pressure calculation algorithm has been de\u00adveloped which attempts to smooth the pressure profile of the ne\u00adtwork thus reducing leakage. In the Warsaw University of Tech\u00adnology we consider solving leakage minimisation problem in two ways. In the first case we consider the following problem:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>J<\/mi><mo>=<\/mo><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mi>N<\/mi><\/munderover><\/mrow><msub><mi>P<\/mi><mi>i<\/mi><\/msub><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">J=min\u2211_{(i=1)}^NP_i \\quad(2)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where: N \u2013 is the number of nodes<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We minimise the sum of pressures over the network in an at\u00adtempt to reduce the overall pressure distribution and hence leakage rates. In the second case we attempt to minimise the formula that models total network leakage, which is essentially a weighted sum of pressures:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>J<\/mi><mo>=<\/mo><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><\/mrow><msub><mi>W<\/mi><mi>i<\/mi><\/msub><mo>\u2219<\/mo><msub><mi>P<\/mi><mi>i<\/mi><\/msub><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">J=min\u2211_{i=1}^NW_i\u2219P_i \\quad(3)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>W<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><mrow><munderover><mo movablelimits=\"false\">\u2211<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>M<\/mi><\/munderover><\/mrow><msub><mi>e<\/mi><mrow><mi>i<\/mi><mi>j<\/mi><\/mrow><\/msub><mo>\u2219<\/mo><msub><mi>l<\/mi><mrow><mi>i<\/mi><mi>j<\/mi><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">W_i=\u2211_{i=1}^Me_{ij} \u2219l_{ij}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">M<sub>i<\/sub> \u2013 represents the number of pipes coincidenced at node i,<br>e<sub>ij<\/sub> \u2013 the leakage rate constant for pipe (i, j),<br>l<sub>ij<\/sub> \u2013 the length of pipe (i, j).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Thus, the problem of pressure minimisation is a special case of leakage minimisation when the weights Wi are taken as unity.<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.3. Constraints<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">In low pressure gas networks, the criteria of satisfactory supply is that the minimum pressure in the network should not fall below a prescribed level. Thus for any load node the pressure constraint may be written:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msubsup><mi>p<\/mi><mi>i<\/mi><mrow><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><\/mrow><\/msubsup><mo>\u2264<\/mo><msub><mi>p<\/mi><mi>i<\/mi><\/msub><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>.<\/mi><mi>.<\/mi><mi>.<\/mi><mo separator=\"true\">,<\/mo><mi>\u03b2<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p_i^{min}\u2264p_i \\quad (i = 1, 2, &#8230;, \\beta)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where: \u03b2 &#8211; is the number of load nodes in a network<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The limit of pressure at the supply governors of a network depends on the availability of pressure in the medium pressure system, the requirement of not exceeding a specified pressure in the network and finally the ability to maintain adequate supply. Hence pressure limits on governors may be written for a supply node k as:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msubsup><mi>p<\/mi><mi>k<\/mi><mrow><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><\/mrow><\/msubsup><mo>\u2264<\/mo><msub><mi>p<\/mi><mi>k<\/mi><\/msub><mo>\u2264<\/mo><msubsup><mi>p<\/mi><mi>k<\/mi><mrow><mi>m<\/mi><mi>a<\/mi><mi>x<\/mi><\/mrow><\/msubsup><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>.<\/mi><mi>.<\/mi><mi>.<\/mi><mo separator=\"true\">,<\/mo><mi>\u03b1<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">p_k^{min}\u2264p_k\u2264p_k^{max} \\quad (k = 1, 2, &#8230;, \\alpha)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where: \u03b1 &#8211; is the number of supply nodes in a network<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">and flow limits as:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msubsup><mi>Q<\/mi><mi>k<\/mi><mrow><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><\/mrow><\/msubsup><mo>\u2264<\/mo><msub><mi>Q<\/mi><mi>k<\/mi><\/msub><mo>\u2264<\/mo><msubsup><mi>Q<\/mi><mi>k<\/mi><mrow><mi>m<\/mi><mi>a<\/mi><mi>x<\/mi><\/mrow><\/msubsup><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mi>.<\/mi><mi>.<\/mi><mi>.<\/mi><mo separator=\"true\">,<\/mo><mi>\u03b1<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">Q_k^{min}\u2264Q_k\u2264Q_k^{max} \\quad (k = 1, 2, &#8230;, \\alpha)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Finally, the network equations themselves are constraints [7]. It means, that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>the flow equation is satisfied for every pipe,<\/li>\n\n\n\n<li>I-st Kirchhoff\u2019s law is fulfilled for each node,<\/li>\n\n\n\n<li>II-nd Kirchhoff\u2019s law is fulfilled for each loop.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The mathematical problem to be solved is large, sparse, nonli\u00adnearly constrained optimisation problem [10]. The optimisation algorithm which has been developed is based on the generalised reduced gradient method (GRG). This approach was found to be generally more suitable than the algorithm which is based on the sequential augmented Lagrangian method. Generally, the pro\u00adblem solved by GRG method is:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>m<\/mi><mi>i<\/mi><mi>n<\/mi><mtext>\u2061<\/mtext><mspace width=\"1em\"><\/mspace><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><menclose notation=\"bottom\" class=\"tml-underline\"><mi>x<\/mi><\/menclose><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mspace width=\"1em\"><\/mspace><menclose notation=\"bottom\" class=\"tml-underline\"><mi>x<\/mi><\/menclose><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>x<\/mi><mi>n<\/mi><\/msub><msup><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mi>T<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">min\u2061 \\quad f( \\underline{x}) \\quad  \\underline{x}=[x_1, x_2, \u2026, x_n]^T<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mi>s<\/mi><mi>u<\/mi><mi>b<\/mi><mi>j<\/mi><mi>e<\/mi><mi>c<\/mi><mi>t<\/mi><mspace width=\"1em\"><\/mspace><mi>t<\/mi><mi>o<\/mi><mspace width=\"1em\"><\/mspace><msub><mi>h<\/mi><mi>j<\/mi><\/msub><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>j<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><mi>m<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">subject \\quad to \\quad  h_j (x)=0 \\quad (j=1, \u2026, m)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>L<\/mi><mi>i<\/mi><\/msub><mo>\u2264<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>\u2264<\/mo><msub><mi>U<\/mi><mi>i<\/mi><\/msub><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo separator=\"true\">,<\/mo><mi>.<\/mi><mi>.<\/mi><mi>.<\/mi><mo separator=\"true\">,<\/mo><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">L_i\u2264x_i\u2264U_i \\quad (i = 1, &#8230;, n)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">where: L<sub>i<\/sub> and U<sub>i<\/sub> are the lower and upper bounds on x<sub>i<\/sub>, respec\u00adtively, the upper and lower bounds are treated as separate vec\u00adtors rather than being classified as inequality constraints becau\u00adse they are treated differently in determining the step length in a search direction.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nonlinear inequality constraints can be accommodated by sub\u00adstracting (or adding, as the case may be) the square of slack va\u00adriables from the inequality constraints thus:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>h<\/mi><mi>j<\/mi><\/msub><mtext>\u2061<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>g<\/mi><mi>j<\/mi><\/msub><mtext>\u2061<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><msubsup><mi>\u03c3<\/mi><mi>j<\/mi><mn>2<\/mn><\/msubsup><mo>=<\/mo><mn>0<\/mn><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">h_j\u2061(x)=g_j\u2061(x)-\u03c3_j^2=0 \\quad(4)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">and permitting the bounds on the \u03c3<sub>j<\/sub>&#8216;s to be:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mi>\u221e<\/mi><mo>\u2264<\/mo><msubsup><mi>\u03c3<\/mi><mi>j<\/mi><mn>2<\/mn><\/msubsup><mo>\u2264<\/mo><mi>\u221e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">-\u221e\u2264\u03c3_j^2\u2264\u221e<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">An alternate method is to substract (or add) the slack variable itself and make the slack variable nonnegative by adding bounds to the problem:<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><msub><mi>h<\/mi><mi>j<\/mi><\/msub><mtext>\u2061<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>g<\/mi><mi>j<\/mi><\/msub><mtext>\u2061<\/mtext><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><msub><mi>\u03c3<\/mi><mi>j<\/mi><\/msub><mo>=<\/mo><mn>0<\/mn><mspace width=\"1em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">h_j\u2061(x)=g_j\u2061(x)-\u03c3_j=0 \\quad (5)<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">subject to: \ud835\udf0e<sub>\ud835\udc57<\/sub> \u2265 0<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">5. Results<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">To check correctness of the algorithm five different gas ne\u00adtworks were tested. It is not a sufficient number of networks to allow rigorous statistical assessment of the benefits of leakage minimisation by pressure control. However, sufficient results are available to reach positive conclusions on the procedure suitable for industrial application. Table 1 lists the test-gas-network used.<\/p>\n\n\n\n<p class=\"has-small-font-size wp-block-paragraph\"><strong>Table 1. The test-gas-network parameters<\/strong><br><strong>Tabela 1. Parametry testowej sieci gazowej<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table has-small-font-size\"><table><tbody><tr><td>Test network<br>number<\/td><td>Number<br>of branches<\/td><td>Number<br>of load nodes<\/td><td>Number<br>of supply governors<\/td><\/tr><tr><td>1<\/td><td>15<\/td><td>9<\/td><td>1<\/td><\/tr><tr><td>2<\/td><td>120<\/td><td>101<\/td><td>3<\/td><\/tr><tr><td>3<\/td><td>350<\/td><td>304<\/td><td>4<\/td><\/tr><tr><td>4<\/td><td>700<\/td><td>608<\/td><td>8<\/td><\/tr><tr><td>5<\/td><td>1200<\/td><td>960<\/td><td>10<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The results of leakage minimisation procedure are the evaluation of new supply governor pressures and the corresponding network solution. The investigation have shown that for every system under test there is some reduction in leakage (see Table 2). It cannot be expected, however, that optimisation will result in a large percenta\u00adge reduction in leakage. What is important is that small reductions in leakage rates represent large sums of money over a period of time.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Table 2. Percentage of peak load for arbitrary and optimum pressure setting<\/strong><br><strong>Tabela 2. Procent obci\u0105\u017cenia szczytowego dla dowolnego i optymalnego ustawienia ci\u015bnienia<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table has-small-font-size\"><table><tbody><tr><td>Test network number<\/td><td>Percentage of peak load for arbitrary pressure setting<\/td><td>Percentage of peak load for optimum pressure setting<\/td><\/tr><tr><td>1<\/td><td>1.69<\/td><td>1.01<\/td><\/tr><tr><td>2<\/td><td>2.27<\/td><td>1.09<\/td><\/tr><tr><td>3<\/td><td>2.97<\/td><td>1.50<\/td><\/tr><tr><td>4<\/td><td>3.91<\/td><td>2.03<\/td><\/tr><tr><td>5<\/td><td>4.01<\/td><td>2.40<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h5 class=\"wp-block-heading\">5.1. Multiple local minima<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">One of the more difficult areas to handle is the situation when it is possible to have more than one solution to the leakage minimi\u00adsation problem. Nearly all optimisation algorithms are designed to only find a local minimum. It is quite possible to have one, two or many local minima to any given problem. We have not exa\u00admined in detail all networks to see if more than one solution is possible. However, we had situations, where two valid solutions were obtained, and different governor pressure settings resul\u00adted. One solution yielded substantially improved leakage savin\u00adgs compared to the other. It is hypothesised that the larger the network (larger number of controlling governors) the increased likelihood of multiple minima. This is, of course, the situation where leakage minimisation would have the greatest benefit.<br>There are several possibilities to deal with this situation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>attempt to find the global minimum; there are a number of techniques around for doing this, though potentially a very large increase in run time can be anticipated,<\/li>\n\n\n\n<li>accept any local minima that the algorithm arrives at; the danger in doing this is that if there were many local mini\u00adma to a given problem, then as a network is updated it is quite possible to flip to another local minimum, which co\u00aduld propose a completely different set of governor settings,<\/li>\n\n\n\n<li>develop a strategy whereby a number of attempts are ma\u00adde to find different local minima from various starting po\u00adints, and then select the least minimum of this set.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Whatever strategy is adopted, some research effort will be ne\u00adcessary in order to resolve this problem to satisfaction.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">6. Conclusions<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Leakage minimisation by mathematical procedures of the type described can only be considered as a supplementary measure. The theoretical solutions of gas networks can be developed and extended to include leakage losses as an implicit part of the solu\u00adtion procedure.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Although leakage rates may be higher in medium pressure ne\u00adtworks, the pipe length involved relative to low pressure networks is such that these systems can generally be treated or repaired mo\u00adre efficiently than low pressure networks. This is why the investi\u00adgation was concerned with low pressure networks but much of the theory would apply to medium pressure systems.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">As for as engineers using the program are concerned there is the question of data requirements. The bulk of the necessary informa\u00adtion consisting of pipe, load and governor data is identical in for\u00admat to that widely used in the Industry. The only additional data required is the loss constant, the governor pressure limits and the required minimum load node pressure [13].<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">7. References<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">[1] Butt, T. J., Amjad, M., Raza, S. F., Riaz, F., Ahmad, S., &amp; Abdollahian, M. (2023). Gas leakage identification and prevention by pressu\u00adre profiling for sustainable supply of natural gas. Sustainability, 15(18), 13604. <a href=\"https:\/\/doi.org\/10.3390\/su151813604\">https:\/\/doi.org\/10.3390\/su151813604<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[2] Lu, L., Zhang, X., Yan, Y., Li, J.-M., &amp; Zhao, X. (2014). Theoretical analysis of natural-gas leakage in urban medium-pressure pipe\u00adlines. Journal of Environment and Human, 1(2).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[3] van Westering, W., Brasjen, B., van der Linden, R. (2014). Natu\u00adral gas distribution network modelling and leak minimization. 2014 IEEE 11th International Conference on Networking, Sen\u00adsing and Control (ICNSC). p. 291\u2013296. <a href=\"https:\/\/doi.org\/10.1109\/icnsc.2014.6819641\">https:\/\/doi.org\/10.1109\/icnsc.2014.6819641<\/a>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[4] Davenport, R. (1992). Minimisation of leakage losses from gas ne\u00adtworks (Ph.D. thesis). The University of Mancheste<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[5] Kim, G., Lourenco, C., Skipper, D., &amp; Xu, L. (2025). Improved mixing and pressure loss formulations for gas network optimization. Optimization and Engineering. <a href=\"https:\/\/doi.org\/10.1007\/s11081-025-09984-x\">https:\/\/doi.org\/10.1007\/s11081-025-09984-x<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[6] Singer, D., Borossay, G., Koltai, T. (1995). An efficient computer me\u00adthod for optimal control of gas networks. Acta Tech (Budapest), 81(3\/4).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[7] Fletcher, R. (1991). Practical methods of optimization. John Wiley and Sons.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[8] Osiadacz, A. J. (1987). Simulation and analysis of gas networks. Gulf Publishing Company.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[9] Koty\u0144ski, \u0141., Murawski, C. (2025). Redukcja strat gazu w sieciach dystrybucyjnych poprzez detekcj\u0119 nieszczelno\u015bci i dynamiczne zarz\u0105dzanie parametrami eksploatacyjnymi. Gaz Woda i Technika Sanitarna, 99(11). <a href=\"https:\/\/doi.org\/10.15199\/17.2025.11.1\">https:\/\/doi.org\/10.15199\/17.2025.11.1<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[10] Osiadacz, A. J., Chaczykowski, M. (2023).Optimisation of the operating parameters of the natural gas compressor station. Gaz Woda i Technika Sanitarna, 97(5), s. 5\u201312. <a href=\"https:\/\/doi.org\/10.15199\/17.2023.5.2\">https:\/\/doi.org\/10.15199\/17.2023.5.2<\/a>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[11] Osiadacz, A. J., &amp; Chaczykowski, M. (2013). Rola stacji gazowych w ogra\u00adniczaniu strat gazu w sieciach dystrybucyjnych [Raport naukowo-ba\u00addawczy]. Polskie Zrzeszenie In\u017cynier\u00f3w i Technik\u00f3w Sanitarnych.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[12] Osiadacz, A. J., Koty\u0144ski, \u0141., &amp; Chaczykowski, M. (2021). SimNet TSGas-pakiet oprogramowania do symulacji dynamicznej sieci gazowych. Gaz Woda i Technika Sanitarna 95(12).&nbsp; <a href=\"https:\/\/doi.org\/10.15199\/17.2021.12.4\">https:\/\/doi.org\/10.15199\/17.2021.12.4<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">[13] Osiadacz, A. J., &amp; Chaczykowski, M. (2022). Calibration of transient pipeline gas flow model. Gaz Woda i Technika Sanitarna 96(6). <a href=\"https:\/\/doi.org\/10.15199\/17.%202022.6.4\">https:\/\/doi.org\/10.15199\/17. 2022.6.4<\/a><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n","protected":false},"excerpt":{"rendered":"<div class=\"mh-excerpt\">Andrzej J. Osiadacz, GWiTS 5\/2026<\/div>\n<p> <a class=\"mh-excerpt-more\" href=\"https:\/\/gwits.pl\/en\/1065545gwits20260504\/\" title=\"Wykrywanie i kontrolowanie wyciek\u00f3w gazu \/ Gas leakage detection and control\"> &#8230;<\/a><\/p>","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17],"tags":[98],"class_list":["post-3390","post","type-post","status-publish","format-standard","category-wszystkie_artykuly","tag-5-2026"],"_links":{"self":[{"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/posts\/3390","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/comments?post=3390"}],"version-history":[{"count":14,"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/posts\/3390\/revisions"}],"predecessor-version":[{"id":3415,"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/posts\/3390\/revisions\/3415"}],"wp:attachment":[{"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/media?parent=3390"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/categories?post=3390"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gwits.pl\/en\/wp-json\/wp\/v2\/tags?post=3390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}