MODELING OF PUMPING STATION OPERATING MODES UNDER VARIABLE OPERATIONAL LOAD CONDITIONS

АННОТАЦИЯ

Статья посвящена моделированию режимов работы насосной станции в условиях переменной эксплуатационной нагрузки. Рассматривается влияние изменяющегося расхода и требуемого напора на выбор числа функционирующих агрегатов, способ регулирования и уровень расходов на энергию. Систематизированы существующие подходы к управлению насосными станциями. Предлагается математическая модель, в которой режим работы станции рассматривается с точки зрения задачи выбора состава оборудования и параметров его функционирования во времени при соблюдении гидравлических ограничений. На условных расчетных данных показано поведение трех типовых режимов и выполнено их сопоставление по затратам энергии, постоянству давления и частоте переключений режимов. На основе полученных результатов сформулированы рекомендации по выбору рационального режима работы насосной станции в зависимости от характера нагрузки.

ABSTRACT

The article is devoted to modeling the operating modes of a pumping station under variable operational load conditions. The study examines the influence of changing flow rate and required head on the choice of the number of operating units, the control method, and the level of energy costs. Existing approaches to pumping station control are systematized. A mathematical model is proposed in which the station operating mode is considered from the standpoint of selecting the equipment configuration and its operating parameters over time while satisfying hydraulic constraints. Using conditional calculated data, the behavior of three typical operating modes is demonstrated, and these modes are compared in terms of energy costs, pressure stability, and switching frequency. Based on the obtained results, recommendations are formulated for selecting a rational pumping station operating mode depending on the nature of the load.

Ключевые слова: насосная станция, переменная эксплуатационная нагрузка, режим работы насосной станции, математическое моделирование работы станции, энергопотребление и регулирование насосов.

Keywords: pumping station, variable operational load, pumping station operating mode, mathematical modeling of station operation, energy consumption and pump control.

Introduction. The operation of a pumping station under real-world conditions depends on continuously changing water demand parameters and the state of the entire network. The variable nature of the load arises from daily peak demand periods, nighttime declines, seasonal fluctuations, and differences between weekdays and weekends, which leads to the fact that calculations based on a single steady-state operating mode lose accuracy and practical relevance. As a result, pressure deviations, excessive electricity consumption, and uneven wear of individual units may occur. For example, in the scientific literature on water distribution systems, a high share of total energy consumption is attributed to water pumping. It is also emphasized that a significant portion of global electricity consumption in water supply systems is associated specifically with pumping equipment, and that reducing these costs is linked to the selection of rational supply regimes and appropriate control methods [7].

The problem described above is further exacerbated in systems where the infrastructure was originally designed with a capacity reserve, while actual consumption has decreased over time. Under such a mismatch between design and actual load, a pumping station may operate outside its best efficiency range for extended periods. As a result, head losses increase, the share of idle and transient operating modes grows, and overall operational efficiency deteriorates. A viewpoint exists that, for oversized systems experiencing a pronounced reduction in actual demand relative to the design values, the use of hydraulic models becomes increasingly justified as a decision-support tool for modernization and operational control [8]. In other words, the task of optimizing pumping station operation, with the selection of appropriate operating modes without compromising service quality, becomes essential. In this context, modeling of pumping station operating modes under variable operational load conditions becomes of direct practical relevance. The scientific relevance of this problem lies in describing the time series of operating modes, while its practical importance is determined by the ability to predict in advance which equipment configuration and control method will ensure the required supply with acceptable energy and operational costs, which has necessitated the development of the author’s methodological approach.

The aim of the study is to develop a methodological approach to modeling the operating modes of a pumping station under variable operational load conditions.

Research methodology. In this study, variable operational load is understood as time-dependent variation in the required flow rate, head, and associated operating parameters of the pumping station. The flow rate, in particular, is influenced by the demand pattern, the distribution of flows across network zones, reservoir levels, the condition of pipelines and valves, as well as the operating rules governing pump unit activation. At the same time, load variability manifests at several temporal scales, including intraday fluctuations, weekly demand patterns, seasonal variations, and short-term transient conditions in which the station must rapidly transition from one operating state to another.

Accordingly, when constructing a load model, it is important to account not for the average flow rate, but for its temporal variation profile. Incorporating variable demand patterns improves the accuracy of describing network behavior over an extended calculation period. For example, using a linear approach, variable patterns can be defined and linked to observational data, making it possible to select a more accurate calculation regime [4]. Within the scope of this study, the classification of pumping station operating modes can be structured based on four criteria: (1) the level of relative load; (2) the degree of temporal stability of the operating mode; (3) the composition of operating equipment; and (4) the control method (Fig. 1).

Figure 1. Classification of pumping station operating modes under variable operational load conditions (compiled by the author)

As a result, operating modes corresponding to low, medium, near-nominal, and peak load levels can be distinguished. In addition, stationary and transient modes are identified, along with single-unit operation, parallel operation of multiple units, standby modes, and others.

It should be noted that variable load affects the pumping station by shifting the operating point along the pump and system characteristic curves, and by determining which unit should be operated at a given moment and whether the flow should be redistributed among the equipment. Variable load also influences the frequency of starts and stops, and consequently the wear of electric motors and valves. Therefore, the problem of selecting an optimal operating mode for a pumping station cannot be reduced solely to ensuring the required flow rate, as it is also necessary to account for both the energy performance and operational aspects of the station.

Taking this into account, it is necessary to conduct a comparative analysis of approaches to the control and modeling of pumping station operating modes (Table 1):

Table 1.

Comparative analysis of existing approaches to the control and modeling of pumping station operating modes (compiled by the author).

Source: compiled by the author based on [1; 2; 3; 5; 6, etc.].

Based on Table 1 and the findings reported in the scientific literature, several conclusions can be drawn:

1) the current research trend is shifting toward time-dependent analysis of pumping station operating modes;

2) there is no universal solution applicable to all types of pumping stations, as the outcome depends on the shape of the load profile, the equipment configuration, the control method, and the selected optimality criterion.

For these reasons, it is advisable for the purposes of further research to employ a mathematical model in which the aforementioned parameters are explicitly defined.

The proposed model is formulated in discrete time. The calculation interval is divided into steps . For each step, the required flow rate  and the required head are specified. Let the station be equipped with m pumping units; for each unit, a binary state variable is introduced, where , if the pump is on and  , if the pump is off. In addition, a variable of relative speed , is introduced, taking values within an admissible range when variable frequency control is available.

The objective function can be written as follows:

where  is the energy consumption at time step ,  is the number of unit switching events between time steps  and , is the penalty for insufficient head,  is the load imbalance indicator for the pumps, and  are weighting coefficients.

The energy component is determined as:

where  s the power of the -th pump at time step , and  is the duration of the time step. The switching indicator can be represented as follows:

The penalty for head deviation is defined by the following expression:

where  is the actual head delivered by the pumping station.

To evaluate load non-uniformity, the following expression is used:

where  is the flow rate through the -th pump, and  is its nominal flow rate.

The system of constraints is defined as follows:

If necessary, a reserve constraint is introduced into the model; in this case, at each time step, a portion of the station’s capacity must remain unused to ensure the possibility of increasing the output when required.

From a practical standpoint, this model makes it possible to address several tasks simultaneously:

1) determine the optimal number of pumps to be operated simultaneously;

2) determine the distribution of flow among the units;

3) evaluate the effect of different control strategies on the overall criterion . Accordingly, the proposed methodological approach appears promising for the analysis of typical pumping station operating modes.

Results and discussion. To demonstrate the capabilities of the model, let us consider a daily profile of relative load expressed in arbitrary units. Suppose that over an interval divided into eight equal segments, the relative demand  takes the following values: 0,32; 0,28; 0,35; 0,52; 0,74; 0,93; 1,00; 0,68. This sequence reflects the nighttime minimum in demand, the increase in load during the morning period, daytime stabilization, and the evening peak. The required head is assumed to remain constant within the assumed calculation zone, and the evaluation is performed for three operating modes:

1) mode A, in which one pump operates at constant speed with throttling;

2) mode B, in which two pumps operate in cascade with threshold-based switching;

3) mode C, in which the pumps operate under variable speed control.

The comparative analysis of the corresponding operating modes is presented in Fig. 2:

Figure 2. Simulation of typical operating modes under variable load conditions (compiled by the author)

Thus, referring to Fig. 2, it illustrates the degree of correspondence between the actual supply and the calculated demand over eight time intervals. The greatest deviation from the required load is observed in Mode A, primarily during periods of low water consumption. Mode B provides a closer approximation to the calculated curve in the medium and high load ranges. Mode C demonstrates the most consistent tracking of demand variations throughout the entire calculation cycle. The obtained relationships confirm that the selected control method has a direct impact on the accuracy of maintaining the required operating mode of the pumping station.

In addition, it is important to consider changes in energy consumption parameters (Fig. 3). For illustration, the following calculated values of the specific energy consumption indicator  expressed in arbitrary units (a.u.) per unit of supply, can be assumed:

Figure 3. Comparative analysis of specific energy consumption under different pumping station operating modes (compiled by the author).

A comparison of the three modes leads to the conclusion that the constant-speed operation with throttling is the simplest in terms of implementation; however, it performs worse with respect to energy consumption in periods of low demand. Cascade operation of pumps improves the match between supply and demand but increases the frequency of switching, whereas variable speed control provides the best performance under uneven load conditions, i.e., in cases where it is necessary to maintain the supply close to the required demand without abrupt transitions between operating states.

It should be noted that the selection of a rational operating mode should be aligned with the characteristics of the load profile. For stations where operation is dominated by extended periods at approximately the same flow rate, it is acceptable to use modes with a limited number of switching events and predefined activation thresholds. For stations with pronounced intraday variability, it is advisable to apply control methods that ensure more accurate tracking of the required flow rate. For facilities where the energy component constitutes a significant share of operating costs, particular importance is attached to minimizing operation outside preferred efficiency ranges. Accordingly, it is necessary to account for the specific operating parameters of individual pumping stations.

Special attention should also be given to transient operating modes, since under frequent and abrupt changes in demand it is important to reduce the total number of starts and stops, as these periods are associated with increased mechanical and electrical stress on the equipment. In the model, this effect is accounted for through a switching penalty, while in the presence of a standby unit it is also necessary to impose a constraint that keeps a portion of the installed capacity unused (which is particularly important for network sections where a rapid increase in flow rate is likely).

When designing the operating mode for the calculation period, it is recommended to follow the procedure outlined below (Fig. 4).

Figure 4. Algorithm for selecting a rational pumping station operating mode over the calculation period (compiled by the author)

Conclusion. Thus, modeling the operating modes of a pumping station under variable operational load conditions should be considered as the problem of temporally matching the station’s supply with the varying demand of the network. The modeling outcome depends on the equipment configuration, the control method, and the shape of the load profile. The proposed mathematical model (the author’s methodological approach) makes it possible to describe the station’s operating mode through a set of energy, hydraulic, and operational indicators. Calculations based on hypothetical data show that the selection of an operating mode should account for intraday demand variability and the permissible switching frequency of units. For stations with pronounced load variability, the most practically relevant mode is the one that ensures close correspondence to the required supply with moderate energy consumption and stable pressure. The obtained results can serve as a basis for further calculations using the actual parameters of a specific pumping station.

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