CN103487186A - Variable frequency water supply system operating efficiency on-line detection method based on grey correlation method - Google Patents

Abstract

The invention provides an online detection method for the operating efficiency of a variable frequency water supply system based on the gray correlation method, which includes: establishing a mathematical model and constraint conditions among the output power, frequency disturbance, and water pressure variation to form an online detection method for the operating efficiency of the water supply system The required mathematical model; the small frequency signal disturbance is performed under steady-state conditions, the actual pressure change value is used as a reference sequence, and the pressure change value is calculated according to the model as a comparison sequence, and the gray correlation algorithm and the gray correlation degree criterion are used for online detection The shaft output power of the system can be used to obtain the efficiency of the frequency conversion water supply system. The invention can realize on-line output power detection without the need of a flow sensor, saves the time and cost required for system installation and debugging, makes the system structure simpler and the system cost lower. The invention can effectively protect the low-efficiency fault caused by the low-frequency operation of the motor and the frequency converter, improve the service life and reliability of the system, and provide reliable guarantee for the safe and efficient operation of the water pump motor.

Description

On-line detection method of frequency conversion water supply system operating efficiency based on gray correlation method

technical field

The invention belongs to the field of mechatronics measurement and control, and in particular relates to an online detection method for the operation efficiency of a frequency conversion water supply system based on a gray correlation method, and is especially suitable for detection of the operation efficiency of a water pump of a frequency conversion water supply device for a pressure tank.

Background technique

As a high-energy-consuming general-purpose machine, water pumps are widely used in various fields of industrial and agricultural production and residents' lives. The annual electric energy consumed by water pump units accounts for more than 21% of the country's total electricity consumption, and accounts for 30% of production costs in water supply enterprises. %~60%. Even if the efficiency of water pumps and water pump systems is only increased by 1%, it will bring huge benefits to my country's energy saving and environmental protection, and 30% to 50% of the electric energy consumed by water pumps can be saved. By adopting frequency conversion control technology, the energy consumption of water pumps can be effectively reduced, and 28.2 billion kWh of electricity can be saved every year, achieving the goal of energy saving and emission reduction. However, the premise of realizing energy saving by frequency conversion control technology is that the water pump is always running in the high efficiency range. However, the water consumption of water supply users is random and uncertain in space and time, and it cannot be guaranteed that the water pump will always run in a high-efficiency range. Especially in the period of low water consumption, because the water consumption is very small, the frequency converter and water pump work at low frequency. At this time, the motor heat loss and low-frequency vibration are serious, the energy consumption of the entire frequency conversion water supply system increases sharply, and the system efficiency is low. Under such working conditions, energy saving and emission reduction cannot be realized, and the long-term low-frequency operation of the water pump motor causes mechanical vibration and serious heating of the motor stator winding, which reduces the safety, reliability and service life of the system, and is detrimental to the safety and reliability of water supply and production costs. Influence, more serious even lead to the occurrence of security accidents. Therefore, the detection of the operating efficiency of the frequency conversion water supply system must be solved.

The real-time detection of the operating efficiency of the frequency conversion water supply system is a key technical issue that needs to be solved in order to achieve energy saving, emission reduction, and safe and reliable water supply in the water supply system. The operating efficiency of the frequency conversion water supply system is the ratio of the output power P _{out of} the pump shaft to the input power P _in . The input power P _in of the power supply can be obtained by measuring the input voltage v _in and the input current i _in of the power supply, and the realization is relatively simple. The output power P _{out of} the water pump shaft needs to be obtained by measuring the water pressure value p(t) and the water flow q(t). The water pressure value p(t) can be measured by a pressure sensor (usually a water pressure gauge), and the measurement of the flow rate q(t) is mainly measured by the following two schemes.

1. Flow sensor scheme: that is, flow detection is realized by installing a flow sensor at the water outlet of the pump; the flow sensor detection mechanism commonly used is mainly the impeller type, but the following three situations will lead to flow detection accuracy and reliability performance indicators not tall:

①When the flow rate of water supply is high, the solid debris in the water will hit the impeller, causing the impeller to be deformed and damaged, causing the moment of inertia of the whole mechanism to be unbalanced, resulting in the failure of flow detection or greatly reducing the accuracy and data reliability; ②The water quality is polluted When it is acidic or alkaline, it will seriously corrode the impeller mechanism, which will also lead to the imbalance of the moment of inertia of the whole mechanism, the failure of flow detection or the greatly reduced accuracy and data reliability; The relevant electromagnetic mechanism acts to convert the rotation signal of the impeller into electrical signals of different frequencies. In the case of a constant cross-sectional area of the pipe, the flow rate of the liquid can be measured by sampling the frequency of the electrical signal. Due to constraints such as mechanism strength, processing accuracy and processing cost, the inertia of rotating parts such as impellers cannot be small, so there is a small flow measurement dead zone range, so small flow conditions such as dripping and water leakage cannot be detected.

2. Dedicated controller + dedicated water pump motor scheme: that is, through repeated testing of the head characteristic curve of the dedicated water pump motor, the characteristic curve of the pump working under different flow conditions is obtained, and the curve is used as a reference value for the system to detect the flow rate , by detecting the similarity analysis between the current operating characteristics of the pump motor and the benchmark reference characteristics, the current flow value of the system can be obtained, but there are serious problems in this method:

(1) High requirements on the hardware and software of the control system. Because the controller needs to collect a large amount of data, perform signal processing and solve the characteristic quantity of the system, and conduct similarity analysis with the reference characteristic quantity stored in the memory, the requirements for the controller are high.

(2) Due to the long-term operation of the system, the liquid medium constant, motor parameters, and frequency converter parameters are affected by environmental factors and aging causes change, causing large changes in the operating characteristics of the system, resulting in poor measurement accuracy and reliability;

(3) Since the water supply system runs continuously for a long time, once the water pump fails, only the motor that is strictly matched with the controller can be used instead of other types of water pump motors, which leads to high operation and maintenance costs of the system and poor practicability and applicability .

Therefore, the detection method of the operation efficiency of the frequency conversion water supply system has a very broad market prospect in the fields of domestic metallurgy, steel, petroleum, chemical industry, water treatment, mines, and residential water.

Contents of the invention

The object of the present invention is to propose a low-cost, good versatility and simple structure online detection method for the operation efficiency of the frequency conversion water supply system based on the gray correlation method.

A method for online detection of the operating efficiency of a variable frequency water supply system based on a gray correlation method, characterized in that it includes the following steps:

(1) The water pressure value of the water supply system pipe network, the output frequency of the inverter circuit and the input power are sampled at intervals of sampling period T _s , and the first sampling values are marked as p(1), f(1) and P _in (1); the current sampling times is k, let k=1;

(2) Establish the water pressure value array {p(i)} composed of M elements, the inverter circuit output frequency array {f(i)}, and the input power array {P _in (i)}; where i={ k-M+1, k-M+2,...k}, M is a preset positive integer greater than 1, k is the current sampling times; p(i)| _i<=0 =0, f( i)| _i<=0 =0, P _in (i)| _i<=0 =0;

(3) Determine whether the water supply system is in a stable state; if yes, go to step (4); otherwise, go to step (13);

逆变电路输出频率平均值 $\stackrel{\&OverBar;}{F}=\frac{1}{M}\underset{i=k-M+1}{\overset{k}{\mathrm{\&Sigma;}}}f\left(i\right)$ 和输入功率平均值 $\stackrel{\&OverBar;}{P_{\mathrm{in}}}=\frac{1}{M}\underset{i=k-M+1}{\overset{k}{\mathrm{\&Sigma;}}}{P}_{\mathrm{in}}\left(i\right);$ (4) Solve the average water pressure

The average value of the output frequency of the inverter circuit $\stackrel{\&OverBar;}{f}=\frac{1}{m}\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}f\left(i\right)$ and average input power $\stackrel{\&OverBar;}{P_{\mathrm{in}}}=\frac{1}{m}\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}{P}_{\mathrm{in}}\left(i\right);$

(5) Mark the current moment as t=0, and give the output frequency a fixed arbitrary disturbance ΔF;

为t＝mT_s时刻的轴输出功率估计值；其中m＝1,2,…,N，

T_d为预先定义的观测时间长度；令

其中为任意设定的轴输出功率估计值的初始值；(6) Definition

is the estimated shaft output power at time t=mT _s ; where m=1,2,…,N,

T _d is the predefined observation time length; let

in is the initial value of the estimated value of shaft output power set arbitrarily;

Let m=1, the second-level minimum difference Δ(min)=0, the second-level maximum difference Δ(max)=1, and the resolution coefficient γ=0.5;

(7) Judging whether mT _s > T _d is true, if it is true, go to step (13); otherwise, at the time t=mT _s , sample the pipe network pressure value p(m); get

是否成立，如果不成立，转入步骤（13）；否则，将估计值

及

ΔF、T_b、ρ、g、P_b、V_b、T和t＝mT_s代入公式 $\frac{\mathrm{\&Delta;}{p}^g\left(m\right)}{\stackrel{\&OverBar;}{P}}=\frac{(2\stackrel{\&OverBar;}{F}\&times;\mathrm{\&Delta;F}+\mathrm{\&Delta;}{F}^2)}{{\stackrel{\&OverBar;}{F}}^2}(1-e^{-\frac{P_{\mathrm{out}}^g\left[m\right]T_b}{\mathrm{\&rho;g}{P}_b{V}_{b}T}t}),$ 求出压力波动估计值Δp^g(m)；(8) judgment

Whether it is true, if not, go to step (13); otherwise, the estimated value

and

ΔF, T _b , ρ, g, P _b , V _b , T and t=mT _s are substituted into the formula $\frac{\mathrm{\&Delta;}{p}^g\left(m\right)}{\stackrel{\&OverBar;}{P}}=\frac{(2\stackrel{\&OverBar;}{f}\&times;\mathrm{\&Delta;F}+\mathrm{\&Delta;}{f}^2)}{{\stackrel{\&OverBar;}{f}}^2}(1-e^{-\frac{P_{\mathrm{out}}^g\left[m\right]T_b}{\mathrm{\&rho;g}{P}_b{V}_{b}T}t}),$ Obtain the estimated value of pressure fluctuation Δp ^g (m);

Among them, P _b is the rated pressure value of the air pressure tank of the water supply system, V _b is the rated volume of the air chamber of the air pressure tank of the water supply system, T _b is the rated temperature of the air pressure tank of the water supply system; T is the ambient temperature, ρ is the liquid density; g is the acceleration of gravity;

(9) Take Δp(m) as the reference sequence and Δp ^g (m) as the comparison sequence, and normalize Δp(m) and Δp ^g (m) to obtain the corresponding normalized sequence Δp ₁ (m) and

求解Δp₁(m)、

的关联系数 ${\mathrm{\&xi;}}_0\left(m\right)=\frac{\mathrm{\&Delta;}\left(\min \right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)}{{\mathrm{\&Delta;}}_0\left(m\right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)};$ (10) Calculate the error sequence

Solve for Δp ₁ (m),

correlation coefficient ${\mathrm{\&xi;}}_0\left(m\right)=\frac{\mathrm{\&Delta;}\left(\min \right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)}{{\mathrm{\&Delta;}}_0\left(m\right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)};$

其中ξ₀(ψ)|_ψ＜＝0＝0；(11) Solve the correlation degree r,

Wherein ξ ₀ (ψ) | _ψ<=0 ＝0;

Judging whether r>=0.95 is true, if true, enter step (12); otherwise, update the variable, let

m=m+1; $P_{\mathrm{out}}^g\left[m\right]=P_{\mathrm{out}}^g[m-1]+\frac{1}{r}\mathrm{sgn}\left(\mathrm{\&Delta;p}\right[(m-1)]-\mathrm{\&Delta;}{p}^g[(m-1)\left]\right),$ Go to step (7);

判断是否满足η＜＝η_min，其中η_min为预先设定的低效率阈值；如果是，则说明系统处于低效率工作状态，逆变器输出关闭，退出；否则，进入步骤（13）；(12) order Calculate System Efficiency

Judging whether η<=η _min is satisfied, where η _min is the preset low efficiency threshold; if yes, it means that the system is in a low efficiency working state, the inverter output is turned off, and exit; otherwise, enter step (13);

(13) Let k=k+1; after the end of this sampling period, carry out the next sampling, and mark the sampling values of water pressure value, inverter circuit output frequency and input power as p(k), f(k) and P _in (k); return to step (2).

A further setting of the present invention is that the definition of the stable state is:

Computes the standard deviation of the array {p(i)} ${\mathrm{\&sigma;}}_p=\sqrt{\frac{m\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}p{\left(i\right)}^2-{\left(\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}p\left(i\right)\right)}^2}{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">m</figure-callout>}}^2}}$ and the standard deviation of the array {f(i)} ${\mathrm{\&sigma;}}_f=\sqrt{\frac{m\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}f{\left(i\right)}^2-{\left(\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}f\left(i\right)\right)}^2}{{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">m</figure-callout>}}^2}},$ Judging whether it is satisfied at the same time: σ _p <ε _p and σ _f <ε _f , where: ε _p and ε _f are preset positive values; if satisfied, the water supply system is considered to be in a stable state; otherwise, the water supply system is considered to be in an unstable state steady state.

The online detection method for the operating efficiency of the frequency conversion water supply system based on the gray correlation method of the present invention has the following beneficial effects:

1. The method for detecting the operating efficiency of the frequency conversion water supply system of the present invention has online output power detection and does not require a flow sensor, which saves the time and cost required for system installation and debugging, makes the system structure simpler and the system cost lower;

2. Compared with the existing special controller + special water pump motor scheme, the present invention is applicable to the operation efficiency detection of frequency conversion water supply system composed of various types of three-phase AC water pump motors, and has wide versatility. Because the shaft output power P _{out of} the frequency conversion water supply system satisfies the formula $\frac{\mathrm{\&Delta;p}\left(t\right)}{P}=\frac{(2f\&times;\mathrm{\&Delta;F}+\mathrm{\&Delta;}{f}^2)}{f^2}(1-e^{-\frac{P_{\mathrm{out}}{T}_b}{\mathrm{\&rho;g}{P}_b{V}_{b}T}t}).$ The formula determines the output power P _out from the parameters Δp(t), P, ΔF, F, T _b , V _b , P _b , ρ, g, T and t (Δp(t), P, ΔF, F, T _b , V _b , P _b , ρ, g, T and t are respectively expressed as the fluctuation amount of the water pressure deviating from the stable value when the frequency ΔF is disturbed, the water pressure value during the stable operation, the frequency disturbance increment, and the output of the inverter circuit during the stable operation Frequency, temperature at rated operation of the air tank, volume size of the air chamber at rated operation of the air tank, nominal pressure at rated operation of the air tank, liquid density, acceleration of gravity, current ambient temperature and time variables, and have nothing to do with the parameters and models of the motor Therefore, the output power P _out detected by this formula can be applied to any type of AC water pump motor, which has wide versatility. On this basis, the efficiency can be obtained by measuring the power input power P _in $\mathrm{\&eta;}=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\&times;100\%.$

3. The method for detecting the operating efficiency of the frequency conversion water supply system of the present invention has the characteristics of fast detection speed, high reliability, and strong practicability; it can effectively protect the low-efficiency faults caused by the low-frequency operation of the motor and the frequency converter, and improve the life and reliability of the system. It provides a reliable guarantee for the safe and efficient operation of the pump motor.

Detailed ways

The invention provides a detection method for the operating efficiency of a frequency conversion water supply system based on a gray correlation method, which mainly adopts a mathematical model of the efficiency of the frequency conversion water supply system.

a) Mathematical model of pump shaft output power

The schematic diagram of the water supply system is shown in Figure 1, mainly including water intake source, check valve 1, water pump motor M, air pressure tank 3, pressure gauge 4, temperature sensor 5, water outlet control valve 2, inverter circuit 6, controller 7, Power input power detection, etc. The bold line in Figure 1 indicates the power line, and the direction of the arrow indicates the direction of power transmission. The source of water intake is mainly tap water pipe network or deep wells, ponds, rivers and lakes, etc.; the main function of check valve 1 is to prevent the water in the user network pipe from flowing back to the water source when the pump stops running; The function of the air pressure tank 3 is to stabilize the water pressure and prevent water hammer accidents from harming the pipe network; the temperature sensor 5 is used to detect the current temperature of the system; the pressure gauge 4 is used to detect the water pressure of the water supply system; the input power The detection device is used to detect the power of the input power supply; the water outlet control valve 2 is used to start or stop the water supply to the user; the controller 7 mainly realizes the input of relevant parameters, the sampling of relevant variables, the display of operating status and the operation of the system control program; The variable circuit 6 receives the control quantity sent by the controller and inverts the output of the input power to realize the frequency conversion speed regulation control of the water pump motor; the input power 8 provides electric energy to the whole system.

The variables are explained as follows: q ₁ (t) is the water intake; q ₂ (t) is the water output; T(t) is the ambient temperature value; p(t) is the water pressure value of the pipe network; P _out (t) is the water pump shaft output power; f(t) is the output frequency value of the inverter circuit; P _in (t) is the input power of the power supply; η is the system efficiency; _{v 1} (t) is the air chamber volume of the air pressure tank; Tank air chamber pressure; v ₂ (t) is the water chamber volume of the air pressure tank; S is the cross-sectional area of the air pressure tank; V _z is the total volume of the air pressure tank; P _b is the rated pressure value of the air chamber of the air pressure tank; V _b is the air chamber of the air pressure tank Rated volume; T _b is the rated temperature of the pressure tank; t is the time variable; ρ is the liquid density; g is the acceleration of gravity.

When the water supply system is in a steady state: the pressure value is P, the output frequency of the inverter circuit is F, the inflow and outflow water flow is Q, the ambient temperature is T, the volume of the air chamber of the air pressure tank is V ₁ , the volume of the water chamber is V ₂ , the air chamber of the air pressure tank is Pressure is p _a (0), and all quantities above are in SI units. Define time t=0 as the last moment when the system operates stably at frequency F, that is, there exists:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; Q</span>}\\ {\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; Q</span>}\\ \mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}\\ {\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&rho;g</span>}\frac{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; S</span>}}\\ \mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; P</span>}\\ {\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ {\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\\ \mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; T</span>}\end{array}\right.$

Assume that the operating frequency of the water pump within [0, T _d ] is: f(t) = F + ΔF, ΔF is the frequency disturbance increment, T _d is the time value greater than 0, according to the power of the water supply system is different in advance It is determined artificially; then the water pressure value is p(t)=P+Δp(t), and Δp(t) is the water pressure fluctuation value caused by ΔF; the water intake of the water pump is q ₁ (t)=Q+Δq ₁ (t ), Δq ₁ (t) is the fluctuation value of the inflow flow caused by ΔF; the water output of the pump is q ₂ (t)=Q+Δq ₂ (t), and Δq ₂ (t) is the fluctuation value of the outlet flow caused by ΔF; According to the master's thesis of Chongqing University "Research and Design of Constant Pressure Sprinkler Irrigation Control System Based on PLC for Tea Garden", the relationship between the water pump inlet flow, water pressure and motor operating frequency is:

$\frac{{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; S</span>}}\mathrm{<span\; class="notranslate">\; f</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="R"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">R</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Among them: η is the efficiency of the water pump, that is, the ratio of the effective power of the motor to the output power of the shaft;

s is the slip rate;

为水泵电机的固有参数；R ₁ ,R ₂ ,X _1σ ,X _2σ ,m ₁ ,

is the intrinsic parameter of the pump motor;

Since the water pump motor adopts frequency conversion speed regulation control, s basically remains unchanged. make:

$\frac{{\mathrm{<span\; class="notranslate">\; m</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; u</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; S</span>}}}{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="R"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">R</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

k is only related to the structural parameters of the motor itself, and has nothing to do with flow and pressure. So the formula can be simplified to:

$\frac{{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; kf</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

Let k'=ηk. Then at t=0, there are:

QP=k′F ² (4)

At t∈[0,T _d ], substitute q ₁ (t)=Q+Δq ₁ (t), f(t)=F+ΔF and p(t)=P+Δp(t) into formula (3) :

(Q+Δq ₁ (t))(P+Δp(t))=k′(F+ΔF) ² (5)

Expand (5), and sort it out:

PQ+QΔp(t)+PΔq ₁ (t)+Δq ₁ (t)Δp(t)=k'(F ² +2FΔF+ΔF ² ) (6)

Substitute (4) into (6) to get:

QΔp(t)+PΔq ₁ (t)+Δq ₁ (t)Δp(t)=k'(2FΔF+ΔF ² ) (7)

Since the values of T _d and ΔF are small, and the mathematical model of the water supply system contains a first-order large inertial link, the water output q ₂ (t) of the system changes very little within the time t∈[0,T _d ], which can be approximated as Constant, that is, q ₂ (t)=Q. Therefore, in the time [0, T _d ], the value of Δp(t) caused by ΔF is small, so there is:

|Δp(t)|＜＜P       (8)

So finishing (7) is:

QΔp(t)+PΔq ₁ (t)=k'(2FΔF+ΔF ² ) (9)

Divide formula (9) by (4) to get:

$\frac{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; Q</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

The dynamic equation of the air tank: at t∈[0,T _d ], the volume change of the water chamber of the air tank is:

${\mathrm{<span\; class="notranslate">\; \&Delta;v</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}$

$<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}$

Therefore, the volume of t∈[0,T _d ] water chamber is:

${\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">V</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

Since V remains constant, the air cell volume is:

${\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

Assuming that the ambient temperature T remains constant within the time t∈[0,T _d ], it can be known from the ideal gas equation:

$\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

Substitute (13) into (14) to get:

$\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$

Let △p _a (t)=p _a (t)-p _a (0) be the pressure change in the air chamber of the air tank, then:

$\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$

The pressure change caused by the volume change of the water chamber is:

$\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&rho;g</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

Therefore, the change in water pressure

$\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>$

$<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; \&rho;g</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

则：If the parameter T _d is selected reasonably, satisfying

but:

$\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&rho;g</span>}\frac{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">S</figure-callout></span><figure-callout\; id="1"\; label="S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">\; </figure-callout>}}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 19</span>}<span\; class="notranslate">\; )</span>$

代入公式(19),并整理得：Will

Substitute into formula (19), and sort out:

$\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&rho;g</span>}\frac{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&rho;g</span>}\frac{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">S</figure-callout></span><figure-callout\; id="1"\; label="S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">\; </figure-callout>}}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 20</span>}<span\; class="notranslate">\; )</span>$

From the formula (20) can get:

$\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&rho;g</span>}\frac{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="V\; S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">V</figure-callout></span><figure-callout\; id="1"\; label="V\; S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">\; </figure-callout>}}{\mathrm{<span\; class="notranslate">\; S</span>}}}{{}_{}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; <</span>\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&rho;g</span>}\frac{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="V\; S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">V</figure-callout></span><figure-callout\; id="1"\; label="V\; S\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">\; </figure-callout>}}{\mathrm{<span\; class="notranslate">\; S</span>}}}{{}_{}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; one</span>}<span\; class="notranslate">\; )</span>$

对应于气压罐垂直高度产生的水压，通常是远小于实际扬程(恒压供水扬程一般在14m以上)，所以

所以有：Wherein: V=V ₁ +V ₂ . because

The water pressure corresponding to the vertical height of the air pressure tank is usually much smaller than the actual head (the constant pressure water supply head is generally above 14m), so

F:

$\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&ap;</span>\frac{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="P\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">P</figure-callout></span><figure-callout\; id="1"\; label="P\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">\; </figure-callout>}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; two</span>}<span\; class="notranslate">\; )</span>$

Substitute (22) into (10) and sort it out:

$\frac{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; Q</span>}}<span\; class="notranslate">\; +</span>\frac{{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="dt\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">dt</figure-callout></span><figure-callout\; id="1"\; label="dt\; V"\; filenames="HDA00003798550500011.png"\; state="\{\{state\}\}">\; </figure-callout>}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$

So Equation (23) is a linear constant coefficient differential equation about Δq ₁ (t), solvable:

$\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; q</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; Q</span>}}{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; four</span>}<span\; class="notranslate">\; )</span>$

Combine formulas (24) and (10) to get:

$\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; Q</span>}}{{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 25</span>}<span\; class="notranslate">\; )</span>$

Assuming that there is no gas leakage in the pressure tank, it can be known from the ideal gas equation:

$\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}}{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; \&times;</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 26</span>}<span\; class="notranslate">\; )</span>$

Simultaneous formulas (25) and (26), and arrange:

$\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; QP</span>}{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}}{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 27</span>}<span\; class="notranslate">\; )</span>$

代入公式(27)并整理得：Due to the output power of the pump shaft

Substitute into formula (27) and sort out:

$\frac{\mathrm{<span\; class="notranslate">\; \&Delta;p</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; out</span>}}{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;g</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 28</span>}<span\; class="notranslate">\; )</span>$

According to inequality (8), at t∈[0,T _d ], the constraints of formula (28) are:

$<span\; class="notranslate">\; |</span>\frac{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; \&Delta;F</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; out</span>}}{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}}{\mathrm{<span\; class="notranslate">\; \&rho;g</span>}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; b</span>}}\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; <</span><span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 29</span>}<span\; class="notranslate">\; )</span>$

Since the parameters P, F, ΔF, ρ, g, P _b , V _b , T _b and T are all observable and known quantities, the pressure disturbance Δp(t) is tested at t∈[0,T _d ] The value of the shaft output power P _out in the steady state can be calculated.

b) Efficiency calculation of frequency conversion water supply system

Substituting the parameters P, F, ΔF, ρ, g, P _b , V _b , T _b , T and Δp(t) into formula (28) can calculate the shaft output power P _out of the system at steady state. So the efficiency η is:

$\mathrm{<span\; class="notranslate">\; \&eta;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; out</span>}}}{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; in</span>}}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 100</span>}<span\; class="notranslate">\; \%</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 30</span>}<span\; class="notranslate">\; )</span>$

By combining (28) and (30), the efficiency η of the frequency conversion water supply system can be obtained.

The present invention provides an online detection method for the operating efficiency of a variable frequency water supply system based on the gray correlation method, which includes the following steps:

(1) The water pressure value of the water supply system pipe network, the output frequency of the inverter circuit and the input power are sampled at intervals of sampling period T _s , and the first sampling values are marked as p(1), f(1) and P _in (1); the current sampling times is k, let k=1;

(2) Establish the water pressure value array {p(i)} composed of M elements, the inverter circuit output frequency array {f(i)}, and the input power array {P _in (i)}; where i={ k-M+1, k-M+2,...k}, M is a preset positive integer greater than 1, k is the current sampling times; p(i)| _i<=0 =0, f( i)| _i<=0 =0, P _in (i)| _i<=0 =0;

以及逆变电路输出频率f(t)的标准差

判断是否同时满足：σ_p＜ε_p和σ_f＜ε_f(其中：ε_p，ε_f为设定正值，可根据实际系统进行设定，比如可以取0.1或者0.2)。如果满足，则认为供水系统处于稳定状态，进入步骤（4）；否则，供水系统处于不稳定状态，转入步骤（13）。(3) Determine whether the water supply system is in a stable state. The steady state is defined as: calculating the standard deviation of M sampled pressure values p(t)

And the standard deviation of the output frequency f(t) of the inverter circuit

Judging whether it is satisfied at the same time: σ _p <ε _p and σ _f <ε _f (where: ε _p and ε _f are set positive values, which can be set according to the actual system, such as 0.1 or 0.2). If it is satisfied, it is considered that the water supply system is in a stable state and enters step (4); otherwise, the water supply system is in an unstable state and proceeds to step (13).

逆变电路输出频率平均值和输入功率平均值 (4) Solve the average water pressure

The average value of the output frequency of the inverter circuit and average input power

(5) Mark this moment as t=0, and give the output frequency a fixed arbitrary disturbance ΔF, namely

其中

为任意设定的轴输出功率估计值的初始值；为不失一般性，

取值较大。N的定义为T_d为预先定义的观测时间长度；(6) Definition is the estimated value of shaft output power at time t=mT _s (m=1,2,…,N), let

in

is the initial value of the estimated value of shaft output power set arbitrarily; without loss of generality,

The value is larger. N is defined as T _d is the predefined observation time length;

Let m=1; define the second-level minimum difference Δ(min)=0, define the second-level maximum difference Δ(max)=1, and define the resolution coefficient γ=0.5.

(7) Judging whether mT _s > T _d is true, if it is true, go to step (13); otherwise, at the time t=mT _s , sample the pipe network pressure value p(m); get

是否成立。不成立，转入步骤（13）；否则，将估计值

及

、ΔF、T_b、ρ、g、P_b、V_b、T和t＝mT_s代入公式 $\frac{\mathrm{\&Delta;}{p}^g\left(m\right)}{\stackrel{\&OverBar;}{P}}=\frac{(2\stackrel{\&OverBar;}{F}\&times;\mathrm{\&Delta;F}+\mathrm{\&Delta;}{F}^2)}{{\stackrel{\&OverBar;}{F}}^2}(1-e^{-\frac{P_{\mathrm{out}}^g\left[m\right]T_b}{\mathrm{\&rho;g}{P}_b{V}_{b}T}t}),$ 求出压力波动估计值Δp^g(m)。(8) judgment

Whether it is established. If not, go to step (13); otherwise, the estimated value

and

, ΔF, T _b , ρ, g, P _b , V _b , T and t=mT _s are substituted into the formula $\frac{\mathrm{\&Delta;}{p}^g\left(m\right)}{\stackrel{\&OverBar;}{P}}=\frac{(2\stackrel{\&OverBar;}{f}\&times;\mathrm{\&Delta;F}+\mathrm{\&Delta;}{f}^2)}{{\stackrel{\&OverBar;}{f}}^2}(1-e^{-\frac{P_{\mathrm{out}}^g\left[m\right]T_b}{\mathrm{\&rho;g}{P}_b{V}_{b}T}t}),$ Find the pressure fluctuation estimate Δp ^g (m).

(9) Take Δp(m) as the reference sequence and Δp ^g (m) as the comparison sequence, and normalize Δp(m) and Δp ^g (m) to obtain the corresponding normalized sequence: Δp ₁ (m )and

(10) Calculate the error sequence Solve for Δp ₁ (m), The correlation coefficient ξ ₀ (m) of ${\mathrm{\&xi;}}_0\left(m\right)=\frac{\mathrm{\&Delta;}\left(\min \right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)}{{\mathrm{\&Delta;}}_0\left(m\right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)}.$

(其中：ξ₀(ψ)|_ψ＜＝0＝0)。(11) Solve the correlation degree r:

(where: ξ ₀ (ψ) | _ψ<=0 =0).

Judging whether r>=0.95 holds true. If it is established, go to step (12); otherwise, update the variable:

m=m+1; $P_{\mathrm{out}}^g\left[m\right]=P_{\mathrm{out}}^g[m-1]+\frac{1}{r}\mathrm{sgn}\left(\mathrm{\&Delta;p}\right[(m-1)]-\mathrm{\&Delta;}{p}^g[(m-1)\left]\right),$ Go to step (7).

就是系统实际轴输出功率P_out。计算系统效率：

判断实际输出效率是否满足η＜＝η_min(其中：η_min为设定的低效率阈值，可根据实际系统进行设定，比如设定为30%或40%等)。如果是，则说明系统处于低效率工作状态，逆变器输出关闭，退出。否则，进入步骤（13）。(12) Estimated value

is the actual shaft output power P _out of the system. Calculate system efficiency:

Determine whether the actual output efficiency satisfies η<=η _min (where: η _min is the set low efficiency threshold, which can be set according to the actual system, such as 30% or 40%). If it is, it means that the system is in a low-efficiency working state, and the inverter output is turned off and exits. Otherwise, go to step (13).

(13) Let k=k+1; after the end of this sampling period, carry out the next sampling, and mark the sampling values of water pressure value, inverter circuit output frequency and input power as p(k), f(k) and P _in (k); return to step (2).

Claims (2)

1. a kind of frequency conversion water supply system operational efficiency online detection method based on gray correlation method, is characterized in that, comprises the steps: (1) The water pressure value of the water supply system pipe network, the output frequency of the inverter circuit and the input power are sampled at intervals of sampling period T _s , and the first sampling values are marked as p(1), f(1) and P _in (1); the current sampling times is k, let k=1; (2) Establish the water pressure value array {p(i)} composed of M elements, the inverter circuit output frequency array {f(i)}, and the input power array {P _in (i)}; where i={ k-M+1, k-M+2,...k}, M is a preset positive integer greater than 1, k is the current sampling times; p(i)| _i<=0 =0, f( i)| _i<=0 =0, P _in (i)| _i<=0 =0; (3) Determine whether the water supply system is in a stable state; if yes, go to step (4); otherwise, go to step (13); (4) Solve the average water pressure

The average value of the output frequency of the inverter circuit $\stackrel{\&OverBar;}{f}=\frac{1}{m}\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}f\left(i\right)$ and average input power $\stackrel{\&OverBar;}{P_{\mathrm{in}}}=\frac{1}{m}\underset{i=k-m+1}{\overset{k}{\mathrm{\&Sigma;}}}{P}_{\mathrm{in}}\left(i\right);$ (5) Mark the current moment as t=0, and give the output frequency a fixed arbitrary disturbance ΔF; (6) Definition

is the estimated shaft output power at time t=mT _s ; where m=1,2,…,N,

T _d is the predefined observation time length; let

in

is the initial value of the estimated value of shaft output power set arbitrarily; Let m=1, the second-level minimum difference Δ(min)=0, the second-level maximum difference Δ(max)=1, and the resolution coefficient γ=0.5; (7) Judging whether mT _s > T _d is true, if it is true, go to step (13); otherwise, at the time t=mT _s , sample the pipe network pressure value p(m); get

(8) judgment Whether it is true, if not, go to step (13); otherwise, the estimated value

and

ΔF, T _b , ρ, g, P _b , V _b , T and t=mT _s are substituted into the formula $\frac{\mathrm{\&Delta;}{p}^g\left(m\right)}{\stackrel{\&OverBar;}{P}}=\frac{(2\stackrel{\&OverBar;}{f}\&times;\mathrm{\&Delta;F}+\mathrm{\&Delta;}{f}^2)}{{\stackrel{\&OverBar;}{f}}^2}(1-e^{-\frac{P_{\mathrm{out}}^g\left[m\right]T_b}{\mathrm{\&rho;g}{P}_b{V}_{b}T}t}),$ Obtain the estimated value of pressure fluctuation Δp ^g (m); Among them, P _b is the rated pressure value of the air pressure tank of the water supply system, V _b is the rated volume of the air chamber of the air pressure tank of the water supply system, T _b is the rated temperature of the air pressure tank of the water supply system; T is the ambient temperature, ρ is the liquid density; g is the acceleration of gravity; (9) Take Δp(m) as the reference sequence and Δp ^g (m) as the comparison sequence, and normalize Δp(m) and Δp ^g (m) to obtain the corresponding normalized sequence Δp ₁ (m) and

(10) Calculate the error sequence

Solve for Δp ₁ (m),

correlation coefficient ${\mathrm{\&xi;}}_0\left(m\right)=\frac{\mathrm{\&Delta;}\left(\min \right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)}{{\mathrm{\&Delta;}}_0\left(m\right)+\mathrm{\&gamma;\&Delta;}\left(\max \right)};$ (11) Solve the correlation degree r,

Wherein ξ ₀ (ψ) | _ψ<=0 ＝0; Judging whether r>=0.95 is true, if true, enter step (12); otherwise, update the variable, let m=m+1; $P_{\mathrm{out}}^g\left[m\right]=P_{\mathrm{out}}^g[m-1]+\frac{1}{r}\mathrm{sgn}\left(\mathrm{\&Delta;p}\right[(m-1)]-\mathrm{\&Delta;}{p}^g[(m-1)\left]\right),$ Go to step (7); (12) order

Calculate System Efficiency

Judging whether η<=η _min is satisfied, where η _min is the preset low efficiency threshold; if yes, it means that the system is in a low efficiency working state, the inverter output is turned off, and exit; otherwise, enter step (13); (13) Let k=k+1; after the end of this sampling period, carry out the next sampling, and mark the sampling values of water pressure value, inverter circuit output frequency and input power as p(k), f(k) and P _in (k); return to step (2). 2. the frequency conversion water supply system operating efficiency online detection method based on gray correlation method according to claim 1, is characterized in that, the definition of described steady state is: Computes the standard deviation of the array {p(i)}

and the standard deviation of the array {f(i)}

Judging whether it is satisfied at the same time: σ _p <ε _p and σ _f <ε _f , where: ε _p and ε _f are preset positive values; if satisfied, the water supply system is considered to be in a stable state; otherwise, the water supply system is considered to be in an unstable state steady state.

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