Abstract
State estimation (SE) usually serves as the basic function of the energy management system (EMS). In this paper, the time-scale characteristics of the integrated heat and electricity networks are studied and an SE model is established. Then, a two-stage iterative algorithm is proposed to estimate the time delay of heat power transportation in the pipeline. Meanwhile, to accommodate the measuring resolutions of the integrated network, a hybrid SE approach is developed based on the two-stage iterative algorithm. Results show that, in both steady and dynamic processes, the two-stage estimator has good accuracy and convergence. The hybrid estimator has good performance on tracking the variation of the states in the heating network, even when the available measurements are limited.
Keywords
Integrated heat and electricity network; measurement resolution; state estimation (SE); time-scale characteristics
ENERGY internet, also named the multi-energy system or the integrated energy system (IES), is proposed to fully release the flexibility of various energy networks such as heating, cooling, natural gas, and even transportation networks, with its advantages in improving energy efficiency, reducing carbon emissions and increasing renewable energy penetration [
Reference [
Electricity networks are monitored by remote terminal units (RTUs), phasor measurement units (PMUs), and other intelligent electronic devices (IEDs), the automation level of which is high. However, the monitoring and communication infrastructures of the DH network are not so advanced as the electricity networks due to technical and economic reasons. In recent years, more attention has been paid to improve the automation level of the DH network. In [
However, errors caused by the measurements are inevitable. To get a complete description of the integrated system state and detect the bad data, it is necessary to make state estimation (SE). SE has been widely used in electric systems [
In this paper, a two-stage SE (TSE) approach and a hybrid estimation algorithm for IHEN is proposed, which considers both the dynamics of the pipeline and different sampling resolutions of two systems.
The remainder of this paper is organized as follows: Section II introduces the model of IHEN. Section III describes the SE model, the TSE approach and the hybrid estimation algorithm. Section IV introduces and discusses the results of numerical simulation and case study. Section V presents the conclusions and directions for future work.
The model of IHEN consists of an electricity network model, a heating network model, and coupling component models. The heating network includes the hydraulic network and thermal network. In the integrated network, three types of energy are considered: the electric energy in the electricity network, the pressure energy in the hydraulic network, and the thermal energy in the thermal network.
The spread speed of thermal energy in the thermal network is much slower than that of electric energy in the electricity network and the pressure energy in the hydraulic network [
The electricity network model is established based on the Kirchhoff’s current law (KCL), Kirchhoff’s voltage law (KVL), and Ohm’s law. In this paper, the AC power flow model for transmission network [
The steady-state power flow equations of electricity network reflect the relationship between electric power injection and bus voltage, which could be represented with the following algebraic equations:
(1) |
(2) |
where Pi, Qi are the active and reactive power injections of bus i, respectively; Ui, θi are the voltage magnitude and phase angle at bus i, respectively; and is the i
A typical DH network is a dual-pipe system where heat is delivered to customers as heated water. The supply pipe takes the hot water to customers and the return pipe brings the cooled water back to the heat source. The water is circulated with pumps which are usually located at the heat source.
When analyzing a DH network, the entire network is usually separated into a hydraulic network and a thermal network [
1) Water flow in the pipeline is one-dimensional, incompressible and fully developed.
2) The properties of water such as density, kinematic viscosity, specific heat and so on do not change with the pressure and temperature.
3) Convective heat transfer from the fluid to the surroundings is in the radial direction only.
The relationship between the head pressure and mass flow rate is considered in the model of the hydraulic network. Mass flowing through pipe and heat exchanger will cause pressure losses, which could be written as [
(3) |
(4) |
where , are the pressure losses of the pipe and heat exchanger, respectively; mpi,i, mhex,i are the mass flow rates of the pipe and heat exchanger, respectively; Kpi,i, Khex,i are the relative resistance coefficients of the pipe and heat exchanger, respectively; and Npi, Nhex are the numbers of the pipes and heat exchangers, respectively.
As for the circulation pump, variable speed pumps are considered in this paper, the relationship between the mass flow rate and head pressure generated by the pump can be expressed as [
(5) |
where is the pressure generation of the circulation pump; is the mass flow rate of the pump; Npu is the number of the pumps; and is the relative coefficient.
Equations
(6) |
where is the pressure loss vector; is the relative resistance coefficient vector; and is the mass flow rate vector.
Considering the hydraulic network operates in the steady state, the conservation of mass and pressure energy is also satisfied, respectively [
(7) |
(8) |
where A is the node-branch incidence matrix; B is the loop-branch incidence matrix of the DH network; and is the nodal flow injection vector.
Note that in [
(9) |
After mq is specified, the hydraulic model could be performed independently [
The model of the thermal network describes the temperature distribution at each node and thermal power flow in the network. In the thermal network, the components that could cause the change of temperature, e.g., the mixer, the heat exchanger at the source or the loads, and the pipeline, are considered, while circulation pumps and valves are assumed to be lossless of heat.
(10) |
where is the mixture temperature of a node; is the mass flow rate within a pipe leaving the node; is the temperature of the flow at the end of an incoming pipe; is the mass flow rate within a pipe coming into the node; and is the number of the nodes.
Equations
(11) |
(12) |
where , , , and are the heat power, mass flow rate, inlet temperature, and outlet temperature at sources, respectively; , , , and are the heat power, mass flow rate, inlet temperature, and outlet temperature at loads, respectively; Cp is the specific heat capacity of the water; and , are the numbers of the heat sources and the heat loads, respectively.
The dynamics of the pipeline are introduced by using a pseudo-transient model [
(13) |
where S is the cross-section area of the pipeline; m is the mass of flow; T is the temperature; is the overall heat transfer coefficient per unit length of the pipeline; is the density of water; Ta is the ambient temperature; and t, x are the time point and space point, respectively.
The characteristic lines of (13) are defined by:
(14) |
Considering the temperature along the characteristic line, the PDE along the line can be written as (15) according to (13) and (14).
(15) |
(16) |

Fig. 1 Scheme of characteristic line of (13).
where mpi,i is the mass flow rate of the pipe; and Spi,i is the cross-section area of the pipe.
Thus the integral equation along the characteristic line of pipe is calculated by (15) and (17), and the relationship of the inlet and outlet temperature of the pipe is represented by (18).
(17) |
(18) |
However, since the measurements have specific sampling resolution, and the SE only considers the state at sampling points, the integral
(19) |
where is the mass flow rate at time and can be obtained from the historically estimated results.
(20) |
where is the virtual column of the pipe, and the relationship can be illustrated as

Fig. 2 Schematic illustration of estimating time delay in a pipeline.
And can be calculated by (21) based on (19):
(21) |
Then, the time delay s of each pipe can be estimated with the virtual volume :
(22) |
For pipe , the historically estimated inlet temperature which is closest to time ts is selected as in (23):
(23) |
According to (21)-(23), the outlet temperature of a pipeline could be determined by the time delay of mass flow and historical inlet temperature:
(24) |
Usually, in the thermal network, the supply temperature at each source and the outlet temperature at each load are specified [
1) The mass flow rate m of each pipe is determined by the hydraulic model.
2) The historical inlet temperature of each pipeline is known.
3) The delay of mass flow in each pipeline could be determined by (22).
Then, the thermal model could be performed independently by solving (10), (11) and (24).
The electricity network and the DH network are coupled with coupling components at the boundary area, as shown in

Fig. 3 Illustration of coupling area and coupling components of integrated network.
In this paper, only a few coupling components are modeled [
(25) |
(26) |
where , are the thermal power, the electric power output of a gas turbine of internal combustion reciprocating engine, respectively; , are the thermal power, the electric power output of an extraction steam turbine, respectively; is the electric power output of the extraction steam turbine in full condensing mode; , are the numbers of the gas turbine of internal combustion reciprocating engine and the extraction steam turbine, respectively.
A circulation pump is propelled by a motor that consumes the electricity to create and maintain a pressure difference. The electricity consumption of the circulation pump is shown in (27).
(27) |
where is the electric power consumption of the circulation pump; and is the efficiency of the pump.
Denoting (28) as the pressure energy supplied by a circulation pump [
(28) |
(29) |
Denoting the power flow at the boundary area and the relative coefficient in matrix form, we can obtain (30)-(32), and represents the transition:
(30) |
(31) |
(32) |
Then we can obtain:
(33) |
SE of the combined heat and electricity network utilizes the measurements from both electricity network and DH networks. As for electricity network, the most commonly-used measurements include the magnitude of bus voltage denoted as U; active power and reactive power at both ends of a transmission line, denoted as and ; and the electrical power injection at the bus bar, denoted as Pinj and Qinj. The state variables are the magnitude and phase angle of the bus voltage, denoted as U and θ, respectively. Note that one of the voltage phases in θ should be chosen as a reference phase angle. The measurement vector of the electricity network and the state vector are represented in (34) and (35), respectively.
(34) |
(35) |
As for the DH network, the measurements are usually located at the heat source or consumer nodes as well as some important pipelines [
In [
(36) |
(37) |
SE of IHEN is to estimate the solution of the following optimization problem with a given measurement set of the integrated network:
(38) |
s.t.
(39) |
(40) |
(41) |
(42) |
(43) |
(44) |
(45) |
Equations
To solve the optimization problem in (38)-(45), a moving horizon estimation approach with a two-stage iterative algorithm is adopted to solve the problem [
Step 1: initialize all the state variables and parameters of the fluid.
Step 2: with real-time measurement metered at time step t, add measurement functions in (39) and (40).
Step 3: initialize the index of outer iteration, i = 1.
Step 4: in the first stage, the time delay s of each pipeline is estimated using (22).
Step 5: in the second stage, the entire SE problem in (38) is modeled and solved. Firstly, select the historically estimated inlet temperature of each pipeline which is closest to time ts, and add the dynamic thermal constraint of each pipeline in (44) into the model.
Step 6: add other steady-state constraints into the model (37), including (41)-(43) of the DH network, and (45) of the coupling area.
Step 7: solve the SE problem by Lagrange multipliers in (46) and get the estimates . An inner iteration algorithm is performed with the correct equation as in (47) [
(46) |
(47) |
where is the vector of Lagrange multipliers; W is the weight matrix of measurement; and F, C are the Jacob matrices of f(x) and c(x), respectively.
Step 8: if the outer iteration is convergent, i.e., , or the index of the iteration i reaches the maximum number, go to the next time step, i.e., , and repeat from Step 2; otherwise, update state variables with newly estimated results, set , and repeat from the first stage in Step 4.
The discrepant measurement resolutions of the electricity network and the DH network should be considered when performing the combined SE. As for the electricity system, the monitoring resolution of SCADA system is usually seconds to minutes. Since the fast dynamics of the electricity system may cause nasty accidents, data are sampled with high frequency and transmitted with a special wireline network. However, in the DH network, the measurement data are still used for billing purposes in most circumstances, with the resolution of minutes to hours [
During the measuring period of
Although there is no available measurement, from the analysis in Section II-A, we found that with some specified conditions, the hydraulic network and thermal network can be solved independently. Therefore, we propose a method that uses only the measurements of the electricity network to estimate the states and dynamics of the DH network. The steps are introduced as follows:
Step 1: perform the independent SE of the electricity system [
Step 2: calculate the power flow of the boundary area of the electricity network as
Step 3: calculate the power flow of the boundary area of the DH network by using coupling constraints in (33), and obtain .
Step 4: estimate the mass flow rate of each circulation pump, i.e., mpu, by using (5), (28), and the power consumption of each circulation pump in
Step 5: solve the hydraulic network by using (6)-(8), and the mass flow rate of each circulation pump, i.e., mpu, and then obtain the mass flow rate of each pipe, source, and consumer.
Step 6: estimate the time delay s of each pipe by using the mass flow rate of each pipe calculated in Step 5, i.e., mpi.
Step 7: supposing that the outlet temperatures of all heat consumers, i.e., , are specified, calculate the nodal temperature of the return network by using (10) and (23) as well as the time delay
Step 8: estimate the supply temperature of each heat source by using (11), the mass flow rate estimated in Step 5, the power injection of each source in
Step 9: calculate the nodal temperature of the supply network by using (10) and (23), the supply temperature of each heat source , and the time delay
Step 10: estimate the heat power of each consumer ΦL by using (12), the mass flow rate estimated in Step 5, the supply temperature estimated in Step 9 and the specified outlet temperature.
When
In this section, we validate the effectiveness of the proposed TSE approach and the HSE approach via numerical simulation on different cases. Firstly, the TSE approach is tested on a simple integrated network compared with the steady SE (SSE) approach proposed in [
In all of the test cases, some conditions in the network are specified, then true values of other states of the heating network are obtained by 12-hour simulation conducted by Bentley sisHY
Following indexes are used to measure the accuracy of the SE:
(48) |
(49) |
where is the number of time points; is the number of measurements; is the measurement; is the true value of the estimates; and is the estimates. Use to evaluate the effects of SE, and a small value of the index means a good estimation.
In this part, the proposed TSE approach is verified via numerical test on a simple integrated network. The integrated network consists of a 6-bus electricity network as shown in

Fig. 4 One-line diagram of 6-bus electricity network.

Fig. 5 Structure of combined network.
The SSE approach proposed in [
Table II gives the estimate result of both approaches. The of the estimates from the TSE approach is lower than the from the SSE approach, which confirms that the TSE approach is more accurate than the SSE approach.
at each time point of both approaches are calculated (), and the results are depicted in

Fig. 6 of estimated results at different time points.

Fig. 7 Measurement and true value of and .
In this part, we illustrate the HSE approach with a 6-bus and 5-node IHEN. The 6-bus electricity network used here is the same one as shown in
1) Heat source 1: from 0-2 hours, 80 ℃; from 2-12 hours, 100 ℃.
2) Heat source 2: from 0-3 hours, 80 ℃; from 3-12 hours, 100 ℃.
The HSE approach introduced in

Fig. 8 Test results of true value, measurements and estimates using HSE. (a) Power output and supply temperature of CHP1. (b) Power consumption and mass flow rate of pump 1. (c) Power output and supply temperature of CHP2. (d) Power consumption and mass flow rate of pump 2. (e) Mass flow rate of pipes 1, 3, 4, and 5. (f) Supply temperature of heat consumers 1, 2, and 3.
The supply temperatures of CHP units are regulated following curves (the third sub-figure of
The other advantage of HSE is to precisely estimate the time delay of heat transport in the pipeline by using the measurements in the electricity network. Here, the performance of estimating the time delay of each pipe is illustrated in

Fig. 9 Estimates of time delay of each pipe using HSE, comparing with TSE. (a) Pipe 1. (b) Pipe 3. (c) Pipe 4. (d) Pipe 5.
In this paper, we firstly study the time-scale characteristics of IHEN. Then a TSE approach is proposed considering time delay caused by the process of heat power transportation in the pipeline. After that, a HSE approach is developed to integrate measurements of different networks with discrepant resolutions. Results show that, in both steady and dynamic processes, the two-stage estimator has good convergence. The hybrid estimator has good performance on tracking the variation of states in the heat network, even when the available measurements are very limited. When the heat network becomes active, the hybrid estimator shows obvious advantages.
The proposed estimation approach could be easily expanded when the dynamics of other components of the network should be considered. In our future work, dynamic properties of other components in the heat network shall be considered such as heat exchangers and buildings. Besides, the bad data identification with the dynamic model should also be considered.
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