AU2020100429A4

AU2020100429A4 - A dynamic optimal energy flow computing method for the combined heat and power system

Info

Publication number: AU2020100429A4
Application number: AU2020100429A
Authority: AU
Inventors: Wei Gu; Shuai YAO; Suyang ZHOU
Original assignee: Southeast University
Current assignee: Southeast University
Priority date: 2020-03-20
Filing date: 2020-03-20
Publication date: 2020-09-10
Anticipated expiration: 2028-03-20

Abstract

The invention provides a dynamic optimal energy flow computing method for the combined heat and power system. Firstly, the dynamic heating network model containing 1) the topological constraints that describe how the water temperatures change during and after the process of nodal fusion, and 2) the partial differential equation constraint that describes the temperature dynamics in each pipe is established. The partial differential equation constraint is converted into a group of linear equality constraints through the finite difference method. Secondly, the optimization model is established and solved with LINGO to select the optimal difference step sizes that can best alleviate the contradiction between model accuracy and computational complexity. Finally, with the optimal difference step sizes, the dynamic optimal energy flow model of the combined heat and power system is established, with its objective function to be minimizing the total fuel costs over a certain period, which is a large-scale mixed integer quadratic programming (MIQP) problem with second-order conic constraints. The dynamic optimal energy flow model is solved with the software CPLEX to provide support for optimal planning, operation and control for the combined heat and power system. 1/2 DRAWING Sourceunits - -+Primary energy ---- Heat -- Electricity C gasbiie~Conversion & | Heat load cHipont storage units DHN Gas C 1P1 H heat battery generator pUMP stage Sunlight ga eelectric heat Wind tank Power grid Electric load Fistationg N Figure1I

Description

1/2

DRAWING

Sourceunits - -+Primary energy ---- Heat -- Electricity

C gasbiie~Conversion & | Heat load cHipont storage units DHN H Gas C 1P1 heat battery generator pUMP stage Sunlight ga eelectric heat Wind tank Power grid Electric load Fistationg N

Figure1I

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A DYNAMIC OPTIMAL ENERGY FLOW COMPUTING METHOD FOR THE COMBINED HEAT AND POWER SYSTEM BACKGROUND OF THE INVENTION

[0001] The contradiction between social development and energy consumption has become

increasingly prominent. Meeting challenging environmental targets and ensuring sustainable

energy supplies to future generations require innovative reforms of energy utilization. Under this

background, the concept of multi-energy system (MES) emerges, the essence of which is to

integrate diverse carriers of energy (e.g. gas, electricity, heat, hydrogen, etc.) as a whole to fully

exploit the synergies and complementation among them for enhancing the overall energy

efficiency, promoting renewables consumption, as well as reducing energy costs and emissions.

[0002] As a typical form of the MES, combined heat and power (CHP) system provides

extensive interactions between heat and electricity through coupling facilities like cogeneration

units, electric boilers and heat pumps. Compared with the traditional decoupled energy system, the

CHP system can improve the overall energy efficiency by making full use of the waste heat

(produced along with power generation) to satisfy part of civil or industrial heating loads.

Moreover, thermal inertial of heating networks and buildings can significantly increase the

flexibility to accommodate more renewable energy.

[0003] The proposed dynamic optimal energy flow computing method consists of three steps,

including:

[0004] 1) The establishment of dynamic heating network model,

[0005] 2) The selection of optimal difference step sizes,

[0006] 3) The solution to the dynamic optimal energy flow model.

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Si The establishment of dynamic heating network model:

[0007] The mathematical model of dynamic heating network contains two parts: 1) the

constraints related to the network topology (denoted as topological constraints), and 2) the

constraints of temperature dynamics in each pipe.

S1.1 The topological constraints

[0008] To maintain stable hydraulic conditions, "quality regulation" mode is adopted in most

cases where the mass flow rate of hot water is fixed but the temperature is adjusted according to

the fluctuated heating loads. The topological constraints that describe how the water temperatures

change during and after the process of nodal fusion are presented in Eq. (1).

I(rh,-T) T - Jrh

I T°4 T Vie(= ",Vje(= S°I S,"S"VSi L"" Vi Equation 1

[0009] where rh is the mass flow rate of the ithpipe;Tn is the temperature of hot water that

flows from the ith pipe into the nth node; Tout n~J is the temperature of hot water that flows from the

nth node into the jth pipe; 7 is the temperature of the nth node; S Sout is the set of allpipes that

are directly connected to the nth node and have water flowing into / out of it.

S1.2 The constraints of temperature dynamics

[0010] The heating network consists of a group of pipe segments that are buried underground.

Typically, each pipe segment is made of steel and wrapped by a layer of heat insulator with a

mantle. The temperature dynamics along each pipeline is expressed in Eq. (2), which can be

derived from the Law of Conservation of Energy through infinitesimal analysis.

8T 8T v +v -+ (T -Ta)=O 9t 9x thcR Equation 2

[0011] where v and care the flow velocity and specific heat capacity of hot water, respectively;

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T a is the ambient temperature outside the pipe; R is the thermal resistance of the pipe segment; t

and x are time and position variables.

[0012] To obtain the determinate solutions of Eq. (2), a set of initial and boundary conditions are

necessary. Typically, the temperature distribution along the pipeline at initial time point (denoted

as the initial condition) and the temperature of hot water at the inlet (denoted as the boundary

condition) need to be given, as formulated in Eq. (3).

FT(x,)=(o(x), x O tT(Ot)=V,(t), t O Equation 3

[0013] where (p(x) is the function of water temperature on space; V(t) isthefunctionof

water temperature on time.

[0014] Considering the partial differential equation (PDE) in Eq. (2) cannot be solved directly in

an optimization model, we employ the finite difference method to discretize it into a group of

linear quality constraints. The developed difference scheme is presented in Eq. (4).

k+ 1 +a-Tk a- ,8fTk+1+ af- .k '1+a+§8 1+a+p ' 1+a+p '

+ Ta (1 i< M, O k N-1) I1+ a +§6 Equation 4

[0015] where a and § are two parameters defined for simplified representation:

VT Vr h thcR Equation 5

[0016] where hand r denote the spatial and temporal step sizes.

[0017] The developed difference scheme in Eq. (4) is unconditionally stable and convergent for

any given h andr, and has the convergent order of O(h 2 + r 2 ).

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S2 The selection of optimal difference step sizes

[0018] In actual practice, the selection of difference step sizes (hand r) is crucial to achieve a balance between the model accuracy and computational complexity of the whole dynamic optimal energy flow problem. Since the truncation error of our developed difference scheme (shown in Eq. (4)) is O(h 2+ r2), small h and r will lead to low level of calculation error, but at the same time elevate the dimensions of constraint set, thus increasing the computational complexity to find the optimal solution. Therefore, it is necessary to select an optimal combination of spatial and temporal step sizes, which contains three steps: 1) determine the functions of simulation error and computational complexity; 2) determine the upper and lower bounds of spatial and temporal step sizes; 3) establish the optimization model to obtain the optimal step sizes.

S2.1 Determine the functions of simulation error and computational complexity:

[0019] The simulation error of heating network is measured by the truncation error of the adopted difference scheme, which can be expressed as follows:

g(h,r)= 6 Equation 6

[0020] where (h,r) denotes the simulation error (determined by the selected h and r) of

heating network.

[0021] The computational complexity of the dynamic optimal energy flow problem is measured by the size of the constraint set:

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11 (h +1)(r +1) Q~h~)=(+ -x (1 + -)= h r hr Equation 7

[0022] Where Q(h,r) denotes the computational complexity that is dependent on the selected

h andr.

S2.2 Determine the upper and lower bounds of spatial and temporal step sizes

[0023] The upper bound of spatial step size h' takes the minimum of all pipe lengths:

h""a = min(LI,L2 ,. - -,L,} Equation 8

[0024] where L,(i =1,2, -, p) denotes the length of the ith pipe in the heating network.

[0025] Let Cax be the maximal simulation error allowed by the system, then the upper bound

of temporal step size r'ax can be determined by Eq. (9).

(hmax) 2 +(r"') 2 max 6 Equation 9

where Rik denotes the truncation error of the adopted difference scheme.

[0026] Let "maxbe the maximal computing time allowed for solving the dynamic energy flow

problem, then the lower bounds of spatial and temporal step sizes (h""" and rm") can be

determined as follows.

F1D(hmax,r"min) o'"a

|1 (hi"n,r'""), r'ax Equation 10

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[0027] where 1(h,r) denotes the computing time required to solve the dynamic energy flow

problem under a certain combination of h and r.

S2.3 Establish the optimization model to obtain the optimal step sizes

[0028] The selection of the optimal step sizes is to seek a tradeoff between the model accuracy

and the solution complexity. First of all, the coefficient of model accuracy wI and that of solution

complexity C02 have to be determined based on the application scenarios, where the following

equation has to be satisfied.

1 2 Equation 11

[0029] Typically, for the off-line optimal energy flow problems, the computing time is sufficient,

and the solution complexity is usually less important than the model accuracy. Therefore, mio

should be greater than C02 . For the on-line optimal energy flow problems, Cio is usually smaller

than cw 2 .

[0030] With h", hx rnin rmax, cio and c2 obtained fromEq.(9)- (11), the optimal

step sizes can be determined through the following optimization model.

hr- h2 + .2 max co x -WLox 2 (h 1)(- 1) 6 s.t. h"n< h h"ax Z""" max ,r ru,. Equation 12

S3 The solution to the dynamic optimal energy flow model

[0031] The solution to the dynamic optimal energy flow model can be divided into three steps: 1)

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establish the objective function; 2) establish the constraint set; 3) give the solution strategy.

S3.1 Establish the objective function

[0032] The objective function of the dynamic energy flow model is to minimize the total fuel

costs over period TN , as formulated in Eq. (13). TV

min (c J2,c° cZJi +C; Jpt)- At t=1 is jESc kc S Equation 13

[0033] where cg , c' and c are the unit prices of natural gas, coal, and electricity; , and

J't are the natural gas and coal consumptions of a certain facility at time t; P,t is the interacted Pc

electric power in tie line k at time t, the sign of which is positive when purchasing power and

negative when selling power; Sg and S° are the sets of all gas and coal facilities; Se is the set

of all tie lines; At is the dispatch interval.

S3.2 Establish the constraint set

[0034] The constraints of power grid are expressed by the following equation: I )+ P. (P,-rT P, -aEm(j) hEv(j)

(Q, -x aE ) a I +Q= + EVJQf)I

-U J .U- I 2(r/P+xQ)±( 'i + xJ)P '' Yii 2P 2Q I -U 2 +U2i

2 Equation 14

[0035] where m(j) and n(j) denote the set of parent nodes and child nodes of node]j; and

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Qj denote the active and reactive power in branch aj; P and Qj denote the net injected active

and reactive power in node]j; r and xaj denote the resistance and reactance of branch aj; U,

and I. denote the voltage of node i and the current in branch ij; •12 is the 2-norm of a vector.

[0036] The constraints of heating network are shown in Eq. (1), (3), (4) and (5). Considering the thermal inertia of heat loads, the supplied and demand thermal power do not need to be balanced in real time as in the power grid. Instead, a slight deviation between them is allowed to provide more flexibility for wind power accommodation, with the requirements that total supplied and demand heat within a certain period should be balanced. Therefore, the constraints of heating loads can be expressed as follows.

FQload( )QSupply <Qload(18

ViE SQL lT (Qi.t At)= (Q°ad - At) t_ t- Equation 15

[0037] where 5 denotes the maximal deviation ratio between the supplied and demand heat

power at load i; T denotes the total number of dispatch intervals within which the heat balance

should be satisfied; SQL denotes the set of all heat loads.

[0038] Typical conversion units like electric boilers and heat pumps are modeled as a linear equality constraint by using a fixed coefficient denoting the efficiency in the process of thermoelectric conversion:

QOlt =77p2hpi Equation 16

[0039] where Pi" and Qut denote the input electric power and output thermal power

respectively; 77p2h is the efficiency coefficient.

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[0040] For each facility, the minimal / maximal power output constraints can be expressed as:

F~ m ~i pllax , ViE Sac

tQimin Q1 Qrax Equation 17

[0041] where Sc is the set of all facilities in the system.

[0042] For facilities like generators and cogeneration units, their power outputs cannot change

arbitrarily within a certain period, which can be described as the ramping up / down constraints:

Fr Fin (t + At)- F(t) FPr"ax .e

, ,VlE 'S Qin (t+At) - QK(t) Qirmax Equation 18

[0043] Where Sfac is the set of facilities with ramping constraints in the system.

S3.3 Solution strategy

[0044] The overall dynamic energy flow problem can be concluded as a typical large-scale

mixed integer quadratic programming problem with second order conic constraints, which can be

formulated as:

min f(x) =xTcx+d x s.t. bib Ax bb A°qX = b°q

x, E{O,1}, ic I

Bx 2 Cx Equation 19

[0045] where is x the decision variable vector consisting of the power output of each facility

and the state variables of power grid and heating network; the superscript T represents the

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transpose operation of a matrix; the superscripts lb and ub represent the lower and upper limits of a

variable, respectively; Iis the set of all 0-1 variables in this problem; A, A°4, B and C are

coefficient matrices; bib, bub and b"q are constant vectors.

[0046] The solution process of this dynamic optimal energy flow problem is: 1) solve Eq. (12) with the software LINGO to get the optimal difference step sizes of heating network (e.g. h and r); 2) with the h and r obtained from step 1), solve Eq. (19) through the software CPLEX to finally get the optimal energy flow of the system.

[0047] The invention may be better understood with reference to the illustrations of embodiments of the invention:

[0048] Figure 1 is a schematic diagram of the combined heat and power system,

[0049] Figure 2 is a flowchart of the implementation procedures of this invention.

Claims

1/2 CLAIMS

1. A dynamic optimal energy flow computing method for the combined heat and power

system is proposed, which is characterized by the following steps: step 1) establish the dynamic

heating network model; step 2) select the optimal difference step sizes that can best alleviate the

contradiction between model accuracy and computational complexity; Step 3) with the optimal

step sizes obtained from step 2), the dynamic optimal energy flow model can be established, which

is a large-scale mixed integer quadratic programming (MIQP) problem with second-order conic

constraints. The dynamic optimal energy flow model is solved with the software CPLEX to

provide support for optimal planning, operation and control for the combined heat and power

system.

2. The dynamic heating network model as stated in claim 1 is characterized by two parts: 1)

the topological constraints that describe how the water temperatures change during and after the

process of nodal fusion; 2) the partial differential equation constraint that describes the

temperature dynamics in each pipe, which is discretized into a group of linear quality constraints

with the finite difference method.

3. The selection of the optimal difference step sizes as stated in claim 1 is characterized by

the following sub-steps: 1) determine the functions of simulation error and computational

complexity; 2) determine the upper and lower bounds of spatial and temporal difference step sizes;

3) establish the optimization model and solve it with the software LINGO to obtain the optimal

step sizes.

4. The solution to the dynamic optimal energy flow model as stated in claim 1 is

characterized by the following sub-steps: 1) establish the objective function --- minimizing the

total fuel costs of the system over a certain period; 2) establish the constraint set that consists of the

constraints of power grid, heating network, facilities and loads; 3) solve the dynamic optimal

energy flow model with the software CPLEX to obtain the optimal operation states of each facility,