Modeling the Transient Security Constraints of Natural Gas Network in Day-ahead Power System Scheduling

Abstract

The rapid deployment of gas-fired generating units makes the power system more vulnerable to failures in the natural gas system. To reduce the risk of gas system failure and to guarantee the security of power system operation, it is necessary to take the security constraints of natural gas pipelines into account in the day-ahead power generation scheduling model. However, the minute- and hour-level dynamic characteristics of gas systems prevents an accurate decision-making simply with the steady-state gas flow model. Although the partial differential equations depict the dynamics of gas flow accurately, they are hard to be embedded into the power system scheduling model, which consists of algebraic equations and inequations. This paper addresses this dilemma by proposing an algebraic transient model of natural gas network which is similar to the branch-node model of power network. Based on the gas flow model, the day-ahead power system scheduling model is then proposed with the solution technique of successive linear programming and Benders decomposition. Tests are conducted to prove the effectiveness of the proposed models.

Full text

Modeling the Transient Security Constraints of
Natural Gas Network in Day-ahead
Power System Scheduling
Jingwei Yang∗, Ning Zhang∗, Chongqing Kang∗, Pierre Pinson†
∗Department of Electrical Engineering, Tsinghua University, Beijing, China
†Department of Electrical Engineering, Technical University of Denmark, Denmark
Abstract—The rapid deployment of gas-fired generating units
makes the power system more vulnerable to failures in the
natural gas system. To reduce the risk of gas system failure and to
guarantee the security of power system operation, it is necessary
to take the security constraints of natural gas pipelines into
account in the day-ahead power generation scheduling model.
However, the minute- and hour-level dynamic characteristics of
gas systems prevents an accurate decision-making simply with
the steady-state gas flow model. Although the partial differential
equations depict the dynamics of gas flow accurately, they are
hard to be embedded into the power system scheduling model,
which consists of algebraic equations and inequations. This paper
addresses this dilemma by proposing an algebraic transient model
of natural gas network which is similar to the branch-node
model of power network. Based on the gas flow model, the day-
ahead power system scheduling model is then proposed with the
solution technique of successive linear programming and Benders
decomposition. Tests are conducted to prove the effectiveness of
the proposed models.
Index Terms—Natural gas, gas network, power system schedul-
ing, transient model, dynamic characteristics, multi-energy sys-
tems
I. INTRODUCTION
The number of natural gas-fired units is increasing rapidly
in many parts of the world. According to the U.S. Energy
Information Administration (EIA), gas-fired units reached up
to 42% of total generation capacity in U.S. at the end of 2014.
The net electricity generation from natural gas accounted for
28% [1]. Even in China, where natural gas is not as rich as
the U.S., gas-fired units play an important role in many places
such as Guangdong and Beijing.
The growing dependency of power generation on natural
gas makes the power system more vulnerable to failures in
natural gas system such as pipeline trip off and pressure trigger
[2]. Taking the pressure trigger problem as an example. Too
much use of gas generation during the peak-load hours will
lead to a deep pressure drop in the gas network which may
damage the pipelines and thus make natural gas unavailable
for gas-fired units. To reduce the risk of gas system failure and
protect the power system operation, it is necessary for ISO to
This work is Sponsored jointly by Major International (Regional) Joint
Research Project of National Science Foundation of China (No. 51620105007)
and State Grid Corporation of China “Research and application of the key
technology of the combined heat and power security and economic dispatch”
consider the security constraints of natural gas systems from
the very beginning of daily operation, namely the day-ahead
power system scheduling.
In recent years, some attention has already been paid on the
operation and scheduling of power system with the consider-
ation of natural gas system. In [2], the natural gas network
was divided into sub-areas and nodal balance equations were
used to model the network gas flow. In [3] and [4], the gas
system was modeled with more details in gas flow character-
istics. Benders decomposition was used to solve the complex
nonlinear problem. It is worth noting that these papers all use
steady-state gas flow model to model the security constraints
of natural gas pipelines. However, the dynamic characteristics
of gas flow differs a lot from power flow. On the time scales of
minutes to hours, though power flow can be seen as stationary,
the steady-state modeling of gas flow will lead to inaccuracy
[5]. The neglect of gas dynamic behavior may also reduce the
feasible region of power scheduling problem as the effect of
“gas flow inertia” is not considered.
Coordinating the dynamics of the natural gas system with
day-ahead power system scheduling is not a easy task since
gas flow is governed by a set of partial differential equations
(PDEs) while power scheduling problem is composed of a
set of algebraic equations and inequations. Although [6] and
[7] applied finite difference method to change the PDEs to
algebraic equations, this method is restricted to modeling one-
dimensional single pipeline and is difficult to be realized
in meshed gas network. [8] employed a reduced network
flow model to simplify the PDEs, however, the model is not
simple enough to be easily incorporated into power system
problems. This leads to a lot of trouble in embedding the
gas dynamic security constraints into the day-ahead power
system scheduling model, particularly at the step of computer
implementation.
This paper addresses the problem of appropriate modeling
of transient gas security constraints in the day-ahead power
scheduling model by making two major contributions:
(1) To propose a transient model of natural gas network which
is as compact as the standard branch-node model of power
flow. The model is accurate compared to the PDEs and
easy to be realized;

(2) To establish a day-ahead power system scheduling model
with consideration of gas flow dynamic constraints under
a uniform time scale. Specific solution steps are explained
as well.
The remainder of the paper is organized as follows. Section
II provides the derivation and details of the standardized
modeling of gas flow dynamics. Section III incorporates the
gas pipeline dynamic security constraints into the day-ahead
power system scheduling model and provides the solution
technique for the complex prolem. Section IV illustrates the
effectiveness of the proposed models with two test cases.
Finally, conclusions are drawn in section V.
II. STAN DA RDIZED DYNAM IC MODELING OF GA S
PIPELINE NETWORK WITH FICTITIOUS NODES
A. Basics of the Gas Flow Dynamics
Gas flow along a pipeline is governed by the law of
conservation of mass and Newton’s second law. For ideal
gas flow, where gas temperature and gas compressor factor
are seen as constant, the following two equations are usually
applied to depict the dynamics of gas flow [9]:
∂π(x, t)
∂t =−C1
∂f (x, t)
∂x (1)
∂π2(x, t)
∂x =−C2f2(x, t)(2)
where π(x, t)and f(x, t)denote the node pressure and volume
flow rate(under standard condition) respectively at location x
and time t.C1and C2are constants depend on the parameters
of piplines.
Equations (1) and (2) are hard to solve anayltically. A
common method is to apply Euler finite difference technique to
replace the derivative expression both in space and time. Thus
the PDEs can be transformed to a set of algebraic equations
at (x, t)with time step δt and spatial step δx:
π(x, t)−π(x, t −δt)
δt =−C1
f(x+δx, t)−f(x, t)
δx (3)
π2(x+δx, t)−π2(x, t)
δx =−C2f2(x, t)(4)
It may be easy to model a single pipeline with equations
(3) and (4) as the pipeline is one-dimensional in space and
there is not any connection to other pipelines. For a multi-
ple input/output meshed pipeline network, however, the one-
dimensional PDEs are not able to handle. Although several sets
of PDEs with respect of different pipelines can be assembled,
the boundary equations of PDEs have to be carefully managed.
Therefore, a simple network model which is able to describe
the dynamics of gas network is required.
B. Transient Model of Gas Network
First of all, we re-define the nodes and branchs in gas
network by introducing the idea of fictitious nodes. Fictitious
nodes are nodes along pipelines with no gas injection. This
concept derives from the Eular finite difference technique and
represents the spatial derivation of PDEs that govern gas flow.
As is shown in Fig.1, the white circles denote normal nodes
which are usually places of gas source/load or conjunction
of pipelines. The black circles are fictitious nodes. In the
proposed model, the term “node” include both normal node
and fictitious node. The term “branch” denotes the section of
pipeline connecting two nodes.
Fig. 1. Nodes and branches in gas network.
Fig.2 illustrates the variables defined in the proposed model
with a general branch. For any node iat time t, the associated
variables are gas injection Ii(t)and nodal pressure πi(t). For
any branch (i, j), the associated variables are “from flow”
f(i,j)(t)and “to flow” f(j,i)(t). The compressor has a pressure
boost ratio of kand the same input/output volume flow rate.
There are four kinds of equations in the proposed model: the
branch characteristics, the nodal balance, the initial condition
and the boundary condition. The study period is set to t=
0, δt, 2δt, . . . , T δt. The system condition is given at t= 0 as
initial value while variables in the other periods are unkonown.
δt is the time resolution and Tis a integer constant.
•Branch Characteristics
The branch characteristic equations are derived from equa-
tion (3) and (4). For the general branch (i, j)shown in figure
2, we have:
πi(t)−πi(t−δt)
δt =C1
f(i,j)(t) + f(j,i)(t)
l(i,j)
(5)
π2
i(t)−π2
j(t)/k2
l(i,j)
=C2f2
(i,j)(t)(6)
In the equations, l(i,j)denotes the length of the pipeline (i, j ).
•Nodal Balance
For any node i, the nodal gas flow balance is expressed as
follows:
X
j(i)
f(i,j)(t) = Ii(t)(7)
The notation j(i)denotes any node jconnected to node i
•Initial Condition
The initial condition provides the initial state of the pipeline
network at time t= 0. The value of variable πi(0) has to be
given in the proposed model.
•Boundary Condition
The boundary condition provides the information of nodal
pressure and gas injection in the study period. The nodes can
be classified into two categories—the “known pressure” nodes
and the “known injection” nodes. For any “known pressure”

Fig. 2. A general branch in gas network.
node i,πi(t)is given at t=δt, 2δt, . . . , T δt. Otherwise,
Ii(t)is known. It should be noted that for fictitious nodes,
the injection is always zero.
C. Further Discussion of the Gas Network Dynamic Model
For a gas pipeline network with Nnodes (including fic-
titious nodes) and Mbranches, the number of variables are
(2N+ 2M)T. If the boundary condition is determined, then
the number of equations is equal to that of variables including
(5), (6), (7) and the equations of boundary condition. If the
boundary is provided with a feasible range, then the number
of equations is smaller than the number of variables, which
produces the possibility to optimize. The pressure boost ratio
of compressor is set as constant in this model as it rarely makes
intra-day change. However, the model is still applicable if the
ratio is adjustable.
This proposed method provides a standardized way to
model gas network dynamics by defining fictitious nodes and
expanding the PDEs from one-dimension to multi-dimension
in space. The branch-node model is as brief as the power
network model. Interestingly, the dynamic model can be easily
degraded to steady-state model by neglecting the items on the
left side of equation (5):
0 = C1
f(i,j)(t) + f(j,i)(t)
l(i,j)
(8)
III. DAY-AHEAD POWER SYSTEM SCHEDULING MODEL
WITH GAS NE TWORK DYNA MI C CONSTRAINTS
A. Model
The day-ahead power system scheduling with gas network
dynamic constraints is to determine the commitment and
dispatch of generation units while keep the gas system in
a secure state. As this paper mainly focuses on modeling
natural gas dynamic constraints, the scheduling model is
simplified to only consider the dispatch of gas-fired units in
a vertically integrated electricity market. Future works may
include more detailed and realistic models considering both
the unit commitment decisions and market factors .
In the following model, set Erepresents the nodes in power
system while set NG denotes the nodes in gas network and
set Grepresents all the gas-fired units. GU(i), i ∈E(orN G)
denotes the gas-fired units connected to node iin power
system(or natural gas system).
1) Objective: The objective is set to minimize the fuel cost
of all the generators in the scheduling periods,
min λX
tX
i∈G
aiP2
i(t) + biPi(t) + ci
+ξX
tX
i∈E
LSi(t)(9)
where λand ξare gas price and load shedding penalty
respectively. Pi(t)is the power output of gas unit iat time
step t.LSi(t)is the load shedding at bus iat time step t. The
generation fuel is considered as a quadratic function of output
power.
2) Unit constraints: For any unit i∈G,
Pi≤Pi(t)≤Pi(10)
−Ri≤Pi(t)−Pi(t−δt)≤Ri(11)
Constraint (10) defines the power generation limit of gas-
fired units. Riis the maximum ramping rate.
3) Power network constraints: The DC power flow con-
straints are used here.
−P Fij ≤θi(t)−θj(t)
xij
≤P Fij (12)
X
j(i)
θi(t)−θj(t)
xij
=X
k∈GU(i)
Pk(t)−Li(t)(13)
i, j ∈E
The θi(t)denotes the bus phase angle. xij is the branch
reactance. P Fij is the line flow limit. j(i)denotes all the
buses jconnected to bus i.
4) Natural gas network dynamic constraints: The natural
gas dynamic constraints consist of equations (5), (6) and (7).
Other constraints, including nodal pressure constraints and gas
injection rate constraints, are represented by:
πi(t)≤πi(t)≤πi(t)(14)
Ii(t)≤Ii(t)≤Ii(t)(15)
For nodes with gas-fired plants, the nodal gas injection is
associated with the power generation:
Ii(t) = −X
k∈GU(i)
akP2
k(t) + bkPk(t) + ck(16)
Eqs (5), (6), (7) and (9)-(16) consist of the day-ahead
power system scheduling model with gas network dynamic
constraints. The model is under a uniform time framework,
which means the time resolution δt is the same for power and
gas system. In practice, δt is suggested not to be greater than
30 minutes because of the need to present gas dynamics.
B. Solution
The proposed scheduling model is nonlinear and non-
convex. To solve the model, Benders decomposition is used to
decompose it into a master power scheduling problem and a
gas network security check subproblem. The master problem
is a quadric optimization with (9) as objective and (10)-(13)
as constraints, which is easy to solve.

The gas network security check subproblem is nonlinear
because of (6). This paper uses the technique of successive
linear programming and forms the incremental model as
follows (in the kth iteration):
min X
tX
i∈NG∩G
∆I(k)
i(t)(17)
s.t. J(k−1)∆x(k)=0(µ)(18)
πi(t)≤π(k−1)
i(t)+∆π(k)
i(t)≤πi(t), i ∈N G (19)
Ii(t)≤I(k−1)
i(t)+∆I(k)
i(t)≤Ii(t), i ∈N G (20)
The objective is to minimize the gas load shedding at
nodes with gas-fired units. x=hfT
f rom fT
to πTITiT
is a
vector composed of “from flow” f(i,j)(t), “to flow” f(j,i)(t),
nodal pressure πi(t)and nodal gas injection Ii(t).∆xis the
incremental of x.Jis the Jacobian matrix, which is the partial
derivative of equations (5), (6), (7) with respect to ∆x.
To solve the gas network check subproblem, iteration has
to be done until ||∆x|| is small enough. Once the iteration is
finished, Benders cuts will be formed and added to the master
problem based on part of the Lagrange multiplier µon nodal
balance equations (7):
X
i∈NG
µiX
j∈GU(i)
2ajˆ
Pj(t) + bjPj(t)−ˆ
Pj(t)
+GS(t)≤0(21)
where ˆ
Pj(t)is the power generation of unit jat time tgiven
by the master problem in the last outer iteration. GS(t)is the
total gas shedding, which is calculated as
GS(t) = −Ii(t) + X
i∈NG
Ii(t)
=X
j∈G
ajˆ
Pj(t)2+bjˆ
Pj(t) + cj+X
i∈NG
Ii(t)(22)
The specific steps of the solution is provided in Fig.3.
Fig. 3. Steps to solve the dynamic gas-constrained power scheduling model.
IV. CAS E STU DY
A. Verification of the Proposed Gas Dynamic Model
In this section, the proposed dynamic gas network model
with fictitious nodes is compared with the solution of PDEs,
using a one-dimensional gas pipeline as test case. The pipeline
is a typical transmission pipeline, which is 60 miles in length
and 24 inches in diameter. The inlet-pressure is fixed at 900
psia while the outlet gas load is changing. Fig.4 shows the
results of different models. Different numbers of fictitious
nodes are added in the proposed model to compare the
performance.
Fig. 4. Results of outlet-pressure using different models.
From Fig.4, it can be clearly seen that the results of
the proposed model with 3 or 4 fictitious nodes (branch
length=15 or 12 miles) is almost the same with the accurate
result (PDEs). Although the result is not that accurate when
only one fictitious node is added in the proposed model
(branch length=30 miles), it does exhibit the dynamic changing
progress of outlet-pressure. In fact, a branch length of 10 miles
is accurate enough to model the dynamics of gas flow.
B. Day-ahead Scheduling Results Considering Dynamic Gas
Network Constraints
The gas-electricity test case is modified from [3] with a 6-
bus power network, a 6-node gas network and three gas-fired
units. The system topology is shown in Fig.5 and Fig.6. Note
that the fictitious nodes have been added on Fig.6.
Fig. 5. Power network topology.
A block-wise electricity load profile with a short morning
peak and a long evening peak is adopted for better illustration.
The scheduling period is 24 hours while the time resolution is
15 minutes. Three cases are compared to show the importance
of the consideration of dynamic gas security constraints.

Fig. 6. Gas network topology.
•Case 1: Scheduling model without gas network security
constraints
•Case 2: Scheduling model with steady-state gas network
security constraints
•Case 3: Scheduling model with dynamic gas network
security constraints (the proposed model)
The day-ahead generation and load shedding schedule is
determined in each case. Based on the generation schedule,
the state of natural gas system is calculated again using the
standardized transient model.
Firstly, we focus on the necessity of taking gas network
constraints into account in the day-ahead scheduling model.
Fig.7 shows the pressure fluctuation of node 2. it can be
clearly seen that the result of case 2 and case 3 is within
the nodal pressure limit. However, without the consideration
of gas network constraints, the pressure of node 2 in case 1
goes beyond the lower limit when the electricity load peak
arrives. This may lead to failure in natural gas system and
thus influence the operation of power system.
Fig. 7. Node 2 pressure fluctuation.
Fig. 8. Load shedding schedule in different cases.
Secondly, we illustrate the importance of considering the
dynamics of gas flow by comparing the load shedding result
in case 2 and case 3. As is shown in Fig.8, the load shedding
in case 3 is nearly half of that in case 2. Note the curve for
case 3 is slightly moved for a clearer vision. The reason for
this phenomenon is that the nodal pressure will not change
immediately when the gas injection changes, which is not
considered in the steady-state model. Back to Fig.7, it can be
found that, during the load peak time, the natural gas system is
more fully utilized in case 3. This indicates that the gas-fired
units take more gas during load peak in case 3 than in case 2
and thus leads to the less load shedding.
V. CONCLUSION
This paper proposes a transient model of gas network flow
by defining fictitious nodes and corresponding variables. The
model is as simple as that of power flow and provides a
standard way to model the gas network. Based on the modeling
technique, this paper establishes the day-ahead power system
scheduling model with consideration of gas network dynamic
security constraints. The dynamic gas flow model is verified
compared to the result of PDEs. The power system scheduling
model is tested in different cases. The results show that to take
the dynamic gas network constraints into account instead of
the steady-state constraints will better improve the security and
effectiveness of power system scheduling.
We envision future works to apply the proposed transient
natural gas model in more realistic applications regarding
to coordinated gas-electricity operation. Factors such as the
unit commitment and market coupling can be taken into
consideration to reveal the realistic advantages of the “slow
transient characteristics” of natural gas systems.
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