1
Full Stochastic Scheduling for Low-carbon
Electricity Systems
Fei Teng, Member, IEEE and Goran Strbac, Member, IEEE
Abstract — High penetration of renewable generation will
increase the requirement for both operating reserve and
frequency response, due to its variability, uncertainty and
limited inertia capability. Although the importance of optimal
scheduling of operating reserve has been widely studied, the
scheduling of frequency response has not yet been fully
investigated. In this context, this paper proposes a
computationally-efficient mixed integer linear programming
formulation for a full stochastic scheduling model that
simultaneously optimizes energy production, operating reserve,
frequency response and under-frequency load shedding. By
using value of lost load as the single security measure, the model
optimally balances the cost associated with the provision of
various ancillary services against the benefit of reduced cost of
load curtailment. The proposed model is applied in a 2030 GB
system to demonstrate its effectiveness. Impact of installed
capacity of wind generation and setting of value of lost load are
also analysed.
𝑃𝑠𝑐𝑚𝑎𝑥
𝑃𝑠𝑑𝑚𝑎𝑥
𝐸𝑠𝑚𝑎𝑥
𝐸𝑠𝑚𝑖𝑛
𝑚𝑎𝑥
𝑅𝑔,𝑠
Note to Practitioners— One of the obstacles for large scale
deployment of wind generation is the challenges it imposes on
the efficient operation of the electricity system. This paper
presents a full stochastic scheduling model. The long-term
uncertainty driven by wind forecasting errors and short-term
uncertainty driven by generation outages are modelled by using
scenario tree and capacity outage probability table, respectively.
The model leads to significant operation cost saving within
reasonable computational time. The proposed model could be
applied in real large-scale power systems to support the costeffective integration of wind generation.
Index Terms—Unit commitment, stochastic programming,
power system dynamics, wind energy
𝑁𝑓
NOMENCLATURE
Constants
∆𝜏(𝑛)
Time interval corresponding to node n (h).
𝜂𝑠𝑐 /𝜂𝑠𝑑
Charge/discharge efficiency for storage unit s.
𝜇(𝑗)
Diurnal adjustment constant corresponding to the jth
time step of the day.
𝜋(𝑛)
Probability of reaching node n
𝑦
𝜎 𝑥 , 𝜎𝑖
Standard deviation of random Gaussian process
𝜑1 , 𝜑2
Autoregressive parameters.
𝐴(𝑛)
Set of nodes that are ancestors of node 𝑛.
𝒢
Set of thermal generators.
𝑆
Set of storage units.
𝒩
Set of nodes on the scenarios tree.
𝑎(𝑛)
Parent node of node 𝑛.
𝑐 𝐿𝑆
Value of lost load (£/MWh).
𝑃𝑔𝑚𝑎𝑥
Maximum generation of unit g (MW).
𝑚𝑠𝑔
𝑃𝑔
Minimum stable generation of unit g (MW).
𝑚𝑜
𝑇𝑔
Minimum off time of unit g (h).
𝑇𝑔𝑚𝑢
Minimum up time of tunit g (h).
𝑇𝑔𝑠𝑡
Startup time of l unit g (h).
𝑓𝑔𝐹
𝐻𝑔
𝐻𝑑
𝑇𝑑
𝑓0
∆𝑓𝑚𝑎𝑥
𝑞(𝑛)
𝑊(∙)
𝑋(𝑘)
𝛷(∙)
𝐷(𝑛)
*
Maximum charge rate of storage s (MW)
Maximum discharge rate of storage s (MW)
Maximum charge level of storage s (MWh).
Minimum charge level of storage s (MWh).
Maximum primary reserve capability of unit g or
storage s (MW).
The proportion of the spinning headroom, which can
contribute to primary reserve provision.
Inertia constant of unit g (s).
Inertia constant of demand (s).
Delivery time of primary reserve (s)
Nominal frequency level (Hz)
Maximum frequency deviation at Nadir (Hz).
Forecast error quantile of branch leading to node n.
Sigmoid-shaped function which transforms the wind
level to an aggregated wind output.
kth element in an autoregressive time series which
represents normalized wind level.
Number of outage levels in COPT
Standard Gaussian CDF
System demand at node n.
Multiplying
Decision variables
𝑃𝐿𝑆 (𝑛) Load shed at node n (MW).
𝑃𝑊𝐶 (𝑛) Wind curtailment at node n (MW).
𝑁𝑔𝑠𝑑 (𝑛) Number of unit g that are shut down at node n.
𝑁𝑔𝑠𝑡 (𝑛) Number of unit g that are started up at node n.
𝑃𝑔 (𝑛)
Power output of unit g at node n (MW).
𝑁𝑠𝐺𝑒𝑛 (𝑛) Operation status (0/1 for Pumping/Generating) of
storage s at node n.
𝑃𝑠𝑐 (𝑛)
Charge rate of storage s at node n (MW)
𝑃𝑠𝑑 (𝑛)
Discharge rate of storage s at node n (MW)
𝑅𝑔,𝑠 (𝑛) Primary reserve provision from unit g or storage s at
node n (MW).
𝑢𝑝
𝑁𝑔 (𝑛) Number of unit g that are online at node n.
𝑜𝑓𝑓
𝑁𝑔 (𝑛) Number of unit g that are offline at node n.
𝑠𝑔
𝑁𝑔 (𝑛) Number of unit g that start generating at node n.
𝐶𝑔 (𝑛)
Operating cost of unit g at node n (£)
𝐸𝑠 (𝑛)
State of change of storage I at node n (MWh)
𝐻(𝑛)
System inertia after generation loss at node n
(MWs/Hz).
𝑅(𝑛)
Total primary reserve provision (MW).
I. INTRODUCTION
D
ecarbonisation of Great Britain (GB) electricity system
is expected to be achieved by the integration of large
share of wind generation. However, due to the variability,
uncertainty and limited inertia capability, high penetration of
wind generation significantly increases the requirements for
2
various ancillary services, including both operating reserve
(OR) and frequency response (FR). These additional ancillary
services are mostly delivered through part-loaded generators
and fast standing plants. This not only leads to a lower
efficiency of the system operation, but may eventually limit
the ability of the system to accommodate wind generation.
A number of advanced scheduling approaches have been
proposed, such as robust optimization [1], interval
optimization [2] and stochastic optimization [3], to optimally
schedule the operation of the future low carbon systems with
high penetration of wind generation. The results in [4] clearly
demonstrates the benefits of stochastic approach over
traditional methods in terms of lower operating cost and
reduced wind curtailment. The authors in [5] develop a
computational-efficient stochastic framework for the
scheduling of large-scale power system. However, all of
these works focus on the optimal scheduling of OR in hourly
or half-hourly time resolution. Recently, unit commitment
(UC) with more frequently updates and finer time resolution
is proposed. The sub-hourly dispatch constraints are
incorporated into stochastic UC (SUC) in [6]. Authors in [7]
propose a multi-time resolution UC with the consideration of
system operation down to 5-min interval. The work in [8]
develops an integrated model to assess the impact of variable
generation at multiple timescales. The finest scheduling
interval is 6 seconds for the operation of AGC.
Moreover, as wind generation displaces conventional
plants, the system inertia provided by rotating mass reduces,
causing concerns over dynamic frequency stability [9] [10].
The reduced system inertia leads to faster decline of system
frequency after generation outages. Due to the limited
governor rate of conventional generators, an increased
amount of FR is required to maintain the post-fault frequency
deviation within certain limits. Extensive researches have
been carried out to directly incorporate the post-fault
dynamic frequency evolution into the scheduling models in
order to guarantee the adequacy of FR. Governor-rate
constrained FR requirements are developed and incorporated
into optimal power flow [11] and economic dispatch [12].
While nonlinear frequency constraints are proposed in [13]
by performing a number of dynamic simulations and in [14]
by analytical solutions of frequency dynamic evolution, both
of which are incorporated into deterministic UC models. In
[15], a set of analytical linear constraints are developed to
ensure the post-fault frequency evolution within the standards
and then incorporated into a SUC model. However, all of
these above FR constraints are formulated to meet the fixed
“N-1” security standard. The optimal scheduling of FR
against Under-Frequency Load Shedding (UFLS) has not yet
been investigated.
UFLS is traditionally used as the last resort to prevent the
system blackouts following loss of a generating plant. As the
increased penetration of wind generation, UFLS may become
a cost-effective alternative to the provision of FR during the
scheduling process. Due to dynamic constraints, the
provision of FR by conventional plants is inevitably
accompanied with the delivery of energy production. This
may lead to wind curtailment especially when high wind
output coincides with low demand. Moreover, with a given
level of forecasted wind generation, the realization of
extreme high wind production, hence low inertia, is
associated with very low probability, while requires large
amount of FR. Combined with the low probability of
generation outage, it may be more cost-effective to reduce the
provision of FR and face increased risk of load curtailment
under certain system conditions. Authors in [16] [17] [18]
have carried out researches in the optimal setting of UFLS to
minimize the amount or the cost of load curtailment, but all
with pre-fixed generation commitment decisions. Although
the authors in [18] claim the proposed formulation could be
applied in the UC problem, the required dynamic simulations
may lead to intractability in the large scale system.
Furthermore, it is challenging to explicitly model both
wind forecasting errors and generation outages within a
single scenario tree. Authors in [19] [20] generate separate
scenarios for all credible contingencies. This approach,
however, may end up with large amount of scenarios and
exaggerate the computational difficulties of SUC when
applied in large scale systems. The method proposed in [21]
[22] could actually be applied to incorporate the optimal
scheduling of FR into SUC model without resorting to the
scenario tree. The authors utilize capacity outage probability
tables (COPTs) to approximate Expected Energy Not Served
(EENS) and then incorporate it into system scheduling model.
To support the efficient operation of the future low carbon
system, this paper proposes a computationally-efficient MILP
formulation for a full SUC model that, for the first time,
simultaneously optimize energy production, OR, FR and
UFLS. The main contributions of this paper could be
summarised as:
1. This paper extends the stochastic scheduling concept to
include the optimal scheduling of FR. Value of lost load
(VOLL) is used as the single security measure so that
both OR and PR are optimized through balancing the cost
of provision and the benefit of reduced penalty of EENS.
2. This paper proposes an efficient method to calculate the
amount of UFLS. Different from traditional methods that
rely on the dynamic simulations, this paper analytically
calculates the amount of ULFS driven by the FR shortage,
which is then used to inform the optimal scheduling of FR.
3. Instead of explicitly modelling of each generation outage
through large amount of scenarios, this paper applies
COPTs to approximate the probability of generation
outages so that the uncertainties associated with both
wind forecasting errors and generation outages could be
modelled in a single scenario tree.
The rest of this paper is organized as follows: Section II
describes the methodology that is used to represent the
system uncertainties and construct the scenario tree. Section
III introduces the full stochastic scheduling model, while the
post-fault FR constraints with UFLS option are developed in
Section IV. Case studies are carried out and discussed in
Section V, and Section VI concludes the paper.
3
II. CONSTRUCTION OF SCENARIOS TREE
The first step to apply stochastic optimization is to build a
scenario tree (as shown in Fig.1) that appropriately describes
the uncertainties in the system. Therefore, this section
presents the models used to represent the uncertainties
associated with wind forecasting error and generation outages.
The net demand t hours ahead is defined as the demand plus
the capacity that is forced out between the current time and t
hours ahead, net of the available wind generation. The
cumulative distribution function (CDF) for the net demand is
calculated and used to construct the scenario tree by using
quantile-based approach. At the same time, COPT in each
node of the tree are developed and used to inform the optimal
scheduling of FR.
Scheduling of operating reserve
24 hours
30 mins
𝑦
𝑍(𝑘, 𝑖) = 𝜎𝑖 𝑌(𝑘, 𝑖)
(4)
𝑦
where the scaling factor 𝜎𝑖 can be calibrated to match a userdefined RMS error profile according to (6) and 𝑌(𝑘, 𝑖) is
generated by an AR(2) model with the same AR parameters
𝜑1𝑥 , 𝜑2𝑥 as in (2):
𝑌(𝑘, 𝑖) = 𝜑1𝑥 𝑌(𝑘, 𝑖 − 1) + 𝜑2𝑥 𝑌(𝑘, 𝑖 − 2) + 𝜖 𝑦 (𝑘, 𝑖) (5)
Considering equation (4) and (5), the normalised wind
forecast error follows normal distribution with zero mean and
standard deviation:
𝑠𝑖𝑧
=
𝑖−1
𝑦
𝑦
𝜎𝑖 √∑(𝜓𝑗 )2
𝑗=0
where
𝑦
Net Demand
Time ahead
Fig 1 A typical scenario tree in SUC with frequency response requirements
A. Modelling of wind forecasting error
The model proposed in [23] is applied to simulate the wind
forecasts and realizations. The normalised wind level 𝑋(𝑘) is
assumed to follow a Gaussian process with half-hourly
timestep as in (1), which is then transformed into a nonGaussian power output 𝑃𝑤 (𝑘) with a range from zero to the
installed wind capacity as in (2).
𝑋(𝑘) = 𝜑1𝑥 𝑋(𝑘 − 1) + 𝜑2𝑥 𝑋(𝑘 − 2) + 𝜎 𝑥 𝜖 𝑥 (𝑘),
𝜖 𝑥 (𝑘)~𝑁(0,1) 𝑖. 𝑖. 𝑑. (1)
𝑃𝑤 (𝑘) = 𝑊(𝑋(𝑘) + 𝜇(𝑘 𝑚𝑜𝑑 𝑁 𝑑 ))
(2)
𝑥
where 𝜑1 , 𝜑2𝑥 are constant auto-regression (AR) parameters,
𝜎 𝑥 is the standard variation of wind level, 𝑃𝑤 (𝑘) is the wind
power converted from wind level 𝑋(𝑘), 𝑁 𝑑 is the number of
timesteps in one day, 𝑊(∙) is a sigmoid-shaped
transformation function and 𝜇(𝑗) is used to represent a
diurnal variation. These parameters are calibrated to match
the distribution as well as the diurnal and seasonal variation
of GB historic data [24].
By Defining 𝑍(𝑘, 𝑖) as the forecasting error in the
normalised wind level, the forecasted wind generation can be
expressed as:
𝑃𝑤𝑓 (𝑘, 𝑖) = 𝑊 (𝑋(𝑘 + 𝑖) + 𝑍(𝑘, 𝑖) + 𝜇((𝑘 + 𝑖)𝑚𝑜𝑑 𝑁 𝑑 )) (3)
𝑍(𝑘, 𝑖) can be decomposed into a horizon-dependent
𝑦
scaling factor 𝜎𝑖 and a time series process 𝑌(𝑘, 𝑖):
𝑦
𝑦
𝜓𝑗 = 𝜑1 𝜓𝑗−1 + 𝜑2 𝜓𝑗−2
(7)
𝑤𝑓 (𝑘,
Given a forecasted wind generation 𝑃
𝑖), CDF of the
realized wind generation i timesteps ahead can be written as
𝐶 𝑤 (𝑥; 𝑘, 𝑖) = 𝛷 (
Scheduling of frequency response with UFLS option
(6)
𝑊 −1 (𝑥) − 𝑊 −1 (𝑃𝑤𝑓 (𝑘, 𝑖))
𝑠𝑖𝑧
) (8)
B. Modelling of generation outages
Generation outages are assumed to follow a Markov
process with forced outage probability 𝜆𝑔 ∆𝑡 for each online
unit and repair probability 𝜇𝑔 ∆𝑡 for each on-outage unit
during time interval ∆𝑡 [25]. If the generation commitment
status is known, COPT in each node can be calculated by
convolving the failure probability distributions for all units
that are running. This can be carried out efficiently in the
simulations when the thermal plants are clustered in a group
of identical units, so that COPT for each group can be written
as a binomial expansion. The COPT for failures in group g
can be expressed as a Probability Mass Function (PMF)
𝑢𝑝
𝑁𝑔
{(𝑉𝑔𝑗 , 𝑝𝑔𝑗 )}
𝑗=0
with
𝑉𝑔𝑗 = 𝑗𝑃𝑔𝑚𝑎𝑥
(9)
𝑢𝑝
𝑢𝑝
𝑁𝑔
𝑗
(𝑁 −𝑗)
𝑝𝑔𝑗 = (
) (𝜆𝑔 ∆𝑡) (1 − 𝜆𝑔 ∆𝑡) 𝑔
(10)
𝑗
Having calculated the COPT for each unit group, we can
convolve the PMFs to generate a combined COPT for the
whole thermal fleet as [22]:
𝑢𝑝
𝑁𝑔
{(𝑉𝑖 , 𝑝𝑖 )}𝑖 =⊗ {(𝑉𝑔𝑗 , 𝑝𝑔𝑗 )}
𝑗=0
(11)
where ⊗ denotes iterative convolution.
Similarly, the cumulative COPT, which captures the
probabilities of capacity outages that accumulate between the
current time and the instant before the time interval spanned
by node n, can be conservatively approximated by assuming
each unit in group g that is scheduled to run in each timestep
prior to node n to be a separate event. The cumulative nodal
COPT for the whole system can then be obtained by
convolving the binomial outage distributions for each unit
group. This cumulative COPT is denoted as
{(𝑉𝑗𝑐 , 𝑝𝑗𝑐 )}
(12)
𝑗
However, COPT can only be built after the commitment
status is known; while the commitment status is known only
4
once the SUC has been solved, which requires COPT.
Although different approaches have been applied in the
literature [21] [20] to solve this issue, a simple iterative
scheme as in [22] is adopted in this paper, with an initial
SUC assuming no outages, the second SUC based on the
COPT implied by the solution to the first SUC, and so on. In
the practical applications, there is no significant improvement
on the commitment decisions by running more than two
iterations.
C. Construction of Scenario Tree
According to (8) and (12), the CDF for the net
demand 𝐶(𝑥; 𝑛), which is the probability that the demand
plus outages net wind power is less than x, can be expressed:
𝐶(𝑥; 𝑛) = ∑ 𝑝𝑗𝑐 (𝑛) (1 − 𝐶 𝑤 (𝑉𝑗𝑐 (𝑛) + 𝐷(𝑛) − 𝑥; 𝜄(𝑛)))
complete SUC calculation with a 24-h horizon in half-hourly
time-step, and only the decisions in the root node are applied.
In the next time step, realizations of some uncertain variables
and updated forecasts become available and a new scenario
tree covering another 24-h time horizon is then built. UC and
ED decisions are adjusted accordingly with the inter-temporal
constraints maintained.
The objective is to minimize the expected operation cost
over all the nodes in the scenario tree, including both the
generation cost and the load curtailment cost:
∑ 𝜋(𝑛) (∆𝜏(𝑛) ∗ 𝑐 𝐿𝑆 ∗ (𝑃 𝐿𝑆 (𝑛) + ∑ 𝑝𝑗 (𝑛) ∗ 𝑃𝑗𝑈𝐹𝐿𝑆 (𝑛))
𝑛∈𝒩
𝑗∈𝑁𝑓
+ ∑ 𝐶𝑔 (𝑛))
(13)
A scenario tree is then built based on this distribution.
Quantile-based approach is applied to construct and weight
each node. For simplicity, scenario trees are constructed with
branching only at the root node, which has been shown to
provide similar scheduling decision as that from more
sophisticated scenario tree structures in [5]. For any userdefined quantile 𝑞(𝑛) at node 𝑛 , the level of net demand
𝑃𝑁𝐸𝑇 (𝑛) can be calculated by inversion of (13):
𝑃𝑁𝐸𝑇 (𝑛) = 𝐶 −1 (𝑞(𝑛); 𝑛)
(14)
which could be solved by applying numerical root-finding
algorithm. The corresponding probability is approximated
assuming linear interpolation of c(·) between the quantiles
(trapezium rule), and linear extrapolation beyond the first and
last ones. More details regarding the appropriate choices for
the scenario quantile levels and the process to construct the
scenarios tree are presented in [5].
The generation cost consists of (i) variable cost which is a
linear function of electricity production, (ii) no-load cost
which is a function of a number of synchronized units and (iii)
start-up cost. The load curtailment cost in each node includes
both load curtailment (𝑃𝐿𝑆 (𝑛)) driven by the shortage of OR
and EENS (∑𝑗∈𝑁𝑓 𝑝𝑗 (𝑛) ∗ 𝑃𝑗𝑈𝐹𝐿𝑆 (𝑛)) driven by the shortage
of FR. According to equation (11) , a COPT at each node can
𝑗
be written as a probability mass function {(∆𝑃𝐿 (𝑛), 𝑝𝑗 (𝑛))}.
Given an amount of available FR (𝑅 = ∑𝑔∈𝐺 𝑅𝑔 + ∑𝑠∈𝑆 𝑅𝑠 )
𝑗
and a capacity of generation outage ( ∆𝑃𝐿 ), the detailed
calculation of the amount of UFLS (𝑃𝑗𝑈𝐹𝐿𝑆 (𝑛)) is presented
in the next section.
The optimization subjects to
Power balance constraint (16):
∑ 𝑃𝑔 (𝑛) + ∑(𝑃𝑠𝑑 (𝑛) − 𝑃𝑠𝑐 (𝑛)) − 𝑃𝑊𝐶 (𝑛)
𝑔∈𝐺
III. STOCHASTIC SCHEDULING MODEL
A computational-efficient MILP formulation is developed
for the scheduling model and solved over a scenario tree. As
shown in Fig.1, the frequency constraints are also
incorporated in each node of the scenario tree to ensure the
minimum frequency during transit period above the predefined levels. In the case of insufficient FR, EENS driven by
UFLS is calculated and the corresponding cost is included in
the objective function. Large-scale power systems may
include hundreds of thermal plants, and to model them all
individually may cause significant computational burden.
Therefore, this paper clusters the thermal plants into groups
and uses a set of integer variables to track the operation status
of each group, instead of using a set of binary variables to
track the operation status of each individual plant. The
proposed formulation uses separate decision variables to
track start-ups and shut-downs so that any combinations of
commitment decisions are allowed in the same way as with
the individual plant based approach and the inter-temporal
constraints can be applied correctly. If the number of units in
each group is set to one, the integer variables become binary
and the plants are modelled individually in similar fashion to
traditional formulations such as [26]. The model is
implemented with rolling planning approach, performing a
(15)
𝑔𝜖𝐺
𝑗
𝑠∈𝑆
(16)
= 𝑃𝑁𝐸𝑇 (𝑛) − 𝑃𝐿𝑆 (𝑛)
Thermal Plants Operating Constraints, include minimum and
maximum generation (17), start-up time (18), minimum up
and down times (19)-(20), ramping rates (21)-(22), primary
reserve provision (23)-(24). The definitions of the sets used in
the constraints are presented in the appendix.
𝑚𝑠𝑔
𝑢𝑝
𝑢𝑝
𝑃𝑔
∗ 𝑁𝑔 (𝑛) ≤ 𝑃𝑔 (𝑛) ≤ 𝑃𝑔𝑚𝑎𝑥 ∗ 𝑁𝑔 (𝑛)
(17)
𝑠𝑔
𝑁𝑔 (𝑛) =
𝑁𝑔𝑠𝑡 (𝑎)
∑
(18)
𝑎∈𝐴𝑠𝑡
𝑔 (𝑛)
𝑢𝑝
𝑁𝑔𝑠𝑑 (𝑛) ≤ 𝑁𝑔 (𝑎(𝑛)) −
𝑠𝑔
𝑁𝑔 (𝑎)
∑
(19)
𝑎∈𝐴𝑚𝑢
𝑔 (𝑛)
𝑜𝑓𝑓
𝑁𝑔𝑠𝑡 (𝑛) ≤ 𝑁𝑔
(𝑎(𝑛)) −
∑
𝑁𝑔𝑠𝑑 (𝑎)
(20)
𝑎∈𝐴𝑚𝑜
𝑔 (𝑛)
𝑢𝑝
𝑃𝑔 (𝑛) − 𝑃𝑔 (𝑎(𝑛)) ≤ ∆𝜏(𝑎(𝑛))∆𝑃𝑔𝑟𝑢 𝑁𝑔 (𝑛)
(21)
𝑟𝑑 𝑢𝑝
(𝑛)
𝑃𝑔
− 𝑃𝑔 (𝑎(𝑛)) ≥ −∆𝜏(𝑎(𝑛))∆𝑃𝑔 𝑁𝑔 (𝑎(𝑛)) (22)
(23)
0 ≤ 𝑅𝑔 (𝑛) ≤ 𝑅𝑔𝑚𝑎𝑥
𝑢𝑝
𝑅𝑔 (𝑛) ≤ 𝑓𝑔𝐹 (𝑁𝑔 (𝑛)𝑃𝑔𝑚𝑎𝑥 − 𝑃𝑔 (𝑛))
(24)
Storage Unit Constraints, include minimum and maximum
stored energy (25), minimum and maximum charge/discharge
rate (26) – (27), the storage energy balance (28) and primary
reserve provision (29)-(30)
(25)
𝐸𝑠𝑚𝑖𝑛 ≤ 𝐸𝑠 (𝑛) ≤ 𝐸𝑠𝑚𝑎𝑥
5
0 ≤ 𝑃𝑠𝑐 (𝑛) ≤ (1 − 𝑁𝑠𝐺𝑒𝑛 (𝑛))𝑃𝑠𝑐𝑚𝑎𝑥
0 ≤ 𝑃𝑠𝑑 (𝑛) ≤ 𝑁𝑠𝐺𝑒𝑛 (𝑛)𝑃𝑠𝑑𝑚𝑎𝑥
(26)
(27)
𝑃𝑠𝑑 (𝑛)
) (28)
𝜂𝑑𝑠
(29)
0 ≤ 𝑅𝑠 (𝑛) ≤ 𝑅𝑠𝑚𝑎𝑥
(30)
𝑅𝑠 (𝑛) ≤ (𝑁𝑠𝐺𝑒𝑛 (𝑛)𝑃𝑠𝑑𝑚𝑎𝑥 − 𝑃𝑠𝑑 (𝑛) + 𝑃𝑠𝑐 (𝑛))
𝐸𝑠 (𝑛) = 𝐸𝑠 (𝑎(𝑛)) + ∆𝜏(𝑛) (𝜂𝑐𝑠 𝑃𝑠𝑐 (𝑛) −
IV. CALCULATION OF UNDER FREQUENCY LOAD SHEDDING
This section develops constraints to ensure minimum level
of the post-fault frequency above the pre-defined limits and
presents the method to analytically calculate the amount of
UFLS in the case of FR shortage. The amount of UFLS is
then added into the cost function proposed in section III. The
proposed constraints below correspond to a single node in the
scenario tree; hence the notation for the node n is suspended.
In a multi-machine power system, the generators with
different inertia time constants and governor droops have
different responses to a contingency. However, by assuming
that all generators swing synchronously at a common
frequency, an approximation to the system frequency
evolution can be obtained as an equivalent single-machine
swing equation:
𝜕∆𝑓(𝑡)
(31)
2𝐻 ∗
= ∆𝑅(𝑡) − ∆𝑃𝐿
𝜕𝑡
where H [MWs/Hz] is the inertia from thermal plants and
load and ∆𝑅 [MW] describes the additional power delivered
through FR after the generation loss ∆𝑃𝐿 [MW].
The system is assumed to be at nominal frequency (𝑓0 ) in
the normal operation state. After a sudden generation loss, the
inherent physical characteristic of the rotating machines is to
draw on the stored kinetic energy to restore the balance
between generation and load, referred as inertia response.
The level of system inertia can be calculated as:
𝑢𝑝
∑𝑔∈𝒢 𝐻𝑔 ∗ 𝑃𝑚𝑎𝑥
∗ 𝑁𝑔 + 𝐻𝑑 ∗ 𝑃𝐷
𝑔
𝐻=
(32)
𝑓0
where 𝐻𝑔 / 𝐻𝑑 is inertia time constant for generator group g
or load and 𝑃𝐷 is the load level.
At the same time, the scheduled FR is activated through
governor response. As shown in (33), this paper applies a
conservative assumption that FR is delivered by linearly
𝑅
increasing the active power [11] [15] with a fixed slope ( )
𝑇𝑑
until delivery time (𝑇𝑑 ) or frequency nadir achieved (𝑇𝑛 ) and
then keeps constant. Different delivery speeds of FR are not
modelled and the drawbacks are discussed in the conclusion.
𝑅
∗ 𝑡 𝑖𝑓 𝑚𝑖𝑛 ( 𝑇𝑑 , 𝑇𝑛 ) ≥ 𝑡 ≥ 0
(33)
∆𝑅(𝑡) = {𝑇𝑑
𝑅
𝑡 ≥ 𝑚𝑖𝑛 ( 𝑇𝑑 , 𝑇𝑛 )
The combined provision of inertia response and FR needs
to limit the frequency above a pre-specified level at
frequency nadir. Otherwise, UFLS will be triggered to
prevent the system from a wider blackout.
A. System Frequency Adequacy Constraints
By integrating (31) with the assumption that the delivery
of FR is described by (33), the evolution of frequency
deviation can be obtained as:
1 𝑅 2
∗ ∗ 𝑡 − ∆𝑃𝐿 ∗ 𝑡
2 𝑇𝑑
∆𝑓(𝑡) =
(34)
2∗𝐻
The time 𝑇𝑛 when the frequency reaches its nadir can be
𝜕|∆𝑓(𝑡)|
= 0:
𝑅
𝑇𝑛 = ∆𝑃𝐿 /( )
(35)
𝑇𝑑
The value of frequency deviation at nadir can be obtained
by substituting (35) into (34) and the maximum frequency
deviation |∆𝑓𝑛𝑎𝑑𝑖𝑟 | should not exceed the predefined
threshold ∆𝑓𝑚𝑎𝑥 :
∆𝑃𝐿 ∗ ∆𝑃𝐿 ∗ 𝑇𝑑
|∆𝑓𝑛𝑎𝑑𝑖𝑟 | =
≤ ∆𝑓𝑚𝑎𝑥
(36)
4∗𝐻∗𝑅
By rearranging (36), a bilinear constraint can be obtained:
∆𝑃𝐿 ∗ ∆𝑃𝐿 ∗ 𝑇𝑑
𝐻∗𝑅 ≥
(37)
4 ∗ ∆𝑓𝑚𝑎𝑥
calculated by setting
𝜕𝑡
Fig 2 Piecewise linearized constraints on frequency nadir
Given that the proposed SUC model is based on MILP
formulation, bilinear constraint (37) needs to be linearized. If
the generators are modelled individually, like the majority of
UC formulations, bilinear constraint (37) can be transferred
into an equivalent linear formulation by using the big-M
method as implemented in [15]. However, this paper models
the clusters of identical units to accelerate the computation.
Application of big-M method would lead to a large amount of
additional integer variables. As shown in Fig 2, constraint (37)
is convex over the interested range (𝐻 ≥ 0, 𝑅 ≥ 0) [27]. it
can be linearized using a K-block piecewise linear function
by using the constrained cost variable (CCV) technique [28].
𝑅 ≥ 𝑎𝑘 ∗ 𝐻 + 𝑏𝑘 , 𝑘 = 1,2, ⋯ , 𝐾
(38)
where
𝑦𝑘 − 𝑦𝑘+1
𝑎𝑘 =
𝑥𝑘 − 𝑥𝑘+1
𝑦𝑘 − 𝑦𝑘+1
𝑏𝑘 = −𝑥𝑘 ∗
+ 𝑦𝑘
𝑥𝑘 − 𝑥𝑘+1
𝐻−𝐻
𝑥𝑘 = 𝐻 + (𝑘 − 1)
𝐾
1 ∆𝑃𝐿 ∗ ∆𝑃𝐿 ∗ 𝑇𝑑
𝑦𝑘 =
∗
𝑥𝑘
4 ∗ ∆𝑓𝑚𝑎𝑥
The range of 𝐻 should be chosen based on the system
specification. For the system like GB size (60GW peak
demand), 𝐻 ∈ [𝐻 = 1, 𝐻 = 6] GWs/Hz is a reasonable
6
option. The number of piecewise blocks 𝐾 should be chosen
to balance the accuracy and the computational time. In this
analysis, 𝐾 = 5 is chosen, which is shown to fully capture
the original feasible regions.
B. Calculation of UFLS
In the case of insufficient FR, UFLS is deployed to
maintain the system frequency within the threshold ∆𝑓𝑚𝑎𝑥 .
As discussed in [12], multiple frequency thresholds may exist
to trigger different sets of under-frequency relays. There are
also other types of triggering mechanism, such as when the
frequency remains below a certain threshold for a specified
time period, or when the rate of change of frequency is above
certain threshold. For simplicity, the theoretically optimal
UFLS 𝑃𝑈𝐹𝐿𝑆 is calculated to approximate the UFLS, which is
assumed to be triggered when system frequency reaches the
low frequency limit. The time 𝑡̂, when frequency reaches the
limit, can be obtained by setting ∆𝑓(𝑡) = −∆𝑓𝑚𝑎𝑥 in (34):
𝑡̂ =
𝑇𝑑
𝑇𝑑
𝑅
∗ ∆𝑃𝐿 − √−4 ∗ 𝐻 ∗ ∗ ∆𝑓𝑚𝑎𝑥 + ∆𝑃𝐿 2
𝑅
𝑅
𝑇𝑑
𝑅
4 ∗ 𝐻 ∗ 𝑅 ∗ ∆𝑓𝑚𝑎𝑥
𝑡̂ = √−
+ ∆𝑃𝐿 2 (40)
𝑇𝑑
𝑇𝑑
By rearranging constraint (40), the following equation can
be obtained:
2
𝐻∗𝑅 ≥
(∆𝑃𝐿 2 − 𝑃𝑈𝐹𝐿𝑆 ) ∗ 𝑇𝑑
(41)
4 ∗ ∆𝑓𝑚𝑎𝑥
The constraint (41) suggests that the requirement on the
system inertia and FR decreases along with increased amount
of UFLS. UFLS could hence be used as an alternative to the
provision of FR during scheduling process.
Constraint (41) involves two non-linear elements (𝐻 ∗ 𝑅
2
and 𝑃𝑈𝐹𝐿𝑆 ), and therefore cannot be directly implemented in
the MILP formulation. However, the right side of constraint
2
(41) is a monotonically decreasing function of 𝑃𝑈𝐹𝐿𝑆 . At the
same time, with any fixed 𝑃𝑈𝐹𝐿𝑆 , constraint (41) shows
exactly the same formulation as (37) and could be linearized
by the same technique. Furthermore, UFLS is normally
scheduled in blocks [16] [18], which could be arranged into
discretized options as described in Table I.
TABLE I OPTIONS OF THE AMOUNT OF UNDER FREQUENCY LOAD SHEDDING
Option 1
Option 2
⋮
Option N
Amount
𝑃1𝑈𝐹𝐿𝑆
𝑃2𝑈𝐹𝐿𝑆
⋮
𝑃𝑁𝑈𝐹𝐿𝑆
Decision Variable
𝑚1𝑈𝐹𝐿𝑆
𝑚2𝑈𝐹𝐿𝑆
⋮
𝑈𝐹𝐿𝑆
𝑚𝑁
Therefore, with the additional binary variables ( 𝑚1𝑈𝐹𝐿𝑆 ,
and the assumption that the amounts of
UFLS are arranged in an increasing order from Option 1 to
Option N, a set of bilinear constraints (42) can be developed
to replace nonlinear constraint (41), which can then be
linearized by using the same technique as constraint (37).
𝑈𝐹𝐿𝑆
𝑚2𝑈𝐹𝐿𝑆 , ⋯ , 𝑚𝑁
)
∆𝑃𝐿 2 ∗ 𝑇𝑑
4 ∗ ∆𝑓𝑚𝑎𝑥
𝑈𝐹𝐿𝑆 )
𝐻 ∗ 𝑅 ≥ (1 − 𝑚2𝑈𝐹𝐿𝑆 ⋯ − 𝑚𝑁
∗
(∆𝑃𝐿 2 − 𝑃𝑈𝐹𝐿𝑆
) ∗ 𝑇𝑑
1
4 ∗ ∆𝑓𝑚𝑎𝑥
(42.0)
2
⋮
2
2
(∆𝑃
−
𝑃𝑈𝐹𝐿𝑆
𝐿
𝑁−1 ) ∗ 𝑇𝑑
𝑈𝐹𝐿𝑆 )
𝐻 ∗ 𝑅 ≥ (1 − 𝑚𝑁
∗
4 ∗ ∆𝑓𝑚𝑎𝑥
2
(42.1)
(42. (𝑁 − 1))
𝑈𝐹𝐿𝑆 2
(∆𝑃𝐿 − 𝑃𝑁 ) ∗ 𝑇𝑑
4 ∗ ∆𝑓𝑚𝑎𝑥
𝑈𝐹𝐿𝑆
≤1
{ 𝑚1𝑈𝐹𝐿𝑆 + 𝑚2𝑈𝐹𝐿𝑆 ⋯ + 𝑚𝑁
𝐻∗𝑅 ≥
(42. 𝑁)
(42. (𝑁 + 1))
𝑗
{(∆𝑃𝐿 , 𝑝𝑗 )}
Given any pair of
from COPTs, the amount of
UFLS (𝑃𝑗𝑈𝐹𝐿𝑆 ) required to ensure the maximum post-fault
frequency deviation within the pre-defined limit (∆𝑓𝑚𝑎𝑥 ) can
𝑗
be calculated as (43), while constraint (42) with ∆𝑃𝐿 = ∆𝑃𝐿 is
𝑈𝐹𝐿𝑆
satisfied. The probability ( 𝑝𝑗 ) and the amount ( 𝑃𝑗
) of
UFLS is then used in the objective function (15).
𝑈𝐹𝐿𝑆
𝑈𝐹𝐿𝑆
𝑈𝐹𝐿𝑆
𝑃𝑗𝑈𝐹𝐿𝑆 = 𝑚𝑗,1
∗ 𝑃1𝑈𝐹𝐿𝑆 + 𝑚𝑗,2
∗ 𝑃2𝑈𝐹𝐿𝑆 ⋯ + 𝑚𝑗,𝑛
∗ 𝑃𝑛𝑈𝐹𝐿𝑆 (43)
V. CASE STUDIES
(39)
The theoretically minimum amount of UFLS (𝑃𝑈𝐹𝐿𝑆 ) is
the imbalance between the load and the generation at time t̂:
𝑃𝑈𝐹𝐿𝑆 ≥ ∆𝑃𝐿 −
𝑈𝐹𝐿𝑆
𝐻 ∗ 𝑅 ≥ (1 − 𝑚1𝑈𝐹𝐿𝑆 ⋯ − 𝑚𝑁
)∗
This section presents the case studies to demonstrate the
effectiveness of the proposed model. The annual system
operation is simulated in a system, designed to represent a
possible configuration for GB 2030 scenario [5]. The
maximum demand and total capacity of conventional plants
are 60 GW and 65 GW, respectively. The installed capacity
of wind generation is 40 GW in the base case. The annualized
failure rate ( = 8760 ∗ 𝜆𝑔 ) and mean time to repair ( =
1/(24 ∗ 𝜇𝑔 ) for each generation type are shown in Table II.
A 5 GW storage plant with 40 GWh energy storage capacity
and 75% round efficiency is also included in the generation
mix. This storage plant can provide up to 1 GW of FR. Table
II summarizes the characteristics of conventional plants. The
inertia time constant from demand is assumed to be 1s. The
value of lost load (VOLL) is set to be 30,000 £/MWh. The
reference setting for delivery time ( 𝑇𝑑 = 10𝑠) is chosen
according to the GB practice [29]. The wind forecasting error
is assumed to be 10% in 4-hour ahead. UFLS option is
assumed to be 100 MW incremental upto 800 MW.
TABLE II
CHARACTERISTICS OF THERMAL PLANTS
Number of units
Rated Power (MW)
Min Stable Gen (MW)
No-load cost (£/h)
Marginal cost (£/MWh)
Startup cost (£)
Startup time (h)
Min up time (h)
Min down time (h)
Inertia Constant (s)
Max FR(MW)
FR Slope
Emission(kgCO2/MWh)
Outage rate (occ/year)
Mean time to repair(days)
Nuclear
6
1800
1800
0
10
n/a
n/a
n/a
n/a
7
0
0
0
1.8
30
Coal
50
500
250
1700
62
48000
4
4
2
5
50
1
925
18
3
CCGT
50
500
250
4500
47
40000
4
4
1
5
50
1
394
18
3
OCGT
50
100
20
3000
200
0
0
0
0
5
50
1
557
n/a
n/a
To further accelerate the computation, we have coarsened
the tree in the time dimension by specifying longer nodal
time intervals towards the end of the tree. The start times of
7
the nodes in each scenario were set to 0, 0.5, 1, 1.5, 2, 2.5, 3,
3.5,4, 5, 7, 10, 14, 18 and 24 hours ahead. In the case of
stochastic scheduling, to choose the appropriate number of
scenarios is always challenging. The balance between
accuracy and computational time needs to be achieved. The
work in [5] [30] have demonstrated that a small number of
properly selected scenarios could capture majority of the
benefit of the stochastic unit commitment while be solved
very efficiently. Therefore, following the suggestion from [5],
9 scenarios are applied in the cast study, covering net demand
quantiles of 0.05, 0.3, 0.6, 0.9, 0.97, 0.99, 0.9967, 0.9989 and
0.9996. The optimization was solved by using FICO Xpress
7.1, which was linked to a C++ simulation application via the
BCL interface [31]. The Mixed-Integer Programming (MIP)
gap is set to be 0.1%. The case studies were carried out on a
four-core Intel 3.46GHz Xeon processor with 12GB RAM.
We firstly explore the benefits of stochastic scheduling of
ancillary services, including both OR and FR. Similar to the
approach presented in [4], analysis of the performance of
different scheduling methods is based on the simulation of
the annual system operation:
1. Deterministic scheduling with “N-1” based FR (DS_1):
the scheduling is performed in a rolling basis, by using a
single scenario with a quantile of 0.96 [4], while FR is
scheduled to cover “N-1” security standard.
2. Deterministic scheduling with optimal FR (DS_2): this
scheduling method differs from DD_1 as the fixed “N-1”
security standard is removed.
3. Stochastic scheduling with “N-1” based FR (SS_1): the
traditional stochastic approach as in [5]. FR is
scheduled to cover “N-1” security standard.
4. Full stochastic scheduling (SS_2): this scheduling
method differs from SS_1 as the fixed “N-1” security
standard is removed.
The system performances with different methods are
summarized in Table III. Both stochastic models could
effectively reduce the operation costs and wind curtailments
compared to DS_1 case. In particular, SS_1 leads to about
250 M£ annual operation cost reduction and 2.3% wind
curtailment reduction, while the proposed method (SS_2) can
further reduce the annual operation cost by about 165 M£ and
wind curtailment by about 2.1%. Moreover, the stochastic
approaches reduce the carbon emission by 1.5 Mt and 2.6 Mt,
respectively. This also reduces the required investment on
low-carbon technologies to achieve the decarbonisation target.
The benefits of stochastic methods are driven by more
efficient utilization of energy storage, OCGT and UFLS. This
reduces the average spinning headroom from conventional
plants by 915 MW under SS_1 and by 2294 under SS_2. The
reduced spinning headroom not only increases the efficiency
of system operation but also leaves more room to integrate
wind generation. Moreover, the results also demonstrate the
benefits of optimal scheduling of FR when combined with
both deterministic scheduling (DS_2) and stochastic
scheduling (SS_2). In the case of stochastic scheduling,
optimal scheduling of FR leads to 488 MW less FR
scheduled in each time interval while in total 2179 MWh
more load shedding over the year. Therefore, although the
cost associated with load curtailment under SS_2 is about 65
M£ higher, the generation cost under SS_2 is 230 M£ lower,
leading to an overall saving of 165 M£ over SS_1.
TABLE III DETAILED RESULTS OF SYSTEM OPERATION
Total Cost (B£)
Generation Cost (B£)
Load Curtailment Cost (M£)
Curtailed Wind (% available wind)
CO2 Emission (Mt)
Annual Energy Not Serviced (MWh)
Average spinning headroom (MW/h)
Average FR Provision(MW/h)
COAL: Annual Production (TWh)
Average FR Provision (MW/h)
CCGT: Annual Production (TWh)
Average FR Provision (MW/h)
OCGT: Annual Production (TWh)
Average FR Provision (MW/h)
Storage: Average state of charge (%)
Average FR Provision (MW/h)
DS_1
11.08
11.08
0
6.25
72.9
0
4680
2309
14.97
194
150
1358
0.22
63.05
80
695
DS_2
10.92
10.89
21.86
3.73
72.1
728
3940
2108
14.86
188
148
1230
0.02
6.62
80
684
SS_1
10.83
10.83
0
3.91
71.4
0
3765
2326
13.51
161
149
1358
0.24
68.30
40
739
SS_2
10.67
10.60
65.38
1.79
70.3
2179
2386
1838
13.11
142
148
1009
0.04
6.62
49
680
To further illustrate the proposed full stochastic approach,
a detailed scheduling of FR for a reprehensive day is shown
in Fig 3. During periods with low demand and high wind
(from hour 0 to hour 7), the amount of scheduled FR is
reduced from over 2700 MW under “N-1” security criterion
(SS_1) to less than 1500 MW under optimized provision
(SS_2), although EENS in the later case is around 0.7 MW
for these time intervals. This is due to the fact that low
system inertia driven by low demand and high wind
condition requires large amount of FR. At the same time,
there is less frequency-responsive plants online and hence
high cost to provide FR. These two factors make it more costeffective to reduce the amount of FR provision and face
slightly higher risk of UFLS. On the other hand, for the
periods with high demand and low wind (from hour 8 to hour
23), there are enough frequency-responsive plants online to
cost-effectively provide required amount of FR and hence no
UFLS is scheduled.
Fig 3 Scheduling of FR for a representative day
The variability, uncertainty and low inertia capability of
wind generation are the key drivers for the implementation of
the advanced stochastic approaches. The impact of installed
capacity of wind generation on the operation costs saving
from stochastic approaches is shown in Fig.4. The savings of
stochastic approaches increase along with increased capacity
of installed wind generation. Moreover, when the capacity of
installed wind generation is moderate, there is no significant
8
economic benefit (difference between dotted and solid) from
the proposed full stochastic approach over traditional
stochastic approach. However, when the capacity of the
installed wind generation reaches 30 GW or above,
significant operation cost saving can be obtained by using the
proposed approach. This is due to the fact that with moderate
capacity of installed wind generation, increased OR
requirement driven by wind forecasting error is the main
system operation challenge. Optimal scheduling of FR shows
limited benefit over simple “N-1” security criterion. As the
increased capacity of wind generation, the reduction of
system inertia and frequency-responsive plants would lead to
a significant challenge in providing sufficient amount of FR
under “N-1” security criterion, and hence optimal scheduling
of FR against UFLS becomes beneficial. In the case of 60
GW installed capacity of wind generation, the operation cost
under optimal scheduling of both OR and FR (SS_2) is more
than 500 M£ lower than that under optimal scheduling of OR
only (SS_1).
Fig 4 Impact of installed capacity of wind generation on the benefits of
stochastic approaches.
The cost of EENS is directly linked with the setting of
VOLL, and therefore case studies are carried out to
investigate its impact on the system operation under full
stochastic scheduling approach (SS_2). As presented in Fig.5,
the overall system operation cost increases along with higher
VOLL. In particular, higher VOLL increases the cost of load
curtailment and therefore increased amount of FR is
scheduled to reduce the risk of UFLS after generation outages.
When VOLL increases from 3 k£/MWh to 300 k£/MWh, the
average amount of FR provision increases from 1384 MW/h
to 2212 MW/h while the annual amount of load curtailment
reduces from 3495 MWh to 286 MWh. The results also
suggest that after reaching 30 k£/MWh, further increase of
VOLL shows limited impact on the system operation.
Fig.5 Impact of the setting of VOLL on the system operation
The computational efficiency of SUC model is a key
element for its practical applications. The average run time of
the proposed model (SS_2) in the system with 156
conventional plants is less than 60s per time step, which is
quite tolerable for the application in the scheduling of largescale power systems over half-hourly time interval. There are
three main reasons for the efficient computational
performance. Firstly, thermal plants are clustered into groups.
The proposed model requires one set of integer variables for
each group of plants, instead of one set of binary variables for
each plant in the framework that models plants individually.
Secondly, we implement COPTs to model the uncertainty
associated with generation outages and apply the quantilebased scenario construction approach, both of which help
avoiding large amount of scenarios. Finally, instead of
modelling every time step in the scenarios tree, we coarsen
the tree in the time dimension by specifying longer nodal
time intervals towards the end of the tree.
To demonstrate the validity and the computational benefits
of the grouped unit representation, we also ran stochastic UC
over a week with each of the 156 thermal plants in Table II
represented individually. It took on average 930s per timestep. We compared the schedule with that obtained for the
same week simulated using the grouped representation,
where the average ran time was only 59s. The resulting
schedules were almost identical, with any differences
attributable to the non-zero MIP gap target. Furthermore, to
justify the selection of 9 scenarios, we compared it with the
results from 81 scenarios [5] over a week time. Although the
latter case reduce the operation cost by another 0.2%, the
computational time increases by more than 7 times.
VI. CONCLUSION
This paper proposes a computationally-efficient MILP
formulation for a full SUC model that simultaneously
optimizes energy production, OR, FR and UFLS. The amount
of UFLS is calculated without carrying our dynamic
simulations. The generation outages are represented by
COPTs so that the uncertainties associated with both wind
forecasting errors and generation outages could be modelled
in a single scenario tree. The case studies demonstrate the
effectiveness of proposed approach over deterministic as well
as traditional stochastic approaches. The benefits are
primarily driven by the optimal scheduling of both OR and
FR through dynamically balancing the cost of provision
against the benefit of reduced load curtailment under various
system conditions. Optimal scheduling of OR dominates the
benefit of stochastic approach when the capacity of installed
wind generation is moderate, while the optimal scheduling of
FR shows significant additional benefit when wind
generation reaches 30 GW or above. Moreover, higher VOLL
tends to increase the provision of FR to reduce the risk of
load curtailment, which leads to an increased overall system
operation cost.
The grouped modelling framework of thermal plants
significantly reduce the computation time without
compromising the accuracy. However, the computational
9
benefit would very much depend on to what extend the
thermal plants could be grouped. For the system with heavily
constrained transmission network and generators with
distinguished characteristics, this computational benefit may
be reduced. In fact, the proposed formulation for the optimal
scheduling of FR is generic and could also be combined with
other existing SUC models to enhance the benefits of
stochastic approaches.
There are some areas that could be improved in the future.
Firstly, transmission constraints are not incorporated in the
model. This can be done by simply adding DC load flow
equations. However, there would be multiple uncertainties
associated with wind/demand forecasting errors and
generation outages at different areas in the system, and hence
advanced scenario generation and construction techniques,
such as importance sampling, may need to be applied to
cover the relevant scenario without greatly increasing the
total number of scenarios. At the same time, decomposition
technique combined with parallel computing may be applied
to tackle computational burden. Furthermore, the developed
model assumes a fixed delivery time (10s) of FR for all the
generators. However, there are fast frequency response
resources e.g. battery storage, which are capable to provide
FR in less than 1s. In order to incentivize the faster delivery
of FR, the different delivery speeds should be explicitly
modelled in the scheduling process. Finally, wind turbines
are assumed not to provide any ancillary services in this
paper. However, previous studies [32] [33] has demonstrated
that a supplementary control loop could be incorporated into
the controller of wind plants to provide frequency response
similar to conventional plants. It is therefore important to
include the synthetic inertia and FR provision from wind
plants and investigate the impacts on the system operation.
APPENDIX
The definitions of the sets used in the inter-temporal
constraints for thermal units follows the work in [5]. For any
node n, A(n) is defined as the set that contains all the nodes
that are ancestors of node n in the same scenario.
If a generator in group 𝑔 starts generating at node 𝑛, then
it must have been started up at a node in the set
𝐴𝑔𝑠𝑡 (𝑛) = 𝐴(𝑛) ∩ {𝑛′ ∈ 𝒩 ∪ 𝑃: 𝜏(𝑎(𝑛)) − 𝑇𝑔𝑠𝑡 < 𝜏 (𝑛′ )
REFERENCE
[1] R. Jiang, J. Wang and Y. Guan, “Robust Unit Commitment With Wind
Power and Pumped Storage Hydro,” IEEE Trans. Power Syst., vol. 27,
no. 2, pp. 800 - 810, 2012.
[2] Y. Dvorkin, H. Pandžić, M. A. Ortega-Vazquez and D. S. Kirschen, “A
Hybrid Stochastic/Interval Approach to Transmission-Constrained Unit
Commitment,” IEEE Trans. Power Syst., vol. 30, no. 2, pp. 621 - 631,
2015.
[3] P.Meibom, R.Barth, B.Hasche, H.Brand, C.Weber and M.O'Malley,
“Stochastic Optimization Model to Study the Operational Impacts of
High Wind Penetrations in Ireland,” IEEE Trans. Power Syst., vol. 26,
no. 3, pp. 1367-1379, 2011.
[4] A.Tuohy, P. Meibom, E. Denny and M. O'Malley, “Unit Commitment
for System With Significant Wind Penetration,” IEEE Trans. Power
Syst., vol. 24, no. 2, pp. 592-601, 2009.
[5] A. Sturt and G. Strbac, “Efficient stochastic scheduling for simulation
of wind-integrated power systems,” IEEE Trans. Power Syst., vol. 27,
no. 3, pp. 323-334, 2012.
[6] J. Wang, J. Wang, C. Liu and J. P. Ruiz, “Stochastic unit commitment
with sub-hourly dispatch constraints,” Applied Energy, vol. 105, pp.
418 - 422, 2013.
[7] E. A. Bakirtzis, P. N. Biskas, D. P. Labridis and A. G. Bakirtzis,
“Multiple Time Resolution Unit Commitment for Short-Term
Operations Scheduling Under High Renewable Penetration,” IEEE
Trans. Power Syst., vol. 29, no. 1, pp. 149 - 159, 2014.
[8] E. Ela and M. O’Malley, “Studying the Variability and Uncertainty
Impacts of Variable Generation at Multiple Timescales,” IEEE Trans.
Power Syst., vol. 27, no. 3, pp. 1324 - 1333, 2012.
[9] National Grid frequency response working group, “Frequency Response
report,” 2013.
[10] J. O’Sullivan, A. Rogers, D. Flynn, P. Smith and M. O’Malley,
“Studying the Maximum Instantaneous Non-Synchronous Generation in
an Island System—Frequency Stability Challenges in Ireland,” IEEE
Trans. Power Syst., vol. 29, no. 6, pp. 2943 - 2951, 2014.
[11] H. Chavez, R. Baldick and S.Sharma, “Governor Rate-Constrained OPF
for Primary Frequency Control Adequacy,” IEEE trans. Power Syst.,
vol. 29, no. 3, pp. 1473-1480, 2014.
[12] Y. Lee and R. Baldick, “ A Frequency-Constrained Stochastic
Economic Dispatch Model,” IEEE Trans. Power Syst., vol. 28, no. 3,
pp. 2301-2312, 2013.
[13] R. Doherty, G. Lalor, and M. O’Malley, “Frequency control in
competitive electricity market dispatch,” IEEE Trans. Power Syst., vol.
20, no. 3, pp. 1588-1596, 2005.
[14] H.Ahmadi and H.Ghasemi, “Security-Constrained Unit Commitment
With Linearized System Frequency Limit Constraints,” IEEE Trans.
Power Syst., vol. 29, no. 4, pp. 1536 - 1545, July 2014.
[15] F. Teng, V. Trovato and G. Strbac, “Stochastic Scheduling with Inertiadependent Fast Frequency Response Requirements,” IEEE Trans.
Power Syst., vol. 31, no. 2, pp. 1557 - 1566, 2016.
(𝐴. 1)
[16] F. Ceja-Gomez, S. S. Qadri and F. D. Galiana, “Under-Frequency Load
Shedding Via Integer Programming,” IEEE Trans. on Power Syst., vol.
27, no. 3, pp. 1387 - 1394, 2012.
If a generator in group g is shut down at node n, it cannot
have started generating at any node in the set
[17] U. Rudez and R. Mihalic, “Monitoring the First Frequency Derivative
to Improve Adaptive Underfrequency Load-Shedding Schemes,” IEEE
Trans. on Power Syst., vol. 26, no. 2, pp. 839 - 846, 2011.
≤ 𝜏(𝑛) −
𝑇𝑔𝑠𝑡 )
𝐴𝑔𝑚𝑢 (𝑛) = 𝐴(𝑛) ∩ {𝑛′ ∈ 𝒩 ∪ 𝑃: 𝜏(𝑛) − 𝑇𝑔𝑚𝑢 < 𝜏 (𝑛′ )
≤ 𝜏(𝑛)}
(𝐴. 2)
If a generator in group g is started up at node n, it cannot
have been shut down at any node in the set
𝐴𝑔𝑚𝑜 (𝑛) = 𝐴(𝑛) ∩ {𝑛′ ∈ 𝒩 ∪ 𝑃: 𝜏(𝑛) − 𝑇𝑔𝑚𝑜 < 𝜏 (𝑛′ )
≤ 𝜏(𝑛)}
(𝐴. 3)
[18] R. Bhana and T. J. Overbye, “The Commitment of Interruptible Load to
Ensure Adequate System Primary Frequency Response,” IEEE Trans.
Power Syst., vol. 31, no. 3, pp. 2055 - 2063, 2016.
[19] F. Bouffard, F. Galiana and A. Conejo, “Market clearing with stochastic
security— Part I: Formulation,” IEEE Transactions on Power Systems,
vol. 20, p. 1818–1826, 2005.
[20] F. Galiana, F. Bouffard, J. Arroyo and J. Restrepo, “Scheduling and
pricing of coupled energy and primary, secondary and tertiaty
reserves,” Proc. IEEE, vol. 93, no. 11, pp. 1970 - 1983, 2005.
[21] M. Ortega-Vazquez, D. Kirschen and D. Pudjianto, “Optimising the
scheduling of spinning reserve considering the cost of interruptions,”
IEE Proceedings - Generation, Transmission and Distribution, vol.
10
153, no. 5, p. 570 – 575, 2006.
[22] A. Strurt and G. Strbac, “Value of stochastic reserve policies in lowcarbon power systems,” Proceedings of the Institution of Mechanical
Engineers, Part O: Journal of Risk and Reliability, vol. 226, pp. 51 64, 2012.
[23] A. Sturt and G. Strbac, “Time-series modelling of power output for
large-scale wind fleets,” Wind Energy, vol. 14, no. 8, pp. 953-966,
2011.
[24] A. Sturt and G. Strbac, “A time series model for the aggregate Great
Britain wind output circa 2030,” IET Renewable Power Generation,
2013.
[25] R. Billiton and R. Allan, Reliability evaluation of power system,
Plenum, Ed., New York, 1996.
[26] M. Carrión and J. Arroyo, “A computationally efficient mixed-integer
linear formulation for the thermal unit commitment problem,” IEEE
Trans. Power Syst, vol. 21, no. 3, p. 1371–1378, 2006.
[27] B. Stephen and L. Vandenberghe, Convex optimization, Cambridge:
Cambridge university press, 2004.
[28] N. Zhang, C. Kang, Q. Xia, Y. Ding, Y. Huang, R. Sun, J. Huang and J.
Bai, “A Convex Model of Risk-Based Unit Commitment for DayAhead Market Clearing Considering Wind Power Uncertainty,” IEEE
Trans on Power Syst., vol. 30, no. 2, pp. 1582 - 1592, 2015.
[29] National Grid, “Security and Quality of Supply Standards,” [Online].
Available: http://www2.nationalgrid.com/UK/Industryinformation/Electricity-codes/System-Security-and-Quality-of-SupplyStandards/.
[30] A. Papavasiliou, S. S. Oren and R. P. O'Neill, “Reserve Requirements
for Wind Power Integration: A Scenario-Based Stochastic
Programming Framework,” IEEE Trans. Power Syst., vol. 26, no. 4, pp.
2197-2206, 2011.
[31] “FICO Xpress optimization suite,” [Online]. Available:
http://optimization.fico.com.
[32] W. Lei and D. Infield, “Towards an assessment of power system
frequency support from wind plant—modeling aggregate inertial
response,” IEEE Trans. Power Syst., vol. 28, pp. 2283-2291, 2013.
[33] N. R. Ullah, T. Thiringer and D. Karlsson, “Temporary Primary
Frequency Control Support by Variable Speed Wind Turbines—
Potential and Applications,” IEEE Trans. Power Syst., vol. 23, no. 2,
pp. 601 - 612, 2008.
Fei Teng (M’15) is a Research Associate at Imperial College London. He
received PhD degree from Imperial College London in 2015. His research
interests include power system operation, integration of renewable energy.
Goran Strbac (M’95) is a Professor of Electrical Energy Systems with
Imperial College London. His research interests are in modelling and
optimization of electricity system operation and investment, economic and
pricing, and integration of new forms of generation and demand
technologies.