Journal of Power and Energy Engineering, 2015, 3, 206-214
Published Online April 2015 in SciRes. http://www.scirp.org/journal/jpee
How to cite this paper: Dai, H., Zhang, N. and Su, W.C. (2015) A Literature Review of Stochastic Programming and Unit
Commitment. Journal of Power and Energy Engineering, 3, 206-214. http://dx.doi.org/10. 4236/jpee. 2015.34029
A Literature Review of Stochastic
Programming and Unit Commitment
Hang Dai, Ni Zhang, Wencong Su
Department of Electrical and Computer Engineering, University of Michigan, Dearborn, USA
Email: email@example.com u, firstname.lastname@example.org, email@example.com
Received January 2015
The study of unit commitment (UC) aims to find reasonable schedules for generators to optimize
power systems’ operation. Many papers have been published th a t s olve UC through different me-
thods. Articles that systematically summarize UC problems’ progress in order to update research-
ers interested in this field are needed. Because of its promising performance, stochastic pro-
gramming (SP) has become increasingly researched. Most papers, however, present SP’s UC solv-
ing approaches differently, which masks their relationships and makes it hard for new research-
ers to quickly obtain a general idea. Therefore, this paper tries to give a structured bibliographic
survey of SP’s applications in UC problems.
Unit Commitment, Stochastic Programming, Review
Unit commitment (UC) refers to the task of finding an optimal schedule and a production level for power systems’
each generating unit over a given period of time while satisfying device and operating constraints . Motivated by
the immense benefit an optimal schedule can provide, scholars have delved into solving the UC problem over the
past half century .
In existing literatures, UC is defined both narrowly and broadly. Narrowly speaking, UC only serves to pinpoint
the commitment status (on/off) of generators for a defined period. Then, economic dispatch (ED) is performed to
specify the production level every few minutes  . Broadly speaking, UC entails determining both the commit-
ment statue as well as the production level  . In this paper, the broad definition is examined. The classifications
of UC can be different. With respect to security, UC can be classified into two categories: traditional UC where net-
work constraints are ignored and security-constrained UC where network constraints like line outage and transmis-
sion line capacity are considered . From the market operation’s perspective, UC can be divided by either schedul-
ing in a vertically integrated environment where minimizing cost is the objective or in a deregulated environment
where maximizing benefit is the goal . With regard to the treatment of future events, UC can be categorized into
deterministic and stochastic UC.
Two types of uncertainties exist in power systems’ operation: departures from forecasts and unreliable equipment
H. Dai et al.
. Traditionally, forecast errors mainly result from load variations. With an increased penetration of renewable
energy, the intermittency and volatility of renewable generation lead to noticeable generation forecast errors. To
achieve minimum cost while satisfying power balance, power systems first determine power plants’ commitment
statuses and output capacities based on forecasts and constraints. Then, re-dispatch is performed in real-time to ad-
just the difference between the actual demand and scheduled output . UC is thus a multi-stage decision process
similar to SP’s solving procedure. It is therefore natural, as well as reasonable, to implement SP to solve UC prob-
The first trial of applying SP to solve UC was done in 1977 . However, due to the lack of computational capa-
bility, the results were unsatisfying. In contrast, recent increases of computational ability have caused investigations
in this field to boom. Numerous papers have formed and modified SP-based methods to solve UC under different
considerations. Regardless of which methods are used, the developing process remains relatively vague. Therefore,
part of this paper’s contribution is to give a systematic review of SP-based UC formulations.
The rest of the paper is organized as follows. Section 2 presents several existing synopsis articles on UC problems
over the past years. Section 3 outlines several commonly used SP-based UC formulations. Section 4 discusses some
commonly encountered issues in formulating and solving UC by SP. Section 5 concludes the paper and proposes
some future work.
2. Synopsis Articles of Unit Commitment
The UC problem dates back to the 1940s . Given the astronomical volume of papers in this field, several re-
view articles have been presented. One representative review of UC in traditional power systems is given by
. The author reviews more than 150 published articles concerning UC in the past 35 years. Several mathe-
matical methods for solving UC problems are outlined. It also points out UC’s different considerations in both
monopolized and deregulated markets. In addition, it urges that a hybrid model combining different methods’
advantages should be implemented. Similar review articles can be found in -.
With an increased penetration of renewable energy and the implementation of energy storage devices, power
systems’ operation strategies are significantly modified. Some articles have given reviews concerning UC in
such novel power systems infrastructure  .
However, most of these review papers are either a general exposition of UC’s mathematical solution methods
or UC’s applications in some specific areas. There are not any papers that give a general survey and structured
explanation of stochastic programming’s applications in UC. Given the situation,  presents a literature re-
view of solving UC by stochastic optimization methods. Stochastic programming, robust optimization and sto-
chastic dynamic programming are all outlined in the paper. This paper gives a particular r e v i ew of stochastic
programming’s applications in UC. In addition, this paper generalizes several commonly encountered issues in
formulating SP-based UC problems.
3. Stochastic Programming’s Application in Unit Commitment
Power systems throughout the world have either a vertically integrated structure or a deregulated structure. In a
vertically integrated environment, customers of generation companies (GENCOs) are set and guaranteed .
The goal of UC for such power systems is minimizing costs associated with scheduling, load loss, etc. To en-
courage competition and improve power supply quality, the electricity market in several countries have trans-
formed into deregulated market where GENCOs bid to sell their generation and independent system operators
(ISOs) establish the day-ahead and real-time market to guarantee the reliable operation of power systems .
In such a scheme, UC’s goal is to maximize profit or social welfare .
Uncertainty in SP is almost always represented by scenarios, which are a finite number of possible realiza-
tions of the uncertain quantity. When scenario-based SP is implemented to solve the UC problem, power sys-
tems’ operating constraints are enforced for all scenarios and the objective function is normally the expected
cost over these representative scenarios. Since the scenario-based modeling method assigns probabilities of oc-
currence to each scenario, it is biased more towards the most likely forecasted conditions and therefore justifies
its reasonability . This section discusses SP’s applications in UC for vertically integrated structure in sub-
sections 3.1 to 3.4 an d for deregulated structure in subsection 3.5. As can be seen later, UC formulations for
both cases are largely the same with only minor alterations.
H. Dai et al.
3.1. Basic Two-Stage Stochastic Programming Formulation for Unit Commitment
“Basic” in this section is used to differentiate these UC formulations from those that have security or risk con-
siderations. These basic UC formulations serve as the basis for SP’s implementation in UC problems.
Two-stage SP is a commonly used approach for capturing uncertainty, and its general form is shown below
, min,.. , 0cxEQxs tAxbx
()( )( )( )
,min, 0Q xqyWyhTxy
denote the first-stage and second-stage decision variables respectively. The
second-stage is a random vector and
is used to return the cost associated with this random vec-
tor’s consequences to the objective function.
According to , power systems’ short-term operation has two stages. In the first stage, units are selected to
meet the expected load during each hour based on generators’ operating costs and constraints. In the second
stage, power outputs of committed units are decided to meet the actual lo ad demand. The similarity between
power systems’ real operation process and the two-stage SP’s formulation naturally leads us to employ the two-
stage SP for UC. In practice, random variables in the second stage are represented by various scenarios. Refer-
ence  uses scenarios to model power generation and load demand, while  models the wind power output.
3.2. Basic Multi-Stage Stochastic Programming Formulation for Unit Commitment
Uncertainty in each scenario of the two-stage SP is in some sense treated only once. Contingencies, however,
are very likely to evolve in power systems’ operational process. Therefore, some papers propose to use multi-
stage models where unfolding uncertainties’ dynamics over time are captured and decisions are adjusted dy-
namically. Early trials of the multistage SP can be found in  -.
Multi-stage SP employs scenario trees to model uncertainty. The number of branches in each node is deter-
mined by real circumstances or the assumption of the number of instances at each time period. A path from the
original (root) node to one end (leaf) node is equivalent to a scenario. The scenario tree can model demand and
unit failure uncertainties, as in  , and fuel and electricity uncertainties , as in [26 ].
3.3. Security Constrained Unit Commitment
In order to simplify computations, early SP-based UC formulations ignore some constraints like network con-
gestion. Actual systems’ operation, however, must consider additional factors such as emission, fuel and trans-
mission constraints. With the improvement of computational ability, formulations including these additional
constraints have gradually evolved since the 1980s and such UC formulations are called security constraint UC
(SCUC)  . “Security” mainly refers to transmission network’s viability, and it is the essential factor of
differentiating SCUC from traditional UC . However, the definition criterion is sometimes not taken rigo-
rously. In [30 ]’s SCUC model, only transmission constraint is added to the UC problem. However, in ’s
SCUC model, fuel, emission and energy constraints are all considered.
3.4. Chance Constrained Two-Stage Stochastic Programming for Unit Commitment
The three models discussed above are used to model random variables like renewable generation, uncertain load
and component outages with constraints enforced in all scenarios. However, constraints in these models are
strict; they are required to be met in each stage of each scenario. Such stringent restrictions may lead to solutions
that are too conservative and therefor e result in undesirable consequences such as large amounts of wind spil-
lage [20 ]. To handle this problem, some papers recourse to chance constraint programming, which was first pro-
posed in . In chance constraint programming, solutions are permitted to violate constraints to some extent
within a certain confidence level and the optimal solution is achieved in a probabilistic sense. A common form
of chance constraint is shown as follows:
H. Dai et al.
is the constraint restricting a certain variable. The formulation represents the probability that
is the confidence level. In contrast, constraints in previous UC formulations are similar
to the following:
, ,1, 2,,
which demands the constraint be satisfied categorically.
One representative paper of applying chance constrained programming to solve the UC problem is ,
where the requirement that generation strictly meets the demand for each hour is replaced by that the condition
is satisfied at a predetermined probability level.
However, chance constraint itself cannot model uncertainties. In many papers employing chance constraints,
uncertain variables are usu ally captured by other methods instead of SP. For example,  uses a correlation
matrix to model uncertain loads and the chance constraint is only used to “relax” the power balance requirement.
A similar approach is observed in  where probabilistic variables are modeled by a Markov process transition
matrix. Since scenario-based SP is suitable for capturing uncertainty, it may be favorable if we combine SP with
chance constraint; i.e., employ SP’s scenarios to model randomness and use chance constraint programming to
represent constraints. Such combined models are first presented in , where wind power uncertainty is cap-
tured by a number of scenarios and wind spillage is restricted by a certain probability from the chance constraint.
Such models are labeled as chance-constraint two-stage SP. Additionally, the same authors propose a refined
CCTS model in  to define the probability that certain amounts of wind power bidding into the market can be
3.4. Stochastic Programming Formulations for Unit Commitment in Deregulated
Power industries throughout the world have experienced a significant transformation since the 1980s to cater for
the needs of higher power production and delivery efficiency. The gradual privatization of generation, trans-
mission, and redistribution have led power systems change into a deregulated market . In such an electricity
market, interactions between GENCOs and ISOs are vital for the performance of the system. The UC schedules
have an indirect influence on electricity price and a direct influence on cost, and the goal of UC schedules
changes to maximize profit or social welfare.
For deregulated markets, UC schedules can be classified as pool-schedule and self-schedule . In pool-
schedule environments, ISOs make UC decisions based on both GENCOs’ biddings and other considerations
similarly employed in vertically integrated markets. In self-schedule environments, GENCOs make their own
UC decisions before submitting bids to ISOs . For ISO, its goal is to maximize the whole system’s social
welfare, which is defined as the sum of producer surplus and consumer surplus. For GENCO, its objective is to
maximize its own profit. In such an environment, GENCOs are no longer bound to serve the given demand in
the open electricity market . As pointed out in , redefining the UC problem for deregulated environment
involves three alterations compared to the formulation in vertically integrated environment: 1) changing the de-
mand constraints from an equality to inequality; 2) changing the objective function from cost minimization to
profit maximization; 3) changing the reserve power and transmission losses to per contract form. Some more
detailed considerations can be found in . Normally, GENCO’s bidding depends on its interpretation of three
uncertainties: load, generation and market price. In , a CCTS-based wind power generation company’s bid-
ding strategy is proposed to maximize the company’s profit. Considering the fact that merely maximizing payoff
may expose GENCOs to undesirable risks in the market,  defines a metric called expected downside risk as
an explicit constraint in a multi-stage SP formulation to obtain GENCOs’ bidding strategy. From the above ex-
planations, it can be seen that SP’s applications in UC for deregulated structure are largely the same as those in
vertically integrated structure. Table 1 pr ovides an overview of literatures involving SP-based formulations for
4. Specific Issues
4.1. Risk Considerations
Similar to the chance constraint discussed in Section 3.4, some other measures are utilized to avoid costly solu-
H. Dai et al.
Table 1. Summary of SP-based formulations in UC.
Electricity Market Types SP-based Formulations Literature References
Vertically Integrated Market
Basic two-stage SP UC ]24[ ]1[
Basic multi-stage SP UC ]28[-]25[ ]5[
Security-constrained UC (SCUC) ]58[ ]52[ ]32[ ]30[ ]29[ ]21[ ]6[
Chance constrained two-stage UC (CCTS) ]57[ ]37[ ]36[
Deregulated Market —————————— ]58[ ]41[ ]39[ ]37[ ]7[
tions while keeping the risk level within an acceptable domain. These include expected load not served (ELNS),
lost of load probability (LOLP), value at risk (VaR) and conditional value at risk (CVaR). They form what is
called risk-averse UC decisions. A detailed mathe matical exposition can be found in .
LOLP is the most widely used system-wide risk measure metric. In fact, the chance constraint is the probabil-
istic constraint restricting LOLP, and it is equivalent to bound a θ-level VaR of the loss of load . However,
VaR can be hard to compute due to its lack of subadditivity. Therefore, a more coherent and conservative risk
measure based on continuous variables called CVaR is proposed. More detailed discussions between VaR and
CVaR can be found in  .
ELNS is evaluated by taking the expectation of total net load minus the total dispatch. In  , it is ar-
gued that ELNS is better than LOLP since it accounts for both the probability of outages as well as the corres-
ponding average load lost. More detailed discussions between ELNS and LOLP can be found in  . Table
2 lists a quick overview of risk-averse UC.
4.2. Explicit vs. Implicit Reserve Setting
Operating reserves are important for power systems to respond to contingencies like load peaks, generator fail-
ures, scheduled outages, regulation and local area protection . It includes spinn ing reserves and non-spinning
reserves. Here, only spinning reserves are considered.
Normally, there are two ways of specifying a system’s reserve. One is through deterministic criterion and set
reserve equals some fraction of peak load  or the capacity of the largest online generator . In this way,
reserve is set explicitly. Another is using probabilistic criterion. Discussion of different probabilistic reserve set-
ting methods can be found in . As noted in [51 ], SP is regarded as a representative and most commonly used
probabilistic reserve setting method. In probabilistic methods, reserve is set implicitly. The rationale that SP-
based methods can set reserve is that uncertainty is explicitly considered in the SP and the system’s reserve
needs are already taken into account by different scenarios. Therefore, reserve is implicitly set. However, limited
number of scenarios in SP may miss out some contingencies. Given this possibility,  combines deterministic
reserve with the reserve set by SP and verifies that the combined model can give more robust solutions in the
case of generation and load uncertainties. Reference  follows this direction and also shows the satisfying
performance of the combined reserve setting model in the case of wind power forecasting errors. Table 3 lists a
quick overview of literatures with reserve considerations in solving UC problems.
4.3. Perfect, Deterministic and Stochastic Cases
Perfect case is the one where uncertain variable can be precisely forecasted and is the same as the actual situa-
tion. It is also called realized case . In SP’s formulation, this means there is only one scenario exists in each
stage. The deterministic case has only one scenario as well. But variables’ values in deterministic case equal the
expectation value of uncertainties in different scenarios and are therefore different from what will actually occur.
Stochastic case is the situation whe re uncertainties are captured by the scenarios discussed previously.
5. Conclusion and Future Work
While numerous literatures on UC are focusing on improving mathematical computation methods to quickly
solve the formulated objective functions, it has been pointed out that UC problems can be inherently simplified
by improving the modeling quality. Reference  argues that in countries like P.R. China where vertically
H. Dai et al.
Table 2. Summary of risk-averse UC.
Risk Considerations Literature References
VaR and CVaR ]44[-]42[
LOLP and ELNS [45-
Table 3. Reserve considerations in UC.
Reserve Considerations secnerefeR erutaretiL
Deterministic reserve models  
Probabilistic reserve models ]55[ ]52[ ]47[-]45[
Combined reserve models  
integrated environment dominates, the maximum load can be as high as 70% of the total installed generation
capacity. This means many generators are kept generating continuously and therefore they can be expelled from
consideration when solving UC. Such reduction of generators’ amounts will significantly alleviate UC prob-
lem’s computational burden. Also,  suggests that relaxing fast-start unit as continuous variables and model-
ing the aggregate generation outage as load increment can speed calculation noticeably.
Until now, several parameters in UC formulations are unsatisfactorily assumed. The spinning reserve re-
quirement given by UC is sensitive to the value of lost load (VOLL) in . On the one hand, this indicates the
lack of robustness of its UC formulation. On the other, this reveals the importance and necessity of a reasonable
estimation of VOLL. Similarly, the change of wind spillage cost is shown to have a significant impact on re-
serve, generation and demand scheduling in . Some of the paper’s unreasonable scheduling may result from
the arbitrary assignment of wind spillage cost. As noted in , wind spillage cost represents the cost of oppor-
tunity to produce using the spilled wind energy. One possible future work is linking the wind spillage cost with
wind farm’s construction and maintenance fees as well as policies regarding wind power utilization. To compare
the performance of different UC formulations, several metrics need to be employed. The most commonly used
metric is expected total cost because it includes the impacts of both service (cost of generation) and reliability
(cost of load shedding). However, expected total cost may be of less interest for other parties like GENCOs. In
addition, it fails to take other vital issues like UC models’ convergence and computation time into account. As
shown in , the SCUC model in the day-ahead market of Texas’s ERCOT electricity market once failed to
give any feasible solution for almost 12 hours. This indicates the importance of UC models’ convergence. To
evaluate the conflicting impact of WPG, a new metric considering WPG’s dual effects of fuel reduction and re-
serve cost increase is proposed in . Similarly, [ 57] defines a fuzzy membership function as the system’s se-
curity level to resolve WPG’s conflicting impact. Nevertheless, these metrics are only evaluated individually
and limited work has been done to further test their performance. More reasonable matrices for evaluating dif-
ferent UC formulations are therefore demanded.
This work is supported by the new faculty start-up fund at University of Michigan-Dearborn and the China
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