IEEE Power & Energy Magazine - January/February 2017 - 50
dimensionality associated with joint decision-making across
different systems at multiple spatiotemporal scales.
Nonlinearity leads to nonconvex problems (i.e., with multiple locally optimal solutions and a globally optimal solution
difficult to find and assess simultaneously) and, oftentimes,
nondeterministic polynomial-time-hard (NP-hard) programs that may be computationally prohibitive to solve even
for local optimization. Nonconvexity stems from the underlying physics governing water, power, and gas flows as well
as heat-transfer models describing the systems' couplings
and energy conversion stages. Further nonconvexities may
arise from the optimization approach or structure, such as
in the original energy-hub formulation that, by aggregating
elements that contain decision variables in a synthetic inputoutput representation, generates a nonlinear optimization
problem structure, where decision variables may serve to
multiply one another.
along the pipes, as well as the hydraulic characteristics of
variable-speed pumps. Computationally intensive problems
in gas (transmission) networks pertain to finding solutions
for highly nonlinear gas-flow equations in both steady-state
and transient (where intertemporal couplings add to the nonlinearity complexity) forms.
Nonconvexity from Flow Physics
Nonconvexity Due to Coupling Factors
The power-balance constraints in power systems are nonconvex due to nonlinear ac power-flow equations; additional sources of nonconvexity include constraints on the
minimum voltage service levels and on branch thermal
limits applied to power flows. In pump-scheduling problems and water-flow problems, nonconvex constraints naturally emerge from the relevant flow mathematical models
to capture the head losses or pressure losses due to friction
For modeling simplicity, the coupling factors among energy
carriers are often assumed to be constant. This is the case for
the fuel-to-power and fuel-to-heat conversion efficiencies of
CHP units, for example. However, a number of coupling factors are, in fact, nonlinear; examples include the efficiency
and power consumed by a variable-speed water pump, which
are nonlinear functions of the pump's frequency. Further, the
operational region of some types of co-generation units or
absorption and compression chillers may be nonconvex.
Discrete Control Decisions
The binary and discrete decision variables required to consider on/off decisions and discrete power consumption levels
(relevant to appliances, HVAC units, water heaters, EHPs,
and water pumps, to mention just a few) exacerbate the problem complexity. The dimension of the search space increases
exponentially with the number of binary variables (especially
if binary decisions are required over a prolonged optimization horizon), thus rendering solutions for the underlying
optimization problems computationally prohibitive.
Min f (Water) + g (Power) + h (Gas)
s.t Water Constraints
Power Constraints
Gas Constraints
+ Coupling Constraints
Tradeoff Between
Complexity and Flexibility
Setpoints
Adjacent
Systems
Industrial
Area
Solar
H1
Residential
Area
H2
Pumped Hydro
(Storage)
Commercial
Area
Electricity
H3
Heating
H4
Natural Gas
Wind
Water
figure 7. A global control problem optimizes a variety of performance objectives
and economic indicators, while acknowledging intrinsic interdependencies among
systems and operational constraints. Flexibility is naturally enabled by such a problem formulation, since the coupling constraints capture shifts in supply and demand
across energy vectors and networks.
50
ieee power & energy magazine
Energy hubs and multigeneration
models based on so-called dispatch
factors, which are used to specify
how energy is split among conversion units and outputs, may introduce other sources of nonconvexity
via bilinear or trilinear terms appearing in equality constraints.
Nonlinearity could be, in this case,
bypassed either by introducing auxiliary variables or by simplify the
hub model. It is, thus, apparent that
tradeoffs between the complexity
of any modeling approach to the
optimization problems and achievable flexibility must be taken into
account in the system's design and
operational processes.
Opportunities
Off-the-shelf solvers for mixedinteger nonlinear programs could,
in principle, be used to find solutions
january/february 2017
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