The effects of the deregulation of the electricity market 1996 in Sweden on short-term hydropower regulation are unknown. This report investigates patterns in subdaily regulation in the Ångerman River Basin during the period 1993-2011. Differences in subdaily flows and zero flow events between the periods 1993-1995 and 1996-2011 were studied by analyzing hourly data from 8 regulated and 8 unregulated locations with four subdaily flow variation indices. No correlations between the market deregulation and the regulation intensity in the Ångerman River basin were detected. The number of days natural ranges of variability were exceeded and the magnitude of subdaily variation were significantly higher at regulated locations. Zero flow events increased substantially when the periods 1993-2007 and 2008-2011 were compared. Significant correlations were detected in the summer periods between dry years and high magnitudes for the indices that measured variation in volume and low magnitudes for subdaily flow reversals. Zero flow events in the summer periods increased during dry years if the period 2008-2011 were excluded. The major conclusions are that the deregulation of the electricity market has not affected the subdaily regulation of the Ångerman River and that the regulated sites show highly unnatural subdaily variations. Regulation intensity and zero flows events increases during summertime in dry years and the latter also increased substantially after 2007. The current regulation regime is considered harmful for riverine ecosystems and the high and increasing levels of subdaily flow alteration shows the urgent need of national directives for subdaily hydropower operation.

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates.

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates.